id sid tid token lemma pos cana-937 1 1 communications communication NOUN cana-937 1 2 on on ADP cana-937 1 3 applied apply VERB cana-937 1 4 nonlinear nonlinear ADJ cana-937 1 5 analysis analysis NOUN cana-937 1 6 issn issn NOUN cana-937 1 7 : : PUNCT cana-937 1 8 1074 1074 NUM cana-937 1 9 - - PUNCT cana-937 1 10 133x 133x NUM cana-937 1 11 vol vol NOUN cana-937 1 12 31 31 NUM cana-937 1 13 no no NOUN cana-937 1 14 . . PUNCT cana-937 2 1 4s 4s NUM cana-937 2 2 ( ( PUNCT cana-937 2 3 2024 2024 NUM cana-937 2 4 ) ) PUNCT cana-937 2 5 433 433 NUM cana-937 2 6 https://internationalpubls.com https://internationalpubls.com X cana-937 2 7 stochastic stochastic ADJ cana-937 2 8 differential differential ADJ cana-937 2 9 equations equation NOUN cana-937 2 10 in in ADP cana-937 2 11 physics physics PROPN cana-937 2 12 dr dr PROPN cana-937 2 13 nand nand PROPN cana-937 2 14 kumar1 kumar1 PROPN cana-937 2 15 assistant assistant PROPN cana-937 2 16 professor professor NOUN cana-937 2 17 , , PUNCT cana-937 2 18 department department NOUN cana-937 2 19 of of ADP cana-937 2 20 physics physics PROPN cana-937 2 21 , , PUNCT cana-937 2 22 jawaharlal jawaharlal PROPN cana-937 2 23 nehru nehru PROPN cana-937 2 24 memorial memorial PROPN cana-937 2 25 pg pg PROPN cana-937 2 26 college college PROPN cana-937 2 27 , , PUNCT cana-937 2 28 barabanki barabanki PROPN cana-937 2 29 , , PUNCT cana-937 2 30 up up ADP cana-937 2 31 , , PUNCT cana-937 2 32 india india PROPN cana-937 2 33 . . PUNCT cana-937 3 1 email:nandkumar350@gmail.com email:nandkumar350@gmail.com X cana-937 3 2 . . PUNCT cana-937 4 1 orcid orcid NOUN cana-937 4 2 : : PUNCT cana-937 4 3 0009 0009 NUM cana-937 4 4 - - SYM cana-937 4 5 0004 0004 NUM cana-937 4 6 - - PUNCT cana-937 4 7 3139 3139 NUM cana-937 4 8 - - PUNCT cana-937 4 9 3385 3385 NUM cana-937 4 10 article article NOUN cana-937 4 11 history history NOUN cana-937 4 12 : : PUNCT cana-937 4 13 received receive VERB cana-937 4 14 : : PUNCT cana-937 4 15 16 16 NUM cana-937 4 16 - - PUNCT cana-937 4 17 04 04 NUM cana-937 4 18 - - PUNCT cana-937 4 19 2024 2024 NUM cana-937 4 20 revised revise VERB cana-937 4 21 : : PUNCT cana-937 4 22 15 15 NUM cana-937 4 23 - - SYM cana-937 4 24 06 06 NUM cana-937 4 25 - - PUNCT cana-937 4 26 2024 2024 NUM cana-937 4 27 accepted accept VERB cana-937 4 28 : : PUNCT cana-937 4 29 20 20 NUM cana-937 4 30 - - SYM cana-937 4 31 06 06 NUM cana-937 4 32 - - PUNCT cana-937 4 33 2024 2024 NUM cana-937 4 34 abstract abstract NOUN cana-937 4 35 : : PUNCT cana-937 4 36 stochastic stochastic ADJ cana-937 4 37 differential differential ADJ cana-937 4 38 equations equation NOUN cana-937 4 39 ( ( PUNCT cana-937 4 40 sdes sde NOUN cana-937 4 41 ) ) PUNCT cana-937 4 42 are be AUX cana-937 4 43 powerful powerful ADJ cana-937 4 44 mathematical mathematical ADJ cana-937 4 45 tools tool NOUN cana-937 4 46 used use VERB cana-937 4 47 to to PART cana-937 4 48 model model VERB cana-937 4 49 systems system NOUN cana-937 4 50 subject subject ADJ cana-937 4 51 to to ADP cana-937 4 52 random random ADJ cana-937 4 53 fluctuations fluctuation NOUN cana-937 4 54 . . PUNCT cana-937 5 1 in in ADP cana-937 5 2 physics physics NOUN cana-937 5 3 , , PUNCT cana-937 5 4 sdes sde NOUN cana-937 5 5 find find VERB cana-937 5 6 widespread widespread ADJ cana-937 5 7 applications application NOUN cana-937 5 8 ranging range VERB cana-937 5 9 from from ADP cana-937 5 10 statistical statistical ADJ cana-937 5 11 mechanics mechanic NOUN cana-937 5 12 to to ADP cana-937 5 13 quantum quantum ADJ cana-937 5 14 field field NOUN cana-937 5 15 theory theory NOUN cana-937 5 16 . . PUNCT cana-937 6 1 this this DET cana-937 6 2 paper paper NOUN cana-937 6 3 provides provide VERB cana-937 6 4 an an DET cana-937 6 5 in in ADP cana-937 6 6 - - PUNCT cana-937 6 7 depth depth NOUN cana-937 6 8 exploration exploration NOUN cana-937 6 9 of of ADP cana-937 6 10 the the DET cana-937 6 11 theoretical theoretical ADJ cana-937 6 12 foundations foundation NOUN cana-937 6 13 of of ADP cana-937 6 14 sdes sde NOUN cana-937 6 15 in in ADP cana-937 6 16 physics physics NOUN cana-937 6 17 , , PUNCT cana-937 6 18 their their PRON cana-937 6 19 applications application NOUN cana-937 6 20 , , PUNCT cana-937 6 21 and and CCONJ cana-937 6 22 their their PRON cana-937 6 23 implications implication NOUN cana-937 6 24 in in ADP cana-937 6 25 understanding understand VERB cana-937 6 26 complex complex ADJ cana-937 6 27 physical physical ADJ cana-937 6 28 phenomena phenomenon NOUN cana-937 6 29 . . PUNCT cana-937 7 1 we we PRON cana-937 7 2 delve delve VERB cana-937 7 3 into into ADP cana-937 7 4 the the DET cana-937 7 5 mathematical mathematical ADJ cana-937 7 6 framework framework NOUN cana-937 7 7 of of ADP cana-937 7 8 sdes sde NOUN cana-937 7 9 , , PUNCT cana-937 7 10 their their PRON cana-937 7 11 numerical numerical ADJ cana-937 7 12 solutions solution NOUN cana-937 7 13 , , PUNCT cana-937 7 14 and and CCONJ cana-937 7 15 their their PRON cana-937 7 16 role role NOUN cana-937 7 17 in in ADP cana-937 7 18 modeling model VERB cana-937 7 19 various various ADJ cana-937 7 20 physical physical ADJ cana-937 7 21 processes process NOUN cana-937 7 22 . . PUNCT cana-937 8 1 furthermore furthermore ADV cana-937 8 2 , , PUNCT cana-937 8 3 we we PRON cana-937 8 4 present present VERB cana-937 8 5 case case NOUN cana-937 8 6 studies study NOUN cana-937 8 7 illustrating illustrate VERB cana-937 8 8 the the DET cana-937 8 9 practical practical ADJ cana-937 8 10 relevance relevance NOUN cana-937 8 11 of of ADP cana-937 8 12 sdes sde NOUN cana-937 8 13 in in ADP cana-937 8 14 different different ADJ cana-937 8 15 branches branch NOUN cana-937 8 16 of of ADP cana-937 8 17 physics physics NOUN cana-937 8 18 . . PUNCT cana-937 9 1 keywords keyword NOUN cana-937 9 2 : : PUNCT cana-937 9 3 stochastic stochastic ADJ cana-937 9 4 processes process NOUN cana-937 9 5 , , PUNCT cana-937 9 6 differential differential ADJ cana-937 9 7 equations equation NOUN cana-937 9 8 , , PUNCT cana-937 9 9 brownian brownian ADJ cana-937 9 10 motion motion NOUN cana-937 9 11 , , PUNCT cana-937 9 12 ito ito PROPN cana-937 9 13 calculus calculus PROPN cana-937 9 14 , , PUNCT cana-937 9 15 stratonovich stratonovich NOUN cana-937 9 16 calculus calculus NOUN cana-937 9 17 , , PUNCT cana-937 9 18 numerical numerical ADJ cana-937 9 19 methods method NOUN cana-937 9 20 , , PUNCT cana-937 9 21 euler euler NOUN cana-937 9 22 - - PUNCT cana-937 9 23 maruyama maruyama NOUN cana-937 9 24 method method NOUN cana-937 9 25 , , PUNCT cana-937 9 26 stochastic stochastic ADJ cana-937 9 27 runge runge NOUN cana-937 9 28 - - PUNCT cana-937 9 29 kutta kutta NOUN cana-937 9 30 methods method NOUN cana-937 9 31 , , PUNCT cana-937 9 32 quantum quantum NOUN cana-937 9 33 mechanics mechanic NOUN cana-937 9 34 , , PUNCT cana-937 9 35 statistical statistical ADJ cana-937 9 36 mechanics mechanic NOUN cana-937 9 37 , , PUNCT cana-937 9 38 condensed condense VERB cana-937 9 39 matter matter NOUN cana-937 9 40 physics physics NOUN cana-937 9 41 , , PUNCT cana-937 9 42 random random ADJ cana-937 9 43 fluctuations fluctuation NOUN cana-937 9 44 , , PUNCT cana-937 9 45 wiener wiener NOUN cana-937 9 46 process process NOUN cana-937 9 47 , , PUNCT cana-937 9 48 quantum quantum ADJ cana-937 9 49 stochastic stochastic NOUN cana-937 9 50 processes process NOUN cana-937 9 51 , , PUNCT cana-937 9 52 open open ADJ cana-937 9 53 quantum quantum NOUN cana-937 9 54 systems system NOUN cana-937 9 55 , , PUNCT cana-937 9 56 coherence coherence NOUN cana-937 9 57 and and CCONJ cana-937 9 58 decoherence decoherence NOUN cana-937 9 59 , , PUNCT cana-937 9 60 diffusion diffusion NOUN cana-937 9 61 processes process NOUN cana-937 9 62 , , PUNCT cana-937 9 63 phase phase NOUN cana-937 9 64 transitions transition NOUN cana-937 9 65 , , PUNCT cana-937 9 66 quantum quantum ADJ cana-937 9 67 harmonic harmonic ADJ cana-937 9 68 oscillator oscillator NOUN cana-937 9 69 , , PUNCT cana-937 9 70 disordered disordered ADJ cana-937 9 71 systems system NOUN cana-937 9 72 . . PUNCT cana-937 10 1 1 1 X cana-937 10 2 . . X cana-937 10 3 introduction introduction NOUN cana-937 10 4 stochastic stochastic NOUN cana-937 10 5 processes process NOUN cana-937 10 6 stand stand VERB cana-937 10 7 as as ADP cana-937 10 8 foundational foundational ADJ cana-937 10 9 pillars pillar NOUN cana-937 10 10 within within ADP cana-937 10 11 the the DET cana-937 10 12 realm realm NOUN cana-937 10 13 of of ADP cana-937 10 14 physics physics NOUN cana-937 10 15 , , PUNCT cana-937 10 16 offering offer VERB cana-937 10 17 profound profound ADJ cana-937 10 18 insights insight NOUN cana-937 10 19 into into ADP cana-937 10 20 the the DET cana-937 10 21 behavior behavior NOUN cana-937 10 22 of of ADP cana-937 10 23 systems system NOUN cana-937 10 24 influenced influence VERB cana-937 10 25 by by ADP cana-937 10 26 random random ADJ cana-937 10 27 fluctuations fluctuation NOUN cana-937 10 28 . . PUNCT cana-937 11 1 their their PRON cana-937 11 2 significance significance NOUN cana-937 11 3 permeates permeate VERB cana-937 11 4 across across ADP cana-937 11 5 various various ADJ cana-937 11 6 branches branch NOUN cana-937 11 7 of of ADP cana-937 11 8 physics physics PROPN cana-937 11 9 , , PUNCT cana-937 11 10 from from ADP cana-937 11 11 the the DET cana-937 11 12 microscopic microscopic ADJ cana-937 11 13 realm realm NOUN cana-937 11 14 of of ADP cana-937 11 15 quantum quantum ADJ cana-937 11 16 mechanics mechanic NOUN cana-937 11 17 to to ADP cana-937 11 18 the the DET cana-937 11 19 macroscopic macroscopic ADJ cana-937 11 20 scale scale NOUN cana-937 11 21 of of ADP cana-937 11 22 statistical statistical ADJ cana-937 11 23 mechanics mechanic NOUN cana-937 11 24 . . PUNCT cana-937 12 1 this this DET cana-937 12 2 introduction introduction NOUN cana-937 12 3 aims aim VERB cana-937 12 4 to to PART cana-937 12 5 elucidate elucidate VERB cana-937 12 6 the the DET cana-937 12 7 profound profound ADJ cana-937 12 8 importance importance NOUN cana-937 12 9 of of ADP cana-937 12 10 stochastic stochastic ADJ cana-937 12 11 processes process NOUN cana-937 12 12 in in ADP cana-937 12 13 shaping shape VERB cana-937 12 14 our our PRON cana-937 12 15 understanding understanding NOUN cana-937 12 16 of of ADP cana-937 12 17 physical physical ADJ cana-937 12 18 phenomena phenomenon NOUN cana-937 12 19 , , PUNCT cana-937 12 20 provide provide VERB cana-937 12 21 a a DET cana-937 12 22 historical historical ADJ cana-937 12 23 backdrop backdrop NOUN cana-937 12 24 highlighting highlight VERB cana-937 12 25 the the DET cana-937 12 26 contributions contribution NOUN cana-937 12 27 of of ADP cana-937 12 28 eminent eminent ADJ cana-937 12 29 physicists physicist NOUN cana-937 12 30 , , PUNCT cana-937 12 31 and and CCONJ cana-937 12 32 outline outline VERB cana-937 12 33 the the DET cana-937 12 34 purpose purpose NOUN cana-937 12 35 and and CCONJ cana-937 12 36 structure structure NOUN cana-937 12 37 of of ADP cana-937 12 38 this this DET cana-937 12 39 paper paper NOUN cana-937 12 40 . . PUNCT cana-937 13 1 significance significance NOUN cana-937 13 2 of of ADP cana-937 13 3 stochastic stochastic ADJ cana-937 13 4 processes process NOUN cana-937 13 5 in in ADP cana-937 13 6 physics physics NOUN cana-937 13 7 : : PUNCT cana-937 13 8 in in ADP cana-937 13 9 physics physics NOUN cana-937 13 10 , , PUNCT cana-937 13 11 deterministic deterministic ADJ cana-937 13 12 models model NOUN cana-937 13 13 often often ADV cana-937 13 14 fall fall VERB cana-937 13 15 short short ADJ cana-937 13 16 in in ADP cana-937 13 17 capturing capture VERB cana-937 13 18 the the DET cana-937 13 19 inherent inherent ADJ cana-937 13 20 randomness randomness NOUN cana-937 13 21 and and CCONJ cana-937 13 22 uncertainty uncertainty NOUN cana-937 13 23 present present ADJ cana-937 13 24 in in ADP cana-937 13 25 many many ADJ cana-937 13 26 natural natural ADJ cana-937 13 27 phenomena phenomenon NOUN cana-937 13 28 . . PUNCT cana-937 14 1 stochastic stochastic ADJ cana-937 14 2 processes process NOUN cana-937 14 3 , , PUNCT cana-937 14 4 on on ADP cana-937 14 5 the the DET cana-937 14 6 other other ADJ cana-937 14 7 hand hand NOUN cana-937 14 8 , , PUNCT cana-937 14 9 provide provide VERB cana-937 14 10 a a DET cana-937 14 11 robust robust ADJ cana-937 14 12 framework framework NOUN cana-937 14 13 for for ADP cana-937 14 14 modeling model VERB cana-937 14 15 systems system NOUN cana-937 14 16 subject subject ADJ cana-937 14 17 to to ADP cana-937 14 18 random random ADJ cana-937 14 19 fluctuations fluctuation NOUN cana-937 14 20 , , PUNCT cana-937 14 21 thereby thereby ADV cana-937 14 22 bridging bridge VERB cana-937 14 23 the the DET cana-937 14 24 gap gap NOUN cana-937 14 25 between between ADP cana-937 14 26 theory theory NOUN cana-937 14 27 and and CCONJ cana-937 14 28 observation observation NOUN cana-937 14 29 . . PUNCT cana-937 15 1 these these DET cana-937 15 2 processes process NOUN cana-937 15 3 find find VERB cana-937 15 4 applications application NOUN cana-937 15 5 in in ADP cana-937 15 6 a a DET cana-937 15 7 myriad myriad NOUN cana-937 15 8 of of ADP cana-937 15 9 phenomena phenomenon NOUN cana-937 15 10 , , PUNCT cana-937 15 11 including include VERB cana-937 15 12 but but CCONJ cana-937 15 13 not not PART cana-937 15 14 limited limit VERB cana-937 15 15 to to ADP cana-937 15 16 , , PUNCT cana-937 15 17 brownian brownian ADJ cana-937 15 18 motion motion NOUN cana-937 15 19 , , PUNCT cana-937 15 20 diffusion diffusion NOUN cana-937 15 21 processes process NOUN cana-937 15 22 , , PUNCT cana-937 15 23 quantum quantum ADJ cana-937 15 24 mechanical mechanical ADJ cana-937 15 25 systems system NOUN cana-937 15 26 , , PUNCT cana-937 15 27 and and CCONJ cana-937 15 28 complex complex ADJ cana-937 15 29 systems system NOUN cana-937 15 30 far far ADV cana-937 15 31 from from ADP cana-937 15 32 equilibrium equilibrium NOUN cana-937 15 33 . . PUNCT cana-937 16 1 the the DET cana-937 16 2 significance significance NOUN cana-937 16 3 of of ADP cana-937 16 4 stochastic stochastic ADJ cana-937 16 5 processes process NOUN cana-937 16 6 in in ADP cana-937 16 7 physics physics NOUN cana-937 16 8 lies lie VERB cana-937 16 9 in in ADP cana-937 16 10 their their PRON cana-937 16 11 ability ability NOUN cana-937 16 12 to to PART cana-937 16 13 unveil unveil VERB cana-937 16 14 the the DET cana-937 16 15 underlying underlying ADJ cana-937 16 16 mechanisms mechanism NOUN cana-937 16 17 governing govern VERB cana-937 16 18 seemingly seemingly ADV cana-937 16 19 random random ADJ cana-937 16 20 behavior behavior NOUN cana-937 16 21 . . PUNCT cana-937 17 1 through through ADP cana-937 17 2 rigorous rigorous ADJ cana-937 17 3 mathematical mathematical ADJ cana-937 17 4 formalism formalism NOUN cana-937 17 5 and and CCONJ cana-937 17 6 statistical statistical ADJ cana-937 17 7 analysis analysis NOUN cana-937 17 8 , , PUNCT cana-937 17 9 physicists physicist NOUN cana-937 17 10 can can AUX cana-937 17 11 elucidate elucidate VERB cana-937 17 12 the the DET cana-937 17 13 emergent emergent ADJ cana-937 17 14 properties property NOUN cana-937 17 15 of of ADP cana-937 17 16 systems system NOUN cana-937 17 17 , , PUNCT cana-937 17 18 unraveling unravel VERB cana-937 17 19 intricate intricate ADJ cana-937 17 20 patterns pattern NOUN cana-937 17 21 hidden hide VERB cana-937 17 22 within within ADP cana-937 17 23 stochasticity stochasticity NOUN cana-937 17 24 . . PUNCT cana-937 18 1 moreover moreover ADV cana-937 18 2 , , PUNCT cana-937 18 3 stochastic stochastic NOUN cana-937 18 4 processes process NOUN cana-937 18 5 offer offer VERB cana-937 18 6 invaluable invaluable ADJ cana-937 18 7 tools tool NOUN cana-937 18 8 for for ADP cana-937 18 9 predictive predictive ADJ cana-937 18 10 modeling modeling NOUN cana-937 18 11 , , PUNCT cana-937 18 12 enabling enable VERB cana-937 18 13 researchers researcher NOUN cana-937 18 14 to to PART cana-937 18 15 forecast forecast VERB cana-937 18 16 the the DET cana-937 18 17 behavior behavior NOUN cana-937 18 18 of of ADP cana-937 18 19 complex complex ADJ cana-937 18 20 systems system NOUN cana-937 18 21 and and CCONJ cana-937 18 22 devise devise VERB cana-937 18 23 strategies strategy NOUN cana-937 18 24 for for ADP cana-937 18 25 control control NOUN cana-937 18 26 and and CCONJ cana-937 18 27 optimization optimization NOUN cana-937 18 28 . . PUNCT cana-937 19 1 communications communication NOUN cana-937 19 2 on on ADP cana-937 19 3 applied apply VERB cana-937 19 4 nonlinear nonlinear ADJ cana-937 19 5 analysis analysis NOUN cana-937 19 6 issn issn NOUN cana-937 19 7 : : PUNCT cana-937 19 8 1074 1074 NUM cana-937 19 9 - - PUNCT cana-937 19 10 133x 133x NUM cana-937 19 11 vol vol NOUN cana-937 19 12 31 31 NUM cana-937 19 13 no no NOUN cana-937 19 14 . . PUNCT cana-937 20 1 4s 4s NUM cana-937 20 2 ( ( PUNCT cana-937 20 3 2024 2024 NUM cana-937 20 4 ) ) PUNCT cana-937 20 5 434 434 NUM cana-937 21 1 https://internationalpubls.com https://internationalpubls.com X cana-937 21 2 historical historical ADJ cana-937 21 3 context context NOUN cana-937 21 4 and and CCONJ cana-937 21 5 contributions contribution NOUN cana-937 21 6 of of ADP cana-937 21 7 eminent eminent ADJ cana-937 21 8 physicists physicist NOUN cana-937 21 9 : : PUNCT cana-937 21 10 the the DET cana-937 21 11 development development NOUN cana-937 21 12 of of ADP cana-937 21 13 stochastic stochastic ADJ cana-937 21 14 processes process NOUN cana-937 21 15 in in ADP cana-937 21 16 physics physics NOUN cana-937 21 17 traces trace NOUN cana-937 21 18 back back ADV cana-937 21 19 to to ADP cana-937 21 20 the the DET cana-937 21 21 early early ADJ cana-937 21 22 20th 20th ADJ cana-937 21 23 century century NOUN cana-937 21 24 , , PUNCT cana-937 21 25 with with ADP cana-937 21 26 pioneering pioneer VERB cana-937 21 27 contributions contribution NOUN cana-937 21 28 from from ADP cana-937 21 29 luminaries luminary NOUN cana-937 21 30 such such ADJ cana-937 21 31 as as ADP cana-937 21 32 albert albert PROPN cana-937 21 33 einstein einstein PROPN cana-937 21 34 , , PUNCT cana-937 21 35 norbert norbert PROPN cana-937 21 36 wiener wiener PROPN cana-937 21 37 , , PUNCT cana-937 21 38 and and CCONJ cana-937 21 39 paul paul PROPN cana-937 21 40 langevin langevin PROPN cana-937 21 41 . . PUNCT cana-937 22 1 albert albert PROPN cana-937 22 2 einstein einstein PROPN cana-937 22 3 's 's PART cana-937 22 4 seminal seminal ADJ cana-937 22 5 work work NOUN cana-937 22 6 on on ADP cana-937 22 7 brownian brownian ADJ cana-937 22 8 motion motion NOUN cana-937 22 9 in in ADP cana-937 22 10 1905 1905 NUM cana-937 22 11 provided provide VERB cana-937 22 12 empirical empirical ADJ cana-937 22 13 evidence evidence NOUN cana-937 22 14 for for ADP cana-937 22 15 the the DET cana-937 22 16 existence existence NOUN cana-937 22 17 of of ADP cana-937 22 18 atoms atom NOUN cana-937 22 19 and and CCONJ cana-937 22 20 molecules molecule NOUN cana-937 22 21 , , PUNCT cana-937 22 22 laying lay VERB cana-937 22 23 the the DET cana-937 22 24 groundwork groundwork NOUN cana-937 22 25 for for ADP cana-937 22 26 the the DET cana-937 22 27 stochastic stochastic ADJ cana-937 22 28 modeling modeling NOUN cana-937 22 29 of of ADP cana-937 22 30 particle particle NOUN cana-937 22 31 dynamics dynamic NOUN cana-937 22 32 . . PUNCT cana-937 23 1 subsequently subsequently ADV cana-937 23 2 , , PUNCT cana-937 23 3 paul paul PROPN cana-937 23 4 langevin langevin PROPN cana-937 23 5 formulated formulate VERB cana-937 23 6 the the DET cana-937 23 7 langevin langevin ADJ cana-937 23 8 equation equation NOUN cana-937 23 9 in in ADP cana-937 23 10 1908 1908 NUM cana-937 23 11 , , PUNCT cana-937 23 12 describing describe VERB cana-937 23 13 the the DET cana-937 23 14 stochastic stochastic ADJ cana-937 23 15 motion motion NOUN cana-937 23 16 of of ADP cana-937 23 17 a a DET cana-937 23 18 particle particle NOUN cana-937 23 19 subjected subject VERB cana-937 23 20 to to ADP cana-937 23 21 random random ADJ cana-937 23 22 collisions collision NOUN cana-937 23 23 in in ADP cana-937 23 24 a a DET cana-937 23 25 fluid fluid ADJ cana-937 23 26 medium medium NOUN cana-937 23 27 . . PUNCT cana-937 24 1 norbert norbert PROPN cana-937 24 2 wiener wiener PROPN cana-937 24 3 's 's PART cana-937 24 4 groundbreaking groundbreake VERB cana-937 24 5 contributions contribution NOUN cana-937 24 6 to to ADP cana-937 24 7 the the DET cana-937 24 8 theory theory NOUN cana-937 24 9 of of ADP cana-937 24 10 stochastic stochastic ADJ cana-937 24 11 processes process NOUN cana-937 24 12 , , PUNCT cana-937 24 13 particularly particularly ADV cana-937 24 14 in in ADP cana-937 24 15 the the DET cana-937 24 16 formulation formulation NOUN cana-937 24 17 of of ADP cana-937 24 18 the the DET cana-937 24 19 wiener wiener NOUN cana-937 24 20 process process NOUN cana-937 24 21 ( ( PUNCT cana-937 24 22 also also ADV cana-937 24 23 known know VERB cana-937 24 24 as as ADP cana-937 24 25 brownian brownian ADJ cana-937 24 26 motion motion NOUN cana-937 24 27 ) ) PUNCT cana-937 24 28 , , PUNCT cana-937 24 29 revolutionized revolutionize VERB cana-937 24 30 the the DET cana-937 24 31 field field NOUN cana-937 24 32 of of ADP cana-937 24 33 probability probability NOUN cana-937 24 34 theory theory NOUN cana-937 24 35 and and CCONJ cana-937 24 36 its its PRON cana-937 24 37 applications application NOUN cana-937 24 38 in in ADP cana-937 24 39 physics physics NOUN cana-937 24 40 . . PUNCT cana-937 25 1 his his PRON cana-937 25 2 work work NOUN cana-937 25 3 laid lay VERB cana-937 25 4 the the DET cana-937 25 5 theoretical theoretical ADJ cana-937 25 6 foundation foundation NOUN cana-937 25 7 for for ADP cana-937 25 8 understanding understand VERB cana-937 25 9 random random ADJ cana-937 25 10 processes process NOUN cana-937 25 11 and and CCONJ cana-937 25 12 their their PRON cana-937 25 13 role role NOUN cana-937 25 14 in in ADP cana-937 25 15 shaping shape VERB cana-937 25 16 the the DET cana-937 25 17 dynamics dynamic NOUN cana-937 25 18 of of ADP cana-937 25 19 physical physical ADJ cana-937 25 20 systems system NOUN cana-937 25 21 . . PUNCT cana-937 26 1 throughout throughout ADP cana-937 26 2 the the DET cana-937 26 3 20th 20th ADJ cana-937 26 4 century century NOUN cana-937 26 5 and and CCONJ cana-937 26 6 into into ADP cana-937 26 7 the the DET cana-937 26 8 present present ADJ cana-937 26 9 day day NOUN cana-937 26 10 , , PUNCT cana-937 26 11 numerous numerous ADJ cana-937 26 12 physicists physicist NOUN cana-937 26 13 have have AUX cana-937 26 14 furthered further VERB cana-937 26 15 our our PRON cana-937 26 16 understanding understanding NOUN cana-937 26 17 of of ADP cana-937 26 18 stochastic stochastic ADJ cana-937 26 19 processes process NOUN cana-937 26 20 , , PUNCT cana-937 26 21 including include VERB cana-937 26 22 ito ito PROPN cana-937 26 23 kiyosi kiyosi PROPN cana-937 26 24 , , PUNCT cana-937 26 25 who who PRON cana-937 26 26 developed develop VERB cana-937 26 27 the the DET cana-937 26 28 stochastic stochastic ADJ cana-937 26 29 calculus calculus NOUN cana-937 26 30 necessary necessary ADJ cana-937 26 31 for for ADP cana-937 26 32 rigorous rigorous ADJ cana-937 26 33 mathematical mathematical ADJ cana-937 26 34 treatment treatment NOUN cana-937 26 35 of of ADP cana-937 26 36 stochastic stochastic ADJ cana-937 26 37 differential differential ADJ cana-937 26 38 equations equation NOUN cana-937 26 39 . . PUNCT cana-937 27 1 the the DET cana-937 27 2 collective collective ADJ cana-937 27 3 efforts effort NOUN cana-937 27 4 of of ADP cana-937 27 5 these these DET cana-937 27 6 luminaries luminary NOUN cana-937 27 7 have have AUX cana-937 27 8 propelled propel VERB cana-937 27 9 stochastic stochastic ADJ cana-937 27 10 processes process NOUN cana-937 27 11 into into ADP cana-937 27 12 the the DET cana-937 27 13 forefront forefront NOUN cana-937 27 14 of of ADP cana-937 27 15 modern modern ADJ cana-937 27 16 physics physics NOUN cana-937 27 17 , , PUNCT cana-937 27 18 ushering usher VERB cana-937 27 19 in in ADP cana-937 27 20 a a DET cana-937 27 21 new new ADJ cana-937 27 22 era era NOUN cana-937 27 23 of of ADP cana-937 27 24 interdisciplinary interdisciplinary ADJ cana-937 27 25 research research NOUN cana-937 27 26 and and CCONJ cana-937 27 27 discovery discovery NOUN cana-937 27 28 . . PUNCT cana-937 28 1 purpose purpose NOUN cana-937 28 2 and and CCONJ cana-937 28 3 structure structure NOUN cana-937 28 4 of of ADP cana-937 28 5 the the DET cana-937 28 6 paper paper NOUN cana-937 28 7 : : PUNCT cana-937 28 8 this this DET cana-937 28 9 paper paper NOUN cana-937 28 10 aims aim VERB cana-937 28 11 to to PART cana-937 28 12 provide provide VERB cana-937 28 13 a a DET cana-937 28 14 comprehensive comprehensive ADJ cana-937 28 15 exploration exploration NOUN cana-937 28 16 of of ADP cana-937 28 17 stochastic stochastic ADJ cana-937 28 18 differential differential ADJ cana-937 28 19 equations equation NOUN cana-937 28 20 ( ( PUNCT cana-937 28 21 sdes sde NOUN cana-937 28 22 ) ) PUNCT cana-937 28 23 in in ADP cana-937 28 24 physics physics NOUN cana-937 28 25 , , PUNCT cana-937 28 26 delving delve VERB cana-937 28 27 into into ADP cana-937 28 28 their their PRON cana-937 28 29 mathematical mathematical ADJ cana-937 28 30 foundations foundation NOUN cana-937 28 31 , , PUNCT cana-937 28 32 diverse diverse ADJ cana-937 28 33 applications application NOUN cana-937 28 34 , , PUNCT cana-937 28 35 numerical numerical ADJ cana-937 28 36 solutions solution NOUN cana-937 28 37 , , PUNCT cana-937 28 38 and and CCONJ cana-937 28 39 practical practical ADJ cana-937 28 40 implications implication NOUN cana-937 28 41 . . PUNCT cana-937 29 1 each each DET cana-937 29 2 section section NOUN cana-937 29 3 of of ADP cana-937 29 4 the the DET cana-937 29 5 paper paper NOUN cana-937 29 6 will will AUX cana-937 29 7 be be AUX cana-937 29 8 dedicated dedicate VERB cana-937 29 9 to to ADP cana-937 29 10 elucidating elucidate VERB cana-937 29 11 specific specific ADJ cana-937 29 12 aspects aspect NOUN cana-937 29 13 of of ADP cana-937 29 14 sdes sde NOUN cana-937 29 15 , , PUNCT cana-937 29 16 from from ADP cana-937 29 17 their their PRON cana-937 29 18 theoretical theoretical ADJ cana-937 29 19 underpinnings underpinning NOUN cana-937 29 20 to to ADP cana-937 29 21 their their PRON cana-937 29 22 practical practical ADJ cana-937 29 23 utility utility NOUN cana-937 29 24 in in ADP cana-937 29 25 modeling modeling NOUN cana-937 29 26 and and CCONJ cana-937 29 27 understanding understand VERB cana-937 29 28 complex complex ADJ cana-937 29 29 physical physical ADJ cana-937 29 30 phenomena phenomenon NOUN cana-937 29 31 . . PUNCT cana-937 30 1 through through ADP cana-937 30 2 detailed detailed ADJ cana-937 30 3 analysis analysis NOUN cana-937 30 4 and and CCONJ cana-937 30 5 illustrative illustrative ADJ cana-937 30 6 examples example NOUN cana-937 30 7 , , PUNCT cana-937 30 8 this this DET cana-937 30 9 paper paper NOUN cana-937 30 10 seeks seek VERB cana-937 30 11 to to PART cana-937 30 12 showcase showcase VERB cana-937 30 13 the the DET cana-937 30 14 profound profound ADJ cana-937 30 15 impact impact NOUN cana-937 30 16 of of ADP cana-937 30 17 sdes sde NOUN cana-937 30 18 on on ADP cana-937 30 19 the the DET cana-937 30 20 advancement advancement NOUN cana-937 30 21 of of ADP cana-937 30 22 physics physics NOUN cana-937 30 23 and and CCONJ cana-937 30 24 inspire inspire VERB cana-937 30 25 further further ADJ cana-937 30 26 research research NOUN cana-937 30 27 in in ADP cana-937 30 28 this this DET cana-937 30 29 captivating captivate VERB cana-937 30 30 field field NOUN cana-937 30 31 . . PUNCT cana-937 31 1 in in ADP cana-937 31 2 subsequent subsequent ADJ cana-937 31 3 sections section NOUN cana-937 31 4 , , PUNCT cana-937 31 5 we we PRON cana-937 31 6 will will AUX cana-937 31 7 delve delve VERB cana-937 31 8 into into ADP cana-937 31 9 the the DET cana-937 31 10 mathematical mathematical ADJ cana-937 31 11 framework framework NOUN cana-937 31 12 of of ADP cana-937 31 13 sdes sde NOUN cana-937 31 14 , , PUNCT cana-937 31 15 their their PRON cana-937 31 16 applications application NOUN cana-937 31 17 across across ADP cana-937 31 18 various various ADJ cana-937 31 19 domains domain NOUN cana-937 31 20 of of ADP cana-937 31 21 physics physics PROPN cana-937 31 22 , , PUNCT cana-937 31 23 numerical numerical ADJ cana-937 31 24 methods method NOUN cana-937 31 25 for for ADP cana-937 31 26 solving solve VERB cana-937 31 27 them they PRON cana-937 31 28 , , PUNCT cana-937 31 29 and and CCONJ cana-937 31 30 present present ADJ cana-937 31 31 case case NOUN cana-937 31 32 studies study NOUN cana-937 31 33 to to PART cana-937 31 34 illustrate illustrate VERB cana-937 31 35 their their PRON cana-937 31 36 utility utility NOUN cana-937 31 37 . . PUNCT cana-937 32 1 by by ADP cana-937 32 2 the the DET cana-937 32 3 conclusion conclusion NOUN cana-937 32 4 of of ADP cana-937 32 5 this this DET cana-937 32 6 paper paper NOUN cana-937 32 7 , , PUNCT cana-937 32 8 readers reader NOUN cana-937 32 9 will will AUX cana-937 32 10 gain gain VERB cana-937 32 11 a a DET cana-937 32 12 deeper deep ADJ cana-937 32 13 appreciation appreciation NOUN cana-937 32 14 for for ADP cana-937 32 15 the the DET cana-937 32 16 versatility versatility NOUN cana-937 32 17 and and CCONJ cana-937 32 18 significance significance NOUN cana-937 32 19 of of ADP cana-937 32 20 stochastic stochastic ADJ cana-937 32 21 processes process NOUN cana-937 32 22 in in ADP cana-937 32 23 shaping shape VERB cana-937 32 24 our our PRON cana-937 32 25 understanding understanding NOUN cana-937 32 26 of of ADP cana-937 32 27 the the DET cana-937 32 28 physical physical ADJ cana-937 32 29 world world NOUN cana-937 32 30 . . PUNCT cana-937 33 1 2 2 X cana-937 33 2 . . X cana-937 33 3 mathematical mathematical ADJ cana-937 33 4 foundation foundation NOUN cana-937 33 5 of of ADP cana-937 33 6 stochastic stochastic ADJ cana-937 33 7 differential differential ADJ cana-937 33 8 equations equation NOUN cana-937 33 9 ( ( PUNCT cana-937 33 10 sdes sde NOUN cana-937 33 11 ) ) PUNCT cana-937 33 12 definition definition NOUN cana-937 33 13 of of ADP cana-937 33 14 stochastic stochastic ADJ cana-937 33 15 differential differential ADJ cana-937 33 16 equations equation NOUN cana-937 33 17 : : PUNCT cana-937 33 18 stochastic stochastic ADJ cana-937 33 19 differential differential ADJ cana-937 33 20 equations equation NOUN cana-937 33 21 ( ( PUNCT cana-937 33 22 sdes sde NOUN cana-937 33 23 ) ) PUNCT cana-937 33 24 serve serve VERB cana-937 33 25 as as ADP cana-937 33 26 powerful powerful ADJ cana-937 33 27 mathematical mathematical ADJ cana-937 33 28 tools tool NOUN cana-937 33 29 for for ADP cana-937 33 30 modeling model VERB cana-937 33 31 the the DET cana-937 33 32 dynamics dynamic NOUN cana-937 33 33 of of ADP cana-937 33 34 systems system NOUN cana-937 33 35 subject subject ADJ cana-937 33 36 to to ADP cana-937 33 37 random random ADJ cana-937 33 38 fluctuations fluctuation NOUN cana-937 33 39 . . PUNCT cana-937 34 1 at at ADP cana-937 34 2 their their PRON cana-937 34 3 core core NOUN cana-937 34 4 , , PUNCT cana-937 34 5 sdes sde NOUN cana-937 34 6 provide provide VERB cana-937 34 7 a a DET cana-937 34 8 rigorous rigorous ADJ cana-937 34 9 framework framework NOUN cana-937 34 10 to to PART cana-937 34 11 describe describe VERB cana-937 34 12 the the DET cana-937 34 13 evolution evolution NOUN cana-937 34 14 of of ADP cana-937 34 15 a a DET cana-937 34 16 stochastic stochastic ADJ cana-937 34 17 process process NOUN cana-937 34 18 over over ADP cana-937 34 19 time time NOUN cana-937 34 20 . . PUNCT cana-937 35 1 the the DET cana-937 35 2 general general ADJ cana-937 35 3 form form NOUN cana-937 35 4 of of ADP cana-937 35 5 an an DET cana-937 35 6 sde sde PROPN cana-937 35 7 can can AUX cana-937 35 8 be be AUX cana-937 35 9 expressed express VERB cana-937 35 10 as as ADP cana-937 35 11 : : PUNCT cana-937 35 12 𝑑𝑋(𝑑 𝑑𝑋(𝑑 NUM cana-937 35 13 ) ) PUNCT cana-937 35 14 = = SYM cana-937 35 15 π‘Ž(𝑋(𝑑 π‘Ž(𝑋(𝑑 X cana-937 35 16 ) ) PUNCT cana-937 35 17 , , PUNCT cana-937 35 18 𝑑)𝑑𝑑 𝑑)𝑑𝑑 PROPN cana-937 35 19 + + NUM cana-937 35 20 𝑏(𝑋(𝑑 𝑏(𝑋(𝑑 NOUN cana-937 35 21 ) ) PUNCT cana-937 35 22 , , PUNCT cana-937 35 23 𝑑)π‘‘π‘Š(𝑑 𝑑)π‘‘π‘Š(𝑑 PROPN cana-937 35 24 ) ) PUNCT cana-937 35 25 where where SCONJ cana-937 35 26 x(t x(t PROPN cana-937 35 27 ) ) PUNCT cana-937 35 28 represents represent VERB cana-937 35 29 the the DET cana-937 35 30 state state NOUN cana-937 35 31 of of ADP cana-937 35 32 the the DET cana-937 35 33 system system NOUN cana-937 35 34 at at ADP cana-937 35 35 time time NOUN cana-937 35 36 a(x(t),t a(x(t),t PROPN cana-937 35 37 ) ) PUNCT cana-937 35 38 and and CCONJ cana-937 35 39 b(x(t),t b(x(t),t PROPN cana-937 35 40 ) ) PUNCT cana-937 35 41 are be AUX cana-937 35 42 deterministic deterministic ADJ cana-937 35 43 functions function NOUN cana-937 35 44 governing govern VERB cana-937 35 45 the the DET cana-937 35 46 drift drift NOUN cana-937 35 47 and and CCONJ cana-937 35 48 diffusion diffusion NOUN cana-937 35 49 coefficients coefficient NOUN cana-937 35 50 , , PUNCT cana-937 35 51 respectively respectively ADV cana-937 35 52 , , PUNCT cana-937 35 53 dt dt PROPN cana-937 35 54 denotes denote VERB cana-937 35 55 an an DET cana-937 35 56 infinitesimal infinitesimal ADJ cana-937 35 57 time time NOUN cana-937 35 58 increment increment NOUN cana-937 35 59 , , PUNCT cana-937 35 60 and and CCONJ cana-937 35 61 dw(t dw(t X cana-937 35 62 ) ) PUNCT cana-937 35 63 denotes denote VERB cana-937 35 64 the the DET cana-937 35 65 increment increment NOUN cana-937 35 66 of of ADP cana-937 35 67 a a DET cana-937 35 68 wiener wiener NOUN cana-937 35 69 process process NOUN cana-937 35 70 or or CCONJ cana-937 35 71 brownian brownian ADJ cana-937 35 72 motion motion NOUN cana-937 35 73 . . PUNCT cana-937 36 1 communications communication NOUN cana-937 36 2 on on ADP cana-937 36 3 applied apply VERB cana-937 36 4 nonlinear nonlinear ADJ cana-937 36 5 analysis analysis NOUN cana-937 36 6 issn issn NOUN cana-937 36 7 : : PUNCT cana-937 36 8 1074 1074 NUM cana-937 36 9 - - PUNCT cana-937 36 10 133x 133x NUM cana-937 36 11 vol vol NOUN cana-937 36 12 31 31 NUM cana-937 36 13 no no NOUN cana-937 36 14 . . PUNCT cana-937 37 1 4s 4s NUM cana-937 37 2 ( ( PUNCT cana-937 37 3 2024 2024 NUM cana-937 37 4 ) ) PUNCT cana-937 37 5 435 435 NUM cana-937 37 6 https://internationalpubls.com https://internationalpubls.com X cana-937 37 7 3 3 X cana-937 37 8 . . NOUN cana-937 37 9 types type NOUN cana-937 37 10 of of ADP cana-937 37 11 stochastic stochastic ADJ cana-937 37 12 differential differential ADJ cana-937 37 13 equations equation NOUN cana-937 37 14 : : PUNCT cana-937 37 15 two two NUM cana-937 37 16 common common ADJ cana-937 37 17 types type NOUN cana-937 37 18 of of ADP cana-937 37 19 sdes sde NOUN cana-937 37 20 are be AUX cana-937 37 21 distinguished distinguish VERB cana-937 37 22 based base VERB cana-937 37 23 on on ADP cana-937 37 24 the the DET cana-937 37 25 interpretation interpretation NOUN cana-937 37 26 of of ADP cana-937 37 27 the the DET cana-937 37 28 stochastic stochastic ADJ cana-937 37 29 integral integral ADJ cana-937 37 30 : : PUNCT cana-937 37 31 ito ito PROPN cana-937 37 32 and and CCONJ cana-937 37 33 stratonovich stratonovich NOUN cana-937 37 34 . . PUNCT cana-937 38 1 in in ADP cana-937 38 2 ito ito PROPN cana-937 38 3 calculus calculus PROPN cana-937 38 4 , , PUNCT cana-937 38 5 the the DET cana-937 38 6 stochastic stochastic ADJ cana-937 38 7 integral integral NOUN cana-937 38 8 is be AUX cana-937 38 9 interpreted interpret VERB cana-937 38 10 in in ADP cana-937 38 11 the the DET cana-937 38 12 stratonovich stratonovich NOUN cana-937 38 13 sense sense NOUN cana-937 38 14 , , PUNCT cana-937 38 15 leading lead VERB cana-937 38 16 to to ADP cana-937 38 17 the the DET cana-937 38 18 well well ADV cana-937 38 19 - - PUNCT cana-937 38 20 known know VERB cana-937 38 21 ito ito PROPN cana-937 38 22 stochastic stochastic NOUN cana-937 38 23 calculus calculus NOUN cana-937 38 24 . . PUNCT cana-937 39 1 the the DET cana-937 39 2 distinction distinction NOUN cana-937 39 3 between between ADP cana-937 39 4 ito ito PROPN cana-937 39 5 and and CCONJ cana-937 39 6 stratonovich stratonovich NOUN cana-937 39 7 sdes sde NOUN cana-937 39 8 arises arise VERB cana-937 39 9 due due ADJ cana-937 39 10 to to ADP cana-937 39 11 the the DET cana-937 39 12 application application NOUN cana-937 39 13 of of ADP cana-937 39 14 ito ito PROPN cana-937 39 15 's 's PART cana-937 39 16 lemma lemma PROPN cana-937 39 17 , , PUNCT cana-937 39 18 which which PRON cana-937 39 19 involves involve VERB cana-937 39 20 the the DET cana-937 39 21 correction correction NOUN cana-937 39 22 term term NOUN cana-937 39 23 1 1 NUM cana-937 39 24 2 2 NUM cana-937 39 25 𝑏2 𝑏2 PROPN cana-937 39 26 πœ•2𝑓 πœ•2𝑓 PROPN cana-937 39 27 πœ•π‘‹2 πœ•π‘‹2 PROPN cana-937 39 28 𝑑𝑑. 𝑑𝑑. PROPN cana-937 39 29 ito ito PROPN cana-937 39 30 's 's PART cana-937 39 31 lemma lemma PROPN cana-937 39 32 : : PUNCT cana-937 39 33 ito ito PROPN cana-937 39 34 's 's PART cana-937 39 35 lemma lemma PROPN cana-937 39 36 is be AUX cana-937 39 37 a a DET cana-937 39 38 fundamental fundamental ADJ cana-937 39 39 result result NOUN cana-937 39 40 in in ADP cana-937 39 41 stochastic stochastic ADJ cana-937 39 42 calculus calculus NOUN cana-937 39 43 , , PUNCT cana-937 39 44 providing provide VERB cana-937 39 45 a a DET cana-937 39 46 method method NOUN cana-937 39 47 for for ADP cana-937 39 48 calculating calculate VERB cana-937 39 49 the the DET cana-937 39 50 differential differential NOUN cana-937 39 51 of of ADP cana-937 39 52 a a DET cana-937 39 53 function function NOUN cana-937 39 54 of of ADP cana-937 39 55 a a DET cana-937 39 56 stochastic stochastic ADJ cana-937 39 57 process process NOUN cana-937 39 58 . . PUNCT cana-937 40 1 for for ADP cana-937 40 2 a a DET cana-937 40 3 function function NOUN cana-937 40 4 f(x(t),t f(x(t),t NOUN cana-937 40 5 ) ) PUNCT cana-937 40 6 , , PUNCT cana-937 40 7 ito ito PROPN cana-937 40 8 's 's PART cana-937 40 9 lemma lemma PROPN cana-937 40 10 states state NOUN cana-937 40 11 : : PUNCT cana-937 41 1 𝑑𝑓 𝑑𝑓 PROPN cana-937 41 2 = = PUNCT cana-937 41 3 ( ( PUNCT cana-937 41 4 πœ•π‘“ πœ•π‘“ INTJ cana-937 41 5 πœ•π‘‘ πœ•π‘‘ NOUN cana-937 41 6 + + NOUN cana-937 42 1 π‘Ž π‘Ž PROPN cana-937 42 2 πœ•π‘“ πœ•π‘“ ADJ cana-937 42 3 πœ•π‘‹ πœ•π‘‹ X cana-937 42 4 + + NOUN cana-937 42 5 1 1 NUM cana-937 42 6 2 2 NUM cana-937 42 7 𝑏2 𝑏2 NOUN cana-937 42 8 πœ•2𝑓 πœ•2𝑓 PROPN cana-937 42 9 πœ•π‘‹2 πœ•π‘‹2 PUNCT cana-937 42 10 ) ) PUNCT cana-937 42 11 𝑑𝑑 𝑑𝑑 ADP cana-937 42 12 + + NOUN cana-937 42 13 𝑏 𝑏 PROPN cana-937 42 14 πœ•π‘“ πœ•π‘“ ADJ cana-937 42 15 πœ•π‘‹ πœ•π‘‹ PROPN cana-937 42 16 π‘‘π‘Š π‘‘π‘Š PROPN cana-937 42 17 this this DET cana-937 42 18 lemma lemma PROPN cana-937 42 19 is be AUX cana-937 42 20 essential essential ADJ cana-937 42 21 for for ADP cana-937 42 22 transforming transform VERB cana-937 42 23 sdes sde NOUN cana-937 42 24 involving involve VERB cana-937 42 25 stochastic stochastic ADJ cana-937 42 26 processes process NOUN cana-937 42 27 into into ADP cana-937 42 28 differential differential ADJ cana-937 42 29 equations equation NOUN cana-937 42 30 involving involve VERB cana-937 42 31 only only ADJ cana-937 42 32 deterministic deterministic ADJ cana-937 42 33 functions function NOUN cana-937 42 34 , , PUNCT cana-937 42 35 facilitating facilitate VERB cana-937 42 36 their their PRON cana-937 42 37 analysis analysis NOUN cana-937 42 38 and and CCONJ cana-937 42 39 solution solution NOUN cana-937 42 40 . . PUNCT cana-937 43 1 stochastic stochastic ADJ cana-937 43 2 calculus calculus NOUN cana-937 43 3 : : PUNCT cana-937 43 4 stochastic stochastic ADJ cana-937 43 5 calculus calculus NOUN cana-937 43 6 , , PUNCT cana-937 43 7 developed develop VERB cana-937 43 8 by by ADP cana-937 43 9 pioneering pioneer VERB cana-937 43 10 mathematicians mathematician NOUN cana-937 43 11 such such ADJ cana-937 43 12 as as ADP cana-937 43 13 kiyosi kiyosi PROPN cana-937 43 14 ito ito PROPN cana-937 43 15 and and CCONJ cana-937 43 16 paul paul PROPN cana-937 43 17 lΓ©vy lΓ©vy PROPN cana-937 43 18 , , PUNCT cana-937 43 19 provides provide VERB cana-937 43 20 the the DET cana-937 43 21 mathematical mathematical ADJ cana-937 43 22 framework framework NOUN cana-937 43 23 necessary necessary ADJ cana-937 43 24 for for ADP cana-937 43 25 analyzing analyze VERB cana-937 43 26 sdes sde NOUN cana-937 43 27 rigorously rigorously ADV cana-937 43 28 . . PUNCT cana-937 44 1 it it PRON cana-937 44 2 extends extend VERB cana-937 44 3 the the DET cana-937 44 4 concepts concept NOUN cana-937 44 5 of of ADP cana-937 44 6 calculus calculus NOUN cana-937 44 7 to to ADP cana-937 44 8 stochastic stochastic ADJ cana-937 44 9 processes process NOUN cana-937 44 10 , , PUNCT cana-937 44 11 allowing allow VERB cana-937 44 12 for for ADP cana-937 44 13 the the DET cana-937 44 14 manipulation manipulation NOUN cana-937 44 15 and and CCONJ cana-937 44 16 differentiation differentiation NOUN cana-937 44 17 of of ADP cana-937 44 18 functions function NOUN cana-937 44 19 of of ADP cana-937 44 20 stochastic stochastic ADJ cana-937 44 21 variables variable NOUN cana-937 44 22 . . PUNCT cana-937 45 1 numerical numerical ADJ cana-937 45 2 methods method NOUN cana-937 45 3 for for ADP cana-937 45 4 solving solve VERB cana-937 45 5 sdes sde NOUN cana-937 45 6 : : PUNCT cana-937 45 7 numerical numerical ADJ cana-937 45 8 solutions solution NOUN cana-937 45 9 play play VERB cana-937 45 10 a a DET cana-937 45 11 crucial crucial ADJ cana-937 45 12 role role NOUN cana-937 45 13 in in ADP cana-937 45 14 tackling tackle VERB cana-937 45 15 sdes sde NOUN cana-937 45 16 , , PUNCT cana-937 45 17 especially especially ADV cana-937 45 18 in in ADP cana-937 45 19 cases case NOUN cana-937 45 20 where where SCONJ cana-937 45 21 analytical analytical ADJ cana-937 45 22 solutions solution NOUN cana-937 45 23 are be AUX cana-937 45 24 intractable intractable ADJ cana-937 45 25 . . PUNCT cana-937 46 1 two two NUM cana-937 46 2 widely widely ADV cana-937 46 3 used use VERB cana-937 46 4 numerical numerical ADJ cana-937 46 5 methods method NOUN cana-937 46 6 for for ADP cana-937 46 7 solving solve VERB cana-937 46 8 sdes sde NOUN cana-937 46 9 are be AUX cana-937 46 10 the the DET cana-937 46 11 euler euler NOUN cana-937 46 12 - - PUNCT cana-937 46 13 maruyama maruyama NOUN cana-937 46 14 method method NOUN cana-937 46 15 and and CCONJ cana-937 46 16 stochastic stochastic ADJ cana-937 46 17 runge runge NOUN cana-937 46 18 - - PUNCT cana-937 46 19 kutta kutta NOUN cana-937 46 20 methods method NOUN cana-937 46 21 . . PUNCT cana-937 47 1 these these DET cana-937 47 2 methods method NOUN cana-937 47 3 discretize discretize VERB cana-937 47 4 the the DET cana-937 47 5 sdes sde NOUN cana-937 47 6 and and CCONJ cana-937 47 7 provide provide VERB cana-937 47 8 approximate approximate ADJ cana-937 47 9 solutions solution NOUN cana-937 47 10 by by ADP cana-937 47 11 iteratively iteratively ADV cana-937 47 12 updating update VERB cana-937 47 13 the the DET cana-937 47 14 system system NOUN cana-937 47 15 's 's PART cana-937 47 16 state state NOUN cana-937 47 17 over over ADP cana-937 47 18 small small ADJ cana-937 47 19 time time NOUN cana-937 47 20 steps step NOUN cana-937 47 21 . . PUNCT cana-937 48 1 in in ADP cana-937 48 2 the the DET cana-937 48 3 subsequent subsequent ADJ cana-937 48 4 section section NOUN cana-937 48 5 , , PUNCT cana-937 48 6 we we PRON cana-937 48 7 will will AUX cana-937 48 8 delve delve VERB cana-937 48 9 deeper deeply ADV cana-937 48 10 into into ADP cana-937 48 11 the the DET cana-937 48 12 applications application NOUN cana-937 48 13 of of ADP cana-937 48 14 sdes sde NOUN cana-937 48 15 in in ADP cana-937 48 16 physics physics NOUN cana-937 48 17 , , PUNCT cana-937 48 18 exploring explore VERB cana-937 48 19 their their PRON cana-937 48 20 role role NOUN cana-937 48 21 in in ADP cana-937 48 22 statistical statistical ADJ cana-937 48 23 mechanics mechanic NOUN cana-937 48 24 , , PUNCT cana-937 48 25 quantum quantum NOUN cana-937 48 26 mechanics mechanic NOUN cana-937 48 27 , , PUNCT cana-937 48 28 and and CCONJ cana-937 48 29 condensed condense VERB cana-937 48 30 matter matter NOUN cana-937 48 31 physics physics NOUN cana-937 48 32 . . PUNCT cana-937 49 1 through through ADP cana-937 49 2 illustrative illustrative ADJ cana-937 49 3 examples example NOUN cana-937 49 4 and and CCONJ cana-937 49 5 case case NOUN cana-937 49 6 studies study NOUN cana-937 49 7 , , PUNCT cana-937 49 8 we we PRON cana-937 49 9 will will AUX cana-937 49 10 elucidate elucidate VERB cana-937 49 11 how how SCONJ cana-937 49 12 sdes sde NOUN cana-937 49 13 contribute contribute VERB cana-937 49 14 to to ADP cana-937 49 15 our our PRON cana-937 49 16 understanding understanding NOUN cana-937 49 17 of of ADP cana-937 49 18 complex complex ADJ cana-937 49 19 physical physical ADJ cana-937 49 20 phenomena phenomenon NOUN cana-937 49 21 and and CCONJ cana-937 49 22 pave pave VERB cana-937 49 23 the the DET cana-937 49 24 way way NOUN cana-937 49 25 for for ADP cana-937 49 26 further further ADJ cana-937 49 27 advancements advancement NOUN cana-937 49 28 in in ADP cana-937 49 29 theoretical theoretical ADJ cana-937 49 30 and and CCONJ cana-937 49 31 experimental experimental ADJ cana-937 49 32 physics physic NOUN cana-937 49 33 . . PUNCT cana-937 50 1 4 4 X cana-937 50 2 . . X cana-937 50 3 applications application NOUN cana-937 50 4 of of ADP cana-937 50 5 stochastic stochastic ADJ cana-937 50 6 differential differential ADJ cana-937 50 7 equations equation NOUN cana-937 50 8 in in ADP cana-937 50 9 physics physics NOUN cana-937 50 10 stochastic stochastic NOUN cana-937 50 11 differential differential ADJ cana-937 50 12 equations equation NOUN cana-937 50 13 ( ( PUNCT cana-937 50 14 sdes sde NOUN cana-937 50 15 ) ) PUNCT cana-937 50 16 find find VERB cana-937 50 17 diverse diverse ADJ cana-937 50 18 applications application NOUN cana-937 50 19 across across ADP cana-937 50 20 various various ADJ cana-937 50 21 branches branch NOUN cana-937 50 22 of of ADP cana-937 50 23 physics physics PROPN cana-937 50 24 , , PUNCT cana-937 50 25 playing play VERB cana-937 50 26 a a DET cana-937 50 27 crucial crucial ADJ cana-937 50 28 role role NOUN cana-937 50 29 in in ADP cana-937 50 30 modeling modeling NOUN cana-937 50 31 and and CCONJ cana-937 50 32 understanding understand VERB cana-937 50 33 complex complex ADJ cana-937 50 34 physical physical ADJ cana-937 50 35 phenomena phenomenon NOUN cana-937 50 36 . . PUNCT cana-937 51 1 in in ADP cana-937 51 2 this this DET cana-937 51 3 section section NOUN cana-937 51 4 , , PUNCT cana-937 51 5 we we PRON cana-937 51 6 will will AUX cana-937 51 7 explore explore VERB cana-937 51 8 some some PRON cana-937 51 9 of of ADP cana-937 51 10 the the DET cana-937 51 11 key key ADJ cana-937 51 12 applications application NOUN cana-937 51 13 of of ADP cana-937 51 14 sdes sde NOUN cana-937 51 15 in in ADP cana-937 51 16 statistical statistical ADJ cana-937 51 17 mechanics mechanic NOUN cana-937 51 18 , , PUNCT cana-937 51 19 quantum quantum NOUN cana-937 51 20 mechanics mechanic NOUN cana-937 51 21 , , PUNCT cana-937 51 22 and and CCONJ cana-937 51 23 condensed condense VERB cana-937 51 24 matter matter NOUN cana-937 51 25 physics physics NOUN cana-937 51 26 . . PUNCT cana-937 52 1 statistical statistical ADJ cana-937 52 2 mechanics mechanic NOUN cana-937 52 3 : : PUNCT cana-937 52 4 in in ADP cana-937 52 5 statistical statistical ADJ cana-937 52 6 mechanics mechanic NOUN cana-937 52 7 , , PUNCT cana-937 52 8 sdes sde NOUN cana-937 52 9 are be AUX cana-937 52 10 employed employ VERB cana-937 52 11 to to PART cana-937 52 12 model model VERB cana-937 52 13 the the DET cana-937 52 14 stochastic stochastic ADJ cana-937 52 15 dynamics dynamic NOUN cana-937 52 16 of of ADP cana-937 52 17 systems system NOUN cana-937 52 18 composed compose VERB cana-937 52 19 of of ADP cana-937 52 20 a a DET cana-937 52 21 large large ADJ cana-937 52 22 number number NOUN cana-937 52 23 of of ADP cana-937 52 24 interacting interact VERB cana-937 52 25 particles particle NOUN cana-937 52 26 . . PUNCT cana-937 53 1 one one NUM cana-937 53 2 of of ADP cana-937 53 3 the the DET cana-937 53 4 fundamental fundamental ADJ cana-937 53 5 applications application NOUN cana-937 53 6 of of ADP cana-937 53 7 sdes sde NOUN cana-937 53 8 in in ADP cana-937 53 9 this this DET cana-937 53 10 context context NOUN cana-937 53 11 is be AUX cana-937 53 12 the the DET cana-937 53 13 description description NOUN cana-937 53 14 of of ADP cana-937 53 15 brownian brownian ADJ cana-937 53 16 motion motion NOUN cana-937 53 17 . . PUNCT cana-937 54 1 by by ADP cana-937 54 2 modeling model VERB cana-937 54 3 the the DET cana-937 54 4 random random ADJ cana-937 54 5 motion motion NOUN cana-937 54 6 of of ADP cana-937 54 7 particles particle NOUN cana-937 54 8 suspended suspend VERB cana-937 54 9 in in ADP cana-937 54 10 a a DET cana-937 54 11 fluid fluid ADJ cana-937 54 12 medium medium NOUN cana-937 54 13 , , PUNCT cana-937 54 14 sdes sde NOUN cana-937 54 15 provide provide VERB cana-937 54 16 insights insight NOUN cana-937 54 17 into into ADP cana-937 54 18 diffusion diffusion NOUN cana-937 54 19 processes process NOUN cana-937 54 20 and and CCONJ cana-937 54 21 thermal thermal ADJ cana-937 54 22 fluctuations fluctuation NOUN cana-937 54 23 . . PUNCT cana-937 55 1 communications communication NOUN cana-937 55 2 on on ADP cana-937 55 3 applied apply VERB cana-937 55 4 nonlinear nonlinear ADJ cana-937 55 5 analysis analysis NOUN cana-937 55 6 issn issn NOUN cana-937 55 7 : : PUNCT cana-937 55 8 1074 1074 NUM cana-937 55 9 - - PUNCT cana-937 55 10 133x 133x NUM cana-937 55 11 vol vol NOUN cana-937 55 12 31 31 NUM cana-937 55 13 no no NOUN cana-937 55 14 . . PUNCT cana-937 56 1 4s 4s NUM cana-937 56 2 ( ( PUNCT cana-937 56 3 2024 2024 NUM cana-937 56 4 ) ) PUNCT cana-937 56 5 436 436 NUM cana-937 56 6 https://internationalpubls.com https://internationalpubls.com X cana-937 56 7 moreover moreover ADV cana-937 56 8 , , PUNCT cana-937 56 9 sdes sde NOUN cana-937 56 10 are be AUX cana-937 56 11 used use VERB cana-937 56 12 to to PART cana-937 56 13 study study VERB cana-937 56 14 phase phase NOUN cana-937 56 15 transitions transition NOUN cana-937 56 16 and and CCONJ cana-937 56 17 critical critical ADJ cana-937 56 18 phenomena phenomenon NOUN cana-937 56 19 in in ADP cana-937 56 20 statistical statistical ADJ cana-937 56 21 mechanics mechanic NOUN cana-937 56 22 . . PUNCT cana-937 57 1 by by ADP cana-937 57 2 incorporating incorporate VERB cana-937 57 3 stochastic stochastic ADJ cana-937 57 4 effects effect NOUN cana-937 57 5 into into ADP cana-937 57 6 theoretical theoretical ADJ cana-937 57 7 models model NOUN cana-937 57 8 , , PUNCT cana-937 57 9 physicists physicist NOUN cana-937 57 10 can can AUX cana-937 57 11 elucidate elucidate VERB cana-937 57 12 the the DET cana-937 57 13 emergence emergence NOUN cana-937 57 14 of of ADP cana-937 57 15 collective collective ADJ cana-937 57 16 behavior behavior NOUN cana-937 57 17 in in ADP cana-937 57 18 systems system NOUN cana-937 57 19 undergoing undergo VERB cana-937 57 20 phase phase NOUN cana-937 57 21 transitions transition NOUN cana-937 57 22 , , PUNCT cana-937 57 23 such such ADJ cana-937 57 24 as as ADP cana-937 57 25 the the DET cana-937 57 26 ising ise VERB cana-937 57 27 model model NOUN cana-937 57 28 for for ADP cana-937 57 29 magnetic magnetic ADJ cana-937 57 30 materials material NOUN cana-937 57 31 or or CCONJ cana-937 57 32 the the DET cana-937 57 33 percolation percolation NOUN cana-937 57 34 model model NOUN cana-937 57 35 for for ADP cana-937 57 36 network network NOUN cana-937 57 37 structures structure NOUN cana-937 57 38 . . PUNCT cana-937 58 1 quantum quantum ADJ cana-937 58 2 mechanics mechanic NOUN cana-937 58 3 : : PUNCT cana-937 58 4 in in ADP cana-937 58 5 quantum quantum ADJ cana-937 58 6 mechanics mechanic NOUN cana-937 58 7 , , PUNCT cana-937 58 8 sdes sde NOUN cana-937 58 9 play play VERB cana-937 58 10 a a DET cana-937 58 11 crucial crucial ADJ cana-937 58 12 role role NOUN cana-937 58 13 in in ADP cana-937 58 14 describing describe VERB cana-937 58 15 the the DET cana-937 58 16 dynamics dynamic NOUN cana-937 58 17 of of ADP cana-937 58 18 open open ADJ cana-937 58 19 quantum quantum NOUN cana-937 58 20 systems system NOUN cana-937 58 21 and and CCONJ cana-937 58 22 quantum quantum NOUN cana-937 58 23 stochastic stochastic NOUN cana-937 58 24 processes process NOUN cana-937 58 25 . . PUNCT cana-937 59 1 these these DET cana-937 59 2 processes process NOUN cana-937 59 3 arise arise VERB cana-937 59 4 when when SCONJ cana-937 59 5 a a DET cana-937 59 6 quantum quantum ADJ cana-937 59 7 system system NOUN cana-937 59 8 interacts interact VERB cana-937 59 9 with with ADP cana-937 59 10 its its PRON cana-937 59 11 environment environment NOUN cana-937 59 12 , , PUNCT cana-937 59 13 leading lead VERB cana-937 59 14 to to ADP cana-937 59 15 decoherence decoherence NOUN cana-937 59 16 and and CCONJ cana-937 59 17 loss loss NOUN cana-937 59 18 of of ADP cana-937 59 19 coherence coherence NOUN cana-937 59 20 . . PUNCT cana-937 60 1 sdes sde NOUN cana-937 60 2 provide provide VERB cana-937 60 3 a a DET cana-937 60 4 framework framework NOUN cana-937 60 5 to to PART cana-937 60 6 model model VERB cana-937 60 7 the the DET cana-937 60 8 stochastic stochastic ADJ cana-937 60 9 evolution evolution NOUN cana-937 60 10 of of ADP cana-937 60 11 quantum quantum ADJ cana-937 60 12 states state NOUN cana-937 60 13 and and CCONJ cana-937 60 14 the the DET cana-937 60 15 emergence emergence NOUN cana-937 60 16 of of ADP cana-937 60 17 classical classical ADJ cana-937 60 18 behavior behavior NOUN cana-937 60 19 from from ADP cana-937 60 20 quantum quantum ADJ cana-937 60 21 principles principle NOUN cana-937 60 22 . . PUNCT cana-937 61 1 for for ADP cana-937 61 2 example example NOUN cana-937 61 3 , , PUNCT cana-937 61 4 in in ADP cana-937 61 5 quantum quantum ADJ cana-937 61 6 optics optic NOUN cana-937 61 7 , , PUNCT cana-937 61 8 sdes sde NOUN cana-937 61 9 are be AUX cana-937 61 10 used use VERB cana-937 61 11 to to PART cana-937 61 12 describe describe VERB cana-937 61 13 the the DET cana-937 61 14 dynamics dynamic NOUN cana-937 61 15 of of ADP cana-937 61 16 quantum quantum ADJ cana-937 61 17 fields field NOUN cana-937 61 18 interacting interact VERB cana-937 61 19 with with ADP cana-937 61 20 optical optical ADJ cana-937 61 21 systems system NOUN cana-937 61 22 . . PUNCT cana-937 62 1 by by ADP cana-937 62 2 incorporating incorporate VERB cana-937 62 3 stochastic stochastic ADJ cana-937 62 4 noise noise NOUN cana-937 62 5 terms term NOUN cana-937 62 6 into into ADP cana-937 62 7 the the DET cana-937 62 8 quantum quantum ADJ cana-937 62 9 field field NOUN cana-937 62 10 equations equation NOUN cana-937 62 11 , , PUNCT cana-937 62 12 physicists physicist NOUN cana-937 62 13 can can AUX cana-937 62 14 simulate simulate VERB cana-937 62 15 the the DET cana-937 62 16 effects effect NOUN cana-937 62 17 of of ADP cana-937 62 18 photon photon NOUN cana-937 62 19 absorption absorption NOUN cana-937 62 20 , , PUNCT cana-937 62 21 emission emission NOUN cana-937 62 22 , , PUNCT cana-937 62 23 and and CCONJ cana-937 62 24 spontaneous spontaneous ADJ cana-937 62 25 emission emission NOUN cana-937 62 26 , , PUNCT cana-937 62 27 leading lead VERB cana-937 62 28 to to ADP cana-937 62 29 a a DET cana-937 62 30 better well ADJ cana-937 62 31 understanding understanding NOUN cana-937 62 32 of of ADP cana-937 62 33 quantum quantum ADJ cana-937 62 34 phenomena phenomenon NOUN cana-937 62 35 such such ADJ cana-937 62 36 as as ADP cana-937 62 37 photon photon NOUN cana-937 62 38 counting counting NOUN cana-937 62 39 statistics statistic NOUN cana-937 62 40 and and CCONJ cana-937 62 41 quantum quantum NOUN cana-937 62 42 fluctuations fluctuation NOUN cana-937 62 43 . . PUNCT cana-937 63 1 condensed condense VERB cana-937 63 2 matter matter NOUN cana-937 63 3 physics physics NOUN cana-937 63 4 : : PUNCT cana-937 63 5 in in ADP cana-937 63 6 condensed condense VERB cana-937 63 7 matter matter NOUN cana-937 63 8 physics physic NOUN cana-937 63 9 , , PUNCT cana-937 63 10 sdes sde NOUN cana-937 63 11 find find VERB cana-937 63 12 applications application NOUN cana-937 63 13 in in ADP cana-937 63 14 modeling model VERB cana-937 63 15 various various ADJ cana-937 63 16 phenomena phenomenon NOUN cana-937 63 17 , , PUNCT cana-937 63 18 including include VERB cana-937 63 19 the the DET cana-937 63 20 dynamics dynamic NOUN cana-937 63 21 of of ADP cana-937 63 22 disordered disordered ADJ cana-937 63 23 systems system NOUN cana-937 63 24 , , PUNCT cana-937 63 25 electron electron NOUN cana-937 63 26 transport transport NOUN cana-937 63 27 in in ADP cana-937 63 28 mesoscopic mesoscopic NOUN cana-937 63 29 devices device NOUN cana-937 63 30 , , PUNCT cana-937 63 31 and and CCONJ cana-937 63 32 fluctuations fluctuation NOUN cana-937 63 33 in in ADP cana-937 63 34 magnetic magnetic ADJ cana-937 63 35 materials material NOUN cana-937 63 36 . . PUNCT cana-937 64 1 for for ADP cana-937 64 2 instance instance NOUN cana-937 64 3 , , PUNCT cana-937 64 4 in in ADP cana-937 64 5 the the DET cana-937 64 6 study study NOUN cana-937 64 7 of of ADP cana-937 64 8 disordered disordered ADJ cana-937 64 9 systems system NOUN cana-937 64 10 , , PUNCT cana-937 64 11 sdes sde NOUN cana-937 64 12 are be AUX cana-937 64 13 used use VERB cana-937 64 14 to to PART cana-937 64 15 describe describe VERB cana-937 64 16 the the DET cana-937 64 17 motion motion NOUN cana-937 64 18 of of ADP cana-937 64 19 electrons electron NOUN cana-937 64 20 in in ADP cana-937 64 21 random random ADJ cana-937 64 22 potential potential ADJ cana-937 64 23 landscapes landscape NOUN cana-937 64 24 , , PUNCT cana-937 64 25 leading lead VERB cana-937 64 26 to to ADP cana-937 64 27 the the DET cana-937 64 28 emergence emergence NOUN cana-937 64 29 of of ADP cana-937 64 30 phenomena phenomenon NOUN cana-937 64 31 such such ADJ cana-937 64 32 as as ADP cana-937 64 33 anderson anderson PROPN cana-937 64 34 localization localization PROPN cana-937 64 35 and and CCONJ cana-937 64 36 the the DET cana-937 64 37 metal metal NOUN cana-937 64 38 - - PUNCT cana-937 64 39 insulator insulator NOUN cana-937 64 40 transition transition NOUN cana-937 64 41 . . PUNCT cana-937 65 1 moreover moreover ADV cana-937 65 2 , , PUNCT cana-937 65 3 sdes sde NOUN cana-937 65 4 are be AUX cana-937 65 5 employed employ VERB cana-937 65 6 to to PART cana-937 65 7 model model VERB cana-937 65 8 the the DET cana-937 65 9 stochastic stochastic ADJ cana-937 65 10 dynamics dynamic NOUN cana-937 65 11 of of ADP cana-937 65 12 magnetic magnetic ADJ cana-937 65 13 domains domain NOUN cana-937 65 14 in in ADP cana-937 65 15 ferromagnetic ferromagnetic ADJ cana-937 65 16 materials material NOUN cana-937 65 17 . . PUNCT cana-937 66 1 by by ADP cana-937 66 2 simulating simulate VERB cana-937 66 3 the the DET cana-937 66 4 random random ADJ cana-937 66 5 motion motion NOUN cana-937 66 6 of of ADP cana-937 66 7 domain domain NOUN cana-937 66 8 walls wall NOUN cana-937 66 9 and and CCONJ cana-937 66 10 magnetic magnetic ADJ cana-937 66 11 moments moment NOUN cana-937 66 12 , , PUNCT cana-937 66 13 physicists physicist NOUN cana-937 66 14 can can AUX cana-937 66 15 gain gain VERB cana-937 66 16 insights insight NOUN cana-937 66 17 into into ADP cana-937 66 18 the the DET cana-937 66 19 dynamics dynamic NOUN cana-937 66 20 of of ADP cana-937 66 21 magnetization magnetization NOUN cana-937 66 22 reversal reversal NOUN cana-937 66 23 processes process NOUN cana-937 66 24 and and CCONJ cana-937 66 25 the the DET cana-937 66 26 emergence emergence NOUN cana-937 66 27 of of ADP cana-937 66 28 magnetic magnetic ADJ cana-937 66 29 domain domain NOUN cana-937 66 30 patterns pattern NOUN cana-937 66 31 . . PUNCT cana-937 67 1 in in ADP cana-937 67 2 the the DET cana-937 67 3 subsequent subsequent ADJ cana-937 67 4 section section NOUN cana-937 67 5 , , PUNCT cana-937 67 6 we we PRON cana-937 67 7 will will AUX cana-937 67 8 discuss discuss VERB cana-937 67 9 numerical numerical ADJ cana-937 67 10 methods method NOUN cana-937 67 11 for for ADP cana-937 67 12 solving solve VERB cana-937 67 13 sdes sde NOUN cana-937 67 14 , , PUNCT cana-937 67 15 including include VERB cana-937 67 16 the the DET cana-937 67 17 eulermaruyama eulermaruyama NOUN cana-937 67 18 method method NOUN cana-937 67 19 and and CCONJ cana-937 67 20 stochastic stochastic ADJ cana-937 67 21 runge runge NOUN cana-937 67 22 - - PUNCT cana-937 67 23 kutta kutta NOUN cana-937 67 24 methods method NOUN cana-937 67 25 . . PUNCT cana-937 68 1 these these DET cana-937 68 2 numerical numerical ADJ cana-937 68 3 techniques technique NOUN cana-937 68 4 play play VERB cana-937 68 5 a a DET cana-937 68 6 crucial crucial ADJ cana-937 68 7 role role NOUN cana-937 68 8 in in ADP cana-937 68 9 simulating simulate VERB cana-937 68 10 the the DET cana-937 68 11 dynamics dynamic NOUN cana-937 68 12 of of ADP cana-937 68 13 stochastic stochastic ADJ cana-937 68 14 systems system NOUN cana-937 68 15 and and CCONJ cana-937 68 16 predicting predict VERB cana-937 68 17 their their PRON cana-937 68 18 behavior behavior NOUN cana-937 68 19 under under ADP cana-937 68 20 different different ADJ cana-937 68 21 conditions condition NOUN cana-937 68 22 . . PUNCT cana-937 69 1 through through ADP cana-937 69 2 numerical numerical ADJ cana-937 69 3 simulations simulation NOUN cana-937 69 4 and and CCONJ cana-937 69 5 computational computational ADJ cana-937 69 6 experiments experiment NOUN cana-937 69 7 , , PUNCT cana-937 69 8 physicists physicist NOUN cana-937 69 9 can can AUX cana-937 69 10 validate validate VERB cana-937 69 11 theoretical theoretical ADJ cana-937 69 12 models model NOUN cana-937 69 13 and and CCONJ cana-937 69 14 test test NOUN cana-937 69 15 hypotheses hypothesis NOUN cana-937 69 16 , , PUNCT cana-937 69 17 thereby thereby ADV cana-937 69 18 advancing advance VERB cana-937 69 19 our our PRON cana-937 69 20 understanding understanding NOUN cana-937 69 21 of of ADP cana-937 69 22 complex complex ADJ cana-937 69 23 physical physical ADJ cana-937 69 24 systems system NOUN cana-937 69 25 . . PUNCT cana-937 70 1 5 5 X cana-937 70 2 . . X cana-937 70 3 numerical numerical ADJ cana-937 70 4 solutions solution NOUN cana-937 70 5 of of ADP cana-937 70 6 stochastic stochastic ADJ cana-937 70 7 differential differential ADJ cana-937 70 8 equations equation NOUN cana-937 70 9 ( ( PUNCT cana-937 70 10 sdes sde NOUN cana-937 70 11 ) ) PUNCT cana-937 70 12 stochastic stochastic ADJ cana-937 70 13 differential differential ADJ cana-937 70 14 equations equation NOUN cana-937 70 15 ( ( PUNCT cana-937 70 16 sdes sde NOUN cana-937 70 17 ) ) PUNCT cana-937 70 18 often often ADV cana-937 70 19 lack lack VERB cana-937 70 20 closed closed ADJ cana-937 70 21 - - PUNCT cana-937 70 22 form form NOUN cana-937 70 23 analytical analytical ADJ cana-937 70 24 solutions solution NOUN cana-937 70 25 , , PUNCT cana-937 70 26 especially especially ADV cana-937 70 27 for for ADP cana-937 70 28 complex complex ADJ cana-937 70 29 systems system NOUN cana-937 70 30 or or CCONJ cana-937 70 31 non non ADJ cana-937 70 32 - - ADJ cana-937 70 33 linear linear ADJ cana-937 70 34 dynamics dynamic NOUN cana-937 70 35 . . PUNCT cana-937 71 1 therefore therefore ADV cana-937 71 2 , , PUNCT cana-937 71 3 numerical numerical ADJ cana-937 71 4 methods method NOUN cana-937 71 5 are be AUX cana-937 71 6 indispensable indispensable ADJ cana-937 71 7 for for ADP cana-937 71 8 approximating approximate VERB cana-937 71 9 solutions solution NOUN cana-937 71 10 and and CCONJ cana-937 71 11 understanding understand VERB cana-937 71 12 the the DET cana-937 71 13 behavior behavior NOUN cana-937 71 14 of of ADP cana-937 71 15 stochastic stochastic ADJ cana-937 71 16 systems system NOUN cana-937 71 17 . . PUNCT cana-937 72 1 in in ADP cana-937 72 2 this this DET cana-937 72 3 section section NOUN cana-937 72 4 , , PUNCT cana-937 72 5 we we PRON cana-937 72 6 will will AUX cana-937 72 7 explore explore VERB cana-937 72 8 two two NUM cana-937 72 9 prominent prominent ADJ cana-937 72 10 numerical numerical ADJ cana-937 72 11 techniques technique NOUN cana-937 72 12 for for ADP cana-937 72 13 solving solve VERB cana-937 72 14 sdes sde NOUN cana-937 72 15 : : PUNCT cana-937 72 16 the the DET cana-937 72 17 euler euler NOUN cana-937 72 18 - - PUNCT cana-937 72 19 maruyama maruyama NOUN cana-937 72 20 method method NOUN cana-937 72 21 and and CCONJ cana-937 72 22 stochastic stochastic ADJ cana-937 72 23 runge runge NOUN cana-937 72 24 - - PUNCT cana-937 72 25 kutta kutta NOUN cana-937 72 26 methods method NOUN cana-937 72 27 . . PUNCT cana-937 73 1 communications communication NOUN cana-937 73 2 on on ADP cana-937 73 3 applied apply VERB cana-937 73 4 nonlinear nonlinear ADJ cana-937 73 5 analysis analysis NOUN cana-937 73 6 issn issn NOUN cana-937 73 7 : : PUNCT cana-937 73 8 1074 1074 NUM cana-937 73 9 - - PUNCT cana-937 73 10 133x 133x NUM cana-937 73 11 vol vol NOUN cana-937 73 12 31 31 NUM cana-937 73 13 no no NOUN cana-937 73 14 . . PUNCT cana-937 74 1 4s 4s NUM cana-937 74 2 ( ( PUNCT cana-937 74 3 2024 2024 NUM cana-937 74 4 ) ) PUNCT cana-937 74 5 437 437 NUM cana-937 74 6 https://internationalpubls.com https://internationalpubls.com X cana-937 74 7 euler euler NOUN cana-937 74 8 - - PUNCT cana-937 74 9 maruyama maruyama NOUN cana-937 74 10 method method NOUN cana-937 74 11 : : PUNCT cana-937 74 12 the the DET cana-937 74 13 euler euler NOUN cana-937 74 14 - - PUNCT cana-937 74 15 maruyama maruyama NOUN cana-937 74 16 method method NOUN cana-937 74 17 is be AUX cana-937 74 18 a a DET cana-937 74 19 simple simple ADJ cana-937 74 20 and and CCONJ cana-937 74 21 widely widely ADV cana-937 74 22 used use VERB cana-937 74 23 numerical numerical ADJ cana-937 74 24 scheme scheme NOUN cana-937 74 25 for for ADP cana-937 74 26 approximating approximate VERB cana-937 74 27 solutions solution NOUN cana-937 74 28 of of ADP cana-937 74 29 sdes sde NOUN cana-937 74 30 . . PUNCT cana-937 75 1 it it PRON cana-937 75 2 discretizes discretize VERB cana-937 75 3 the the DET cana-937 75 4 sde sde PROPN cana-937 75 5 over over ADP cana-937 75 6 small small ADJ cana-937 75 7 time time NOUN cana-937 75 8 intervals interval NOUN cana-937 75 9 and and CCONJ cana-937 75 10 updates update VERB cana-937 75 11 the the DET cana-937 75 12 system system NOUN cana-937 75 13 's 's PART cana-937 75 14 state state NOUN cana-937 75 15 iteratively iteratively ADV cana-937 75 16 . . PUNCT cana-937 76 1 the the DET cana-937 76 2 basic basic ADJ cana-937 76 3 iterative iterative NOUN cana-937 76 4 formula formula NOUN cana-937 76 5 for for ADP cana-937 76 6 the the DET cana-937 76 7 euler euler NOUN cana-937 76 8 - - PUNCT cana-937 76 9 maruyama maruyama NOUN cana-937 76 10 method method NOUN cana-937 76 11 is be AUX cana-937 76 12 given give VERB cana-937 76 13 by by ADP cana-937 76 14 : : PUNCT cana-937 76 15 𝑋𝑛 𝑋𝑛 NOUN cana-937 76 16 + + NOUN cana-937 76 17 1 1 NUM cana-937 76 18 = = SYM cana-937 76 19 𝑋𝑛 𝑋𝑛 NOUN cana-937 76 20 + + CCONJ cana-937 76 21 π‘Ž(𝑋𝑛 π‘Ž(𝑋𝑛 PROPN cana-937 76 22 , , PUNCT cana-937 76 23 𝑑𝑛)π›₯𝑑 𝑑𝑛)π›₯𝑑 ADJ cana-937 76 24 + + CCONJ cana-937 76 25 𝑏(𝑋𝑛 𝑏(𝑋𝑛 NOUN cana-937 76 26 , , PUNCT cana-937 76 27 𝑑𝑛)π›₯π‘Šπ‘› 𝑑𝑛)π›₯π‘Šπ‘› PRON cana-937 76 28 where where SCONJ cana-937 76 29 : : PUNCT cana-937 77 1 β€’ β€’ NOUN cana-937 77 2 xn xn PROPN cana-937 77 3 is be AUX cana-937 77 4 the the DET cana-937 77 5 state state NOUN cana-937 77 6 of of ADP cana-937 77 7 the the DET cana-937 77 8 system system NOUN cana-937 77 9 at at ADP cana-937 77 10 time time NOUN cana-937 77 11 tn tn PROPN cana-937 77 12 , , PUNCT cana-937 77 13 β€’ β€’ NUM cana-937 77 14 Ξ΄t Ξ΄t X cana-937 77 15 is be AUX cana-937 77 16 the the DET cana-937 77 17 time time NOUN cana-937 77 18 step step NOUN cana-937 77 19 , , PUNCT cana-937 77 20 β€’ β€’ NUM cana-937 77 21 Ξ΄wn Ξ΄wn NOUN cana-937 77 22 is be AUX cana-937 77 23 the the DET cana-937 77 24 increment increment NOUN cana-937 77 25 of of ADP cana-937 77 26 the the DET cana-937 77 27 wiener wiener NOUN cana-937 77 28 process process NOUN cana-937 77 29 at at ADP cana-937 77 30 time time NOUN cana-937 77 31 tn tn PROPN cana-937 77 32 . . PUNCT cana-937 78 1 despite despite SCONJ cana-937 78 2 its its PRON cana-937 78 3 simplicity simplicity NOUN cana-937 78 4 , , PUNCT cana-937 78 5 the the DET cana-937 78 6 euler euler NOUN cana-937 78 7 - - PUNCT cana-937 78 8 maruyama maruyama NOUN cana-937 78 9 method method NOUN cana-937 78 10 can can AUX cana-937 78 11 suffer suffer VERB cana-937 78 12 from from ADP cana-937 78 13 numerical numerical ADJ cana-937 78 14 instability instability NOUN cana-937 78 15 , , PUNCT cana-937 78 16 especially especially ADV cana-937 78 17 for for ADP cana-937 78 18 stiff stiff ADJ cana-937 78 19 systems system NOUN cana-937 78 20 or or CCONJ cana-937 78 21 when when SCONJ cana-937 78 22 using use VERB cana-937 78 23 large large ADJ cana-937 78 24 time time NOUN cana-937 78 25 steps step NOUN cana-937 78 26 . . PUNCT cana-937 79 1 however however ADV cana-937 79 2 , , PUNCT cana-937 79 3 it it PRON cana-937 79 4 remains remain VERB cana-937 79 5 a a DET cana-937 79 6 valuable valuable ADJ cana-937 79 7 tool tool NOUN cana-937 79 8 for for ADP cana-937 79 9 preliminary preliminary ADJ cana-937 79 10 simulations simulation NOUN cana-937 79 11 and and CCONJ cana-937 79 12 quick quick ADJ cana-937 79 13 assessments assessment NOUN cana-937 79 14 of of ADP cana-937 79 15 stochastic stochastic ADJ cana-937 79 16 systems system NOUN cana-937 79 17 . . PUNCT cana-937 80 1 stochastic stochastic ADJ cana-937 80 2 runge runge NOUN cana-937 80 3 - - PUNCT cana-937 80 4 kutta kutta NOUN cana-937 80 5 methods method NOUN cana-937 80 6 : : PUNCT cana-937 80 7 stochastic stochastic ADJ cana-937 80 8 runge runge NOUN cana-937 80 9 - - PUNCT cana-937 80 10 kutta kutta NOUN cana-937 80 11 methods method NOUN cana-937 80 12 extend extend VERB cana-937 80 13 the the DET cana-937 80 14 classical classical ADJ cana-937 80 15 runge runge NOUN cana-937 80 16 - - PUNCT cana-937 80 17 kutta kutta NOUN cana-937 80 18 methods method NOUN cana-937 80 19 to to ADP cana-937 80 20 the the DET cana-937 80 21 numerical numerical ADJ cana-937 80 22 integration integration NOUN cana-937 80 23 of of ADP cana-937 80 24 sdes sde NOUN cana-937 80 25 . . PUNCT cana-937 81 1 these these DET cana-937 81 2 methods method NOUN cana-937 81 3 offer offer VERB cana-937 81 4 higher high ADJ cana-937 81 5 - - PUNCT cana-937 81 6 order order NOUN cana-937 81 7 accuracy accuracy NOUN cana-937 81 8 and and CCONJ cana-937 81 9 improved improved ADJ cana-937 81 10 stability stability NOUN cana-937 81 11 compared compare VERB cana-937 81 12 to to ADP cana-937 81 13 the the DET cana-937 81 14 euler euler NOUN cana-937 81 15 - - PUNCT cana-937 81 16 maruyama maruyama NOUN cana-937 81 17 method method NOUN cana-937 81 18 , , PUNCT cana-937 81 19 making make VERB cana-937 81 20 them they PRON cana-937 81 21 suitable suitable ADJ cana-937 81 22 for for ADP cana-937 81 23 more more ADV cana-937 81 24 accurate accurate ADJ cana-937 81 25 simulations simulation NOUN cana-937 81 26 of of ADP cana-937 81 27 stochastic stochastic ADJ cana-937 81 28 systems system NOUN cana-937 81 29 . . PUNCT cana-937 82 1 one one NUM cana-937 82 2 commonly commonly ADV cana-937 82 3 used use VERB cana-937 82 4 stochastic stochastic ADJ cana-937 82 5 runge runge NOUN cana-937 82 6 - - PUNCT cana-937 82 7 kutta kutta NOUN cana-937 82 8 method method NOUN cana-937 82 9 is be AUX cana-937 82 10 the the DET cana-937 82 11 milstein milstein ADJ cana-937 82 12 method method NOUN cana-937 82 13 , , PUNCT cana-937 82 14 which which PRON cana-937 82 15 incorporates incorporate VERB cana-937 82 16 additional additional ADJ cana-937 82 17 terms term NOUN cana-937 82 18 to to PART cana-937 82 19 improve improve VERB cana-937 82 20 the the DET cana-937 82 21 accuracy accuracy NOUN cana-937 82 22 of of ADP cana-937 82 23 the the DET cana-937 82 24 numerical numerical ADJ cana-937 82 25 approximation approximation NOUN cana-937 82 26 . . PUNCT cana-937 83 1 the the DET cana-937 83 2 milstein milstein ADJ cana-937 83 3 method method NOUN cana-937 83 4 is be AUX cana-937 83 5 given give VERB cana-937 83 6 by by ADP cana-937 83 7 : : PUNCT cana-937 83 8 𝑋𝑛 𝑋𝑛 NOUN cana-937 83 9 + + NOUN cana-937 83 10 1 1 NUM cana-937 83 11 = = SYM cana-937 83 12 𝑋𝑛 𝑋𝑛 NOUN cana-937 83 13 + + CCONJ cana-937 83 14 π‘Ž(𝑋𝑛 π‘Ž(𝑋𝑛 PROPN cana-937 83 15 , , PUNCT cana-937 83 16 𝑑𝑛)π›₯𝑑 𝑑𝑛)π›₯𝑑 ADJ cana-937 83 17 + + CCONJ cana-937 83 18 𝑏(𝑋𝑛 𝑏(𝑋𝑛 NOUN cana-937 83 19 , , PUNCT cana-937 83 20 𝑑𝑛)π›₯π‘Šπ‘› 𝑑𝑛)π›₯π‘Šπ‘› PRON cana-937 83 21 + + NOUN cana-937 83 22 21𝑏(𝑋𝑛 21𝑏(𝑋𝑛 NOUN cana-937 83 23 , , PUNCT cana-937 83 24 𝑑𝑛)𝑏′(𝑋𝑛 𝑑𝑛)𝑏′(𝑋𝑛 NOUN cana-937 83 25 , , PUNCT cana-937 83 26 𝑑𝑛)(π›₯π‘Šπ‘›2 𝑑𝑛)(π›₯π‘Šπ‘›2 NOUN cana-937 83 27 βˆ’ βˆ’ PROPN cana-937 83 28 π›₯𝑑 π›₯𝑑 NOUN cana-937 83 29 ) ) PUNCT cana-937 83 30 where where SCONJ cana-937 83 31 : : PUNCT cana-937 83 32 β€’ β€’ NUM cana-937 83 33 bβ€²(xn bβ€²(xn PROPN cana-937 83 34 , , PUNCT cana-937 83 35 tn tn PROPN cana-937 83 36 ) ) PUNCT cana-937 83 37 denotes denote VERB cana-937 83 38 the the DET cana-937 83 39 derivative derivative NOUN cana-937 83 40 of of ADP cana-937 83 41 the the DET cana-937 83 42 diffusion diffusion NOUN cana-937 83 43 coefficient coefficient NOUN cana-937 83 44 with with ADP cana-937 83 45 respect respect NOUN cana-937 83 46 to to ADP cana-937 83 47 the the DET cana-937 83 48 state state NOUN cana-937 83 49 variable variable NOUN cana-937 83 50 . . PUNCT cana-937 84 1 stochastic stochastic ADJ cana-937 84 2 runge runge NOUN cana-937 84 3 - - PUNCT cana-937 84 4 kutta kutta NOUN cana-937 84 5 methods method NOUN cana-937 84 6 , , PUNCT cana-937 84 7 including include VERB cana-937 84 8 the the DET cana-937 84 9 milstein milstein ADJ cana-937 84 10 method method NOUN cana-937 84 11 , , PUNCT cana-937 84 12 offer offer VERB cana-937 84 13 better well ADJ cana-937 84 14 convergence convergence NOUN cana-937 84 15 properties property NOUN cana-937 84 16 and and CCONJ cana-937 84 17 accuracy accuracy NOUN cana-937 84 18 compared compare VERB cana-937 84 19 to to ADP cana-937 84 20 the the DET cana-937 84 21 euler euler NOUN cana-937 84 22 - - PUNCT cana-937 84 23 maruyama maruyama NOUN cana-937 84 24 method method NOUN cana-937 84 25 , , PUNCT cana-937 84 26 especially especially ADV cana-937 84 27 for for ADP cana-937 84 28 stiff stiff ADJ cana-937 84 29 systems system NOUN cana-937 84 30 or or CCONJ cana-937 84 31 when when SCONJ cana-937 84 32 dealing deal VERB cana-937 84 33 with with ADP cana-937 84 34 small small ADJ cana-937 84 35 time time NOUN cana-937 84 36 steps step NOUN cana-937 84 37 . . PUNCT cana-937 85 1 6 6 NUM cana-937 85 2 . . X cana-937 85 3 applications application NOUN cana-937 85 4 of of ADP cana-937 85 5 numerical numerical ADJ cana-937 85 6 solutions solution NOUN cana-937 85 7 : : PUNCT cana-937 85 8 numerical numerical ADJ cana-937 85 9 solutions solution NOUN cana-937 85 10 of of ADP cana-937 85 11 sdes sde NOUN cana-937 85 12 play play VERB cana-937 85 13 a a DET cana-937 85 14 crucial crucial ADJ cana-937 85 15 role role NOUN cana-937 85 16 in in ADP cana-937 85 17 various various ADJ cana-937 85 18 fields field NOUN cana-937 85 19 of of ADP cana-937 85 20 physics physics NOUN cana-937 85 21 , , PUNCT cana-937 85 22 including include VERB cana-937 85 23 statistical statistical ADJ cana-937 85 24 mechanics mechanic NOUN cana-937 85 25 , , PUNCT cana-937 85 26 quantum quantum NOUN cana-937 85 27 mechanics mechanic NOUN cana-937 85 28 , , PUNCT cana-937 85 29 and and CCONJ cana-937 85 30 condensed condense VERB cana-937 85 31 matter matter NOUN cana-937 85 32 physics physics NOUN cana-937 85 33 . . PUNCT cana-937 86 1 these these DET cana-937 86 2 numerical numerical ADJ cana-937 86 3 techniques technique NOUN cana-937 86 4 enable enable VERB cana-937 86 5 physicists physicist NOUN cana-937 86 6 to to PART cana-937 86 7 simulate simulate VERB cana-937 86 8 the the DET cana-937 86 9 behavior behavior NOUN cana-937 86 10 of of ADP cana-937 86 11 stochastic stochastic ADJ cana-937 86 12 systems system NOUN cana-937 86 13 , , PUNCT cana-937 86 14 validate validate VERB cana-937 86 15 theoretical theoretical ADJ cana-937 86 16 models model NOUN cana-937 86 17 , , PUNCT cana-937 86 18 and and CCONJ cana-937 86 19 make make VERB cana-937 86 20 predictions prediction NOUN cana-937 86 21 about about ADP cana-937 86 22 the the DET cana-937 86 23 system system NOUN cana-937 86 24 's 's PART cana-937 86 25 evolution evolution NOUN cana-937 86 26 under under ADP cana-937 86 27 different different ADJ cana-937 86 28 conditions condition NOUN cana-937 86 29 . . PUNCT cana-937 87 1 in in ADP cana-937 87 2 the the DET cana-937 87 3 subsequent subsequent ADJ cana-937 87 4 section section NOUN cana-937 87 5 , , PUNCT cana-937 87 6 we we PRON cana-937 87 7 will will AUX cana-937 87 8 present present VERB cana-937 87 9 case case NOUN cana-937 87 10 studies study NOUN cana-937 87 11 illustrating illustrate VERB cana-937 87 12 the the DET cana-937 87 13 practical practical ADJ cana-937 87 14 relevance relevance NOUN cana-937 87 15 of of ADP cana-937 87 16 sdes sde NOUN cana-937 87 17 and and CCONJ cana-937 87 18 numerical numerical ADJ cana-937 87 19 solutions solution NOUN cana-937 87 20 in in ADP cana-937 87 21 physics physics NOUN cana-937 87 22 . . PUNCT cana-937 88 1 through through ADP cana-937 88 2 these these DET cana-937 88 3 examples example NOUN cana-937 88 4 , , PUNCT cana-937 88 5 we we PRON cana-937 88 6 will will AUX cana-937 88 7 demonstrate demonstrate VERB cana-937 88 8 how how SCONJ cana-937 88 9 sdes sde NOUN cana-937 88 10 and and CCONJ cana-937 88 11 numerical numerical ADJ cana-937 88 12 techniques technique NOUN cana-937 88 13 contribute contribute VERB cana-937 88 14 to to ADP cana-937 88 15 our our PRON cana-937 88 16 understanding understanding NOUN cana-937 88 17 of of ADP cana-937 88 18 complex complex ADJ cana-937 88 19 physical physical ADJ cana-937 88 20 phenomena phenomenon NOUN cana-937 88 21 and and CCONJ cana-937 88 22 pave pave VERB cana-937 88 23 the the DET cana-937 88 24 way way NOUN cana-937 88 25 for for ADP cana-937 88 26 further further ADJ cana-937 88 27 advancements advancement NOUN cana-937 88 28 in in ADP cana-937 88 29 theoretical theoretical ADJ cana-937 88 30 and and CCONJ cana-937 88 31 experimental experimental ADJ cana-937 88 32 physics physic NOUN cana-937 88 33 . . PUNCT cana-937 89 1 communications communication NOUN cana-937 89 2 on on ADP cana-937 89 3 applied apply VERB cana-937 89 4 nonlinear nonlinear ADJ cana-937 89 5 analysis analysis NOUN cana-937 89 6 issn issn NOUN cana-937 89 7 : : PUNCT cana-937 89 8 1074 1074 NUM cana-937 89 9 - - PUNCT cana-937 89 10 133x 133x NUM cana-937 89 11 vol vol NOUN cana-937 89 12 31 31 NUM cana-937 89 13 no no NOUN cana-937 89 14 . . PUNCT cana-937 90 1 4s 4s NUM cana-937 90 2 ( ( PUNCT cana-937 90 3 2024 2024 NUM cana-937 90 4 ) ) PUNCT cana-937 90 5 438 438 NUM cana-937 90 6 https://internationalpubls.com https://internationalpubls.com X cana-937 90 7 7 7 X cana-937 90 8 . . PUNCT cana-937 90 9 case case NOUN cana-937 90 10 studies study NOUN cana-937 90 11 : : PUNCT cana-937 90 12 practical practical ADJ cana-937 90 13 applications application NOUN cana-937 90 14 of of ADP cana-937 90 15 stochastic stochastic ADJ cana-937 90 16 differential differential ADJ cana-937 90 17 equations equation NOUN cana-937 90 18 in in ADP cana-937 90 19 this this DET cana-937 90 20 section section NOUN cana-937 90 21 , , PUNCT cana-937 90 22 we we PRON cana-937 90 23 will will AUX cana-937 90 24 explore explore VERB cana-937 90 25 two two NUM cana-937 90 26 case case NOUN cana-937 90 27 studies study NOUN cana-937 90 28 to to PART cana-937 90 29 demonstrate demonstrate VERB cana-937 90 30 the the DET cana-937 90 31 practical practical ADJ cana-937 90 32 applications application NOUN cana-937 90 33 of of ADP cana-937 90 34 stochastic stochastic ADJ cana-937 90 35 differential differential ADJ cana-937 90 36 equations equation NOUN cana-937 90 37 ( ( PUNCT cana-937 90 38 sdes sde NOUN cana-937 90 39 ) ) PUNCT cana-937 90 40 in in ADP cana-937 90 41 physics physics NOUN cana-937 90 42 . . PUNCT cana-937 91 1 we we PRON cana-937 91 2 will will AUX cana-937 91 3 examine examine VERB cana-937 91 4 how how SCONJ cana-937 91 5 sdes sde NOUN cana-937 91 6 and and CCONJ cana-937 91 7 their their PRON cana-937 91 8 numerical numerical ADJ cana-937 91 9 solutions solution NOUN cana-937 91 10 are be AUX cana-937 91 11 employed employ VERB cana-937 91 12 to to PART cana-937 91 13 model model VERB cana-937 91 14 and and CCONJ cana-937 91 15 analyze analyze VERB cana-937 91 16 real real ADJ cana-937 91 17 - - PUNCT cana-937 91 18 world world NOUN cana-937 91 19 phenomena phenomenon NOUN cana-937 91 20 , , PUNCT cana-937 91 21 providing provide VERB cana-937 91 22 insights insight NOUN cana-937 91 23 into into ADP cana-937 91 24 the the DET cana-937 91 25 stochastic stochastic ADJ cana-937 91 26 behavior behavior NOUN cana-937 91 27 of of ADP cana-937 91 28 physical physical ADJ cana-937 91 29 systems system NOUN cana-937 91 30 . . PUNCT cana-937 92 1 case case NOUN cana-937 92 2 study study NOUN cana-937 92 3 1 1 NUM cana-937 92 4 : : PUNCT cana-937 92 5 brownian brownian ADJ cana-937 92 6 motion motion NOUN cana-937 92 7 brownian brownian PROPN cana-937 92 8 motion motion NOUN cana-937 92 9 serves serve VERB cana-937 92 10 as as ADP cana-937 92 11 a a DET cana-937 92 12 classic classic ADJ cana-937 92 13 example example NOUN cana-937 92 14 of of ADP cana-937 92 15 stochastic stochastic ADJ cana-937 92 16 behavior behavior NOUN cana-937 92 17 observed observe VERB cana-937 92 18 in in ADP cana-937 92 19 various various ADJ cana-937 92 20 physical physical ADJ cana-937 92 21 systems system NOUN cana-937 92 22 , , PUNCT cana-937 92 23 such such ADJ cana-937 92 24 as as ADP cana-937 92 25 the the DET cana-937 92 26 random random ADJ cana-937 92 27 motion motion NOUN cana-937 92 28 of of ADP cana-937 92 29 particles particle NOUN cana-937 92 30 suspended suspend VERB cana-937 92 31 in in ADP cana-937 92 32 a a DET cana-937 92 33 fluid fluid NOUN cana-937 92 34 . . PUNCT cana-937 93 1 sdes sde NOUN cana-937 93 2 provide provide VERB cana-937 93 3 a a DET cana-937 93 4 mathematical mathematical ADJ cana-937 93 5 framework framework NOUN cana-937 93 6 to to PART cana-937 93 7 describe describe VERB cana-937 93 8 the the DET cana-937 93 9 stochastic stochastic ADJ cana-937 93 10 dynamics dynamic NOUN cana-937 93 11 of of ADP cana-937 93 12 brownian brownian ADJ cana-937 93 13 motion motion NOUN cana-937 93 14 , , PUNCT cana-937 93 15 offering offer VERB cana-937 93 16 insights insight NOUN cana-937 93 17 into into ADP cana-937 93 18 diffusion diffusion NOUN cana-937 93 19 processes process NOUN cana-937 93 20 and and CCONJ cana-937 93 21 thermal thermal ADJ cana-937 93 22 fluctuations fluctuation NOUN cana-937 93 23 . . PUNCT cana-937 94 1 in in ADP cana-937 94 2 this this DET cana-937 94 3 case case NOUN cana-937 94 4 study study NOUN cana-937 94 5 , , PUNCT cana-937 94 6 we we PRON cana-937 94 7 will will AUX cana-937 94 8 simulate simulate VERB cana-937 94 9 the the DET cana-937 94 10 trajectory trajectory NOUN cana-937 94 11 of of ADP cana-937 94 12 a a DET cana-937 94 13 brownian brownian ADJ cana-937 94 14 particle particle NOUN cana-937 94 15 using use VERB cana-937 94 16 sdes sde NOUN cana-937 94 17 and and CCONJ cana-937 94 18 numerical numerical ADJ cana-937 94 19 methods method NOUN cana-937 94 20 . . PUNCT cana-937 95 1 we we PRON cana-937 95 2 will will AUX cana-937 95 3 employ employ VERB cana-937 95 4 the the DET cana-937 95 5 euler euler NOUN cana-937 95 6 - - PUNCT cana-937 95 7 maruyama maruyama NOUN cana-937 95 8 method method NOUN cana-937 95 9 or or CCONJ cana-937 95 10 stochastic stochastic ADJ cana-937 95 11 runge runge NOUN cana-937 95 12 - - PUNCT cana-937 95 13 kutta kutta NOUN cana-937 95 14 methods method NOUN cana-937 95 15 to to PART cana-937 95 16 approximate approximate VERB cana-937 95 17 the the DET cana-937 95 18 solution solution NOUN cana-937 95 19 of of ADP cana-937 95 20 the the DET cana-937 95 21 sde sde PROPN cana-937 95 22 governing govern VERB cana-937 95 23 brownian brownian ADJ cana-937 95 24 motion motion NOUN cana-937 95 25 . . PUNCT cana-937 96 1 by by ADP cana-937 96 2 analyzing analyze VERB cana-937 96 3 the the DET cana-937 96 4 simulated simulate VERB cana-937 96 5 trajectory trajectory NOUN cana-937 96 6 , , PUNCT cana-937 96 7 we we PRON cana-937 96 8 can can AUX cana-937 96 9 calculate calculate VERB cana-937 96 10 statistical statistical ADJ cana-937 96 11 properties property NOUN cana-937 96 12 such such ADJ cana-937 96 13 as as ADP cana-937 96 14 the the DET cana-937 96 15 mean mean ADJ cana-937 96 16 square square ADJ cana-937 96 17 displacement displacement NOUN cana-937 96 18 and and CCONJ cana-937 96 19 diffusion diffusion NOUN cana-937 96 20 coefficient coefficient NOUN cana-937 96 21 , , PUNCT cana-937 96 22 comparing compare VERB cana-937 96 23 them they PRON cana-937 96 24 with with ADP cana-937 96 25 theoretical theoretical ADJ cana-937 96 26 predictions prediction NOUN cana-937 96 27 derived derive VERB cana-937 96 28 from from ADP cana-937 96 29 sdes sde NOUN cana-937 96 30 . . PUNCT cana-937 97 1 case case NOUN cana-937 97 2 study study NOUN cana-937 97 3 2 2 NUM cana-937 97 4 : : PUNCT cana-937 97 5 quantum quantum ADJ cana-937 97 6 harmonic harmonic NOUN cana-937 97 7 oscillator oscillator NOUN cana-937 97 8 the the DET cana-937 97 9 quantum quantum ADJ cana-937 97 10 harmonic harmonic NOUN cana-937 97 11 oscillator oscillator NOUN cana-937 97 12 represents represent VERB cana-937 97 13 a a DET cana-937 97 14 fundamental fundamental ADJ cana-937 97 15 quantum quantum ADJ cana-937 97 16 mechanical mechanical ADJ cana-937 97 17 system system NOUN cana-937 97 18 with with ADP cana-937 97 19 applications application NOUN cana-937 97 20 in in ADP cana-937 97 21 various various ADJ cana-937 97 22 fields field NOUN cana-937 97 23 , , PUNCT cana-937 97 24 including include VERB cana-937 97 25 quantum quantum NOUN cana-937 97 26 optics optic NOUN cana-937 97 27 and and CCONJ cana-937 97 28 condensed condense VERB cana-937 97 29 matter matter NOUN cana-937 97 30 physics physics NOUN cana-937 97 31 . . PUNCT cana-937 98 1 when when SCONJ cana-937 98 2 subjected subject VERB cana-937 98 3 to to ADP cana-937 98 4 stochastic stochastic ADJ cana-937 98 5 perturbations perturbation NOUN cana-937 98 6 , , PUNCT cana-937 98 7 such such ADJ cana-937 98 8 as as ADP cana-937 98 9 external external ADJ cana-937 98 10 noise noise NOUN cana-937 98 11 or or CCONJ cana-937 98 12 decoherence decoherence NOUN cana-937 98 13 effects effect NOUN cana-937 98 14 , , PUNCT cana-937 98 15 the the DET cana-937 98 16 dynamics dynamic NOUN cana-937 98 17 of of ADP cana-937 98 18 the the DET cana-937 98 19 quantum quantum ADJ cana-937 98 20 harmonic harmonic NOUN cana-937 98 21 oscillator oscillator NOUN cana-937 98 22 can can AUX cana-937 98 23 be be AUX cana-937 98 24 described describe VERB cana-937 98 25 by by ADP cana-937 98 26 sdes sde NOUN cana-937 98 27 . . PUNCT cana-937 99 1 in in ADP cana-937 99 2 this this DET cana-937 99 3 case case NOUN cana-937 99 4 study study NOUN cana-937 99 5 , , PUNCT cana-937 99 6 we we PRON cana-937 99 7 will will AUX cana-937 99 8 model model VERB cana-937 99 9 the the DET cana-937 99 10 stochastic stochastic ADJ cana-937 99 11 dynamics dynamic NOUN cana-937 99 12 of of ADP cana-937 99 13 a a DET cana-937 99 14 quantum quantum ADJ cana-937 99 15 harmonic harmonic NOUN cana-937 99 16 oscillator oscillator NOUN cana-937 99 17 using use VERB cana-937 99 18 sdes sde NOUN cana-937 99 19 and and CCONJ cana-937 99 20 numerical numerical ADJ cana-937 99 21 techniques technique NOUN cana-937 99 22 . . PUNCT cana-937 100 1 by by ADP cana-937 100 2 incorporating incorporate VERB cana-937 100 3 stochastic stochastic ADJ cana-937 100 4 terms term NOUN cana-937 100 5 into into ADP cana-937 100 6 the the DET cana-937 100 7 schrΓΆdinger schrΓΆdinger ADJ cana-937 100 8 equation equation NOUN cana-937 100 9 or or CCONJ cana-937 100 10 the the DET cana-937 100 11 heisenberg heisenberg PROPN cana-937 100 12 equations equation NOUN cana-937 100 13 of of ADP cana-937 100 14 motion motion NOUN cana-937 100 15 , , PUNCT cana-937 100 16 we we PRON cana-937 100 17 can can AUX cana-937 100 18 simulate simulate VERB cana-937 100 19 the the DET cana-937 100 20 evolution evolution NOUN cana-937 100 21 of of ADP cana-937 100 22 the the DET cana-937 100 23 oscillator oscillator NOUN cana-937 100 24 's 's PART cana-937 100 25 wave wave NOUN cana-937 100 26 function function NOUN cana-937 100 27 or or CCONJ cana-937 100 28 operators operator NOUN cana-937 100 29 over over ADP cana-937 100 30 time time NOUN cana-937 100 31 . . PUNCT cana-937 101 1 we we PRON cana-937 101 2 will will AUX cana-937 101 3 use use VERB cana-937 101 4 numerical numerical ADJ cana-937 101 5 methods method NOUN cana-937 101 6 such such ADJ cana-937 101 7 as as ADP cana-937 101 8 the the DET cana-937 101 9 euler euler NOUN cana-937 101 10 - - PUNCT cana-937 101 11 maruyama maruyama NOUN cana-937 101 12 method method NOUN cana-937 101 13 or or CCONJ cana-937 101 14 stochastic stochastic ADJ cana-937 101 15 runge runge NOUN cana-937 101 16 - - PUNCT cana-937 101 17 kutta kutta NOUN cana-937 101 18 methods method NOUN cana-937 101 19 to to PART cana-937 101 20 approximate approximate VERB cana-937 101 21 the the DET cana-937 101 22 solution solution NOUN cana-937 101 23 of of ADP cana-937 101 24 the the DET cana-937 101 25 sdes sde NOUN cana-937 101 26 and and CCONJ cana-937 101 27 analyze analyze VERB cana-937 101 28 the the DET cana-937 101 29 effects effect NOUN cana-937 101 30 of of ADP cana-937 101 31 stochastic stochastic ADJ cana-937 101 32 perturbations perturbation NOUN cana-937 101 33 on on ADP cana-937 101 34 the the DET cana-937 101 35 oscillator oscillator NOUN cana-937 101 36 's 's PART cana-937 101 37 energy energy NOUN cana-937 101 38 spectrum spectrum NOUN cana-937 101 39 , , PUNCT cana-937 101 40 coherence coherence NOUN cana-937 101 41 properties property NOUN cana-937 101 42 , , PUNCT cana-937 101 43 and and CCONJ cana-937 101 44 quantum quantum NOUN cana-937 101 45 dynamics dynamic NOUN cana-937 101 46 . . PUNCT cana-937 102 1 8 8 X cana-937 102 2 . . X cana-937 102 3 conclusion conclusion NOUN cana-937 102 4 : : PUNCT cana-937 102 5 in in ADP cana-937 102 6 this this DET cana-937 102 7 paper paper NOUN cana-937 102 8 , , PUNCT cana-937 102 9 we we PRON cana-937 102 10 have have AUX cana-937 102 11 explored explore VERB cana-937 102 12 the the DET cana-937 102 13 profound profound ADJ cana-937 102 14 significance significance NOUN cana-937 102 15 of of ADP cana-937 102 16 stochastic stochastic ADJ cana-937 102 17 differential differential ADJ cana-937 102 18 equations equation NOUN cana-937 102 19 ( ( PUNCT cana-937 102 20 sdes sde NOUN cana-937 102 21 ) ) PUNCT cana-937 102 22 in in ADP cana-937 102 23 physics physics NOUN cana-937 102 24 , , PUNCT cana-937 102 25 delving delve VERB cana-937 102 26 into into ADP cana-937 102 27 their their PRON cana-937 102 28 mathematical mathematical ADJ cana-937 102 29 foundation foundation NOUN cana-937 102 30 , , PUNCT cana-937 102 31 diverse diverse ADJ cana-937 102 32 applications application NOUN cana-937 102 33 , , PUNCT cana-937 102 34 numerical numerical ADJ cana-937 102 35 solutions solution NOUN cana-937 102 36 , , PUNCT cana-937 102 37 and and CCONJ cana-937 102 38 practical practical ADJ cana-937 102 39 implications implication NOUN cana-937 102 40 . . PUNCT cana-937 103 1 stochastic stochastic ADJ cana-937 103 2 processes process NOUN cana-937 103 3 are be AUX cana-937 103 4 ubiquitous ubiquitous ADJ cana-937 103 5 in in ADP cana-937 103 6 nature nature NOUN cana-937 103 7 , , PUNCT cana-937 103 8 and and CCONJ cana-937 103 9 sdes sde NOUN cana-937 103 10 provide provide VERB cana-937 103 11 a a DET cana-937 103 12 rigorous rigorous ADJ cana-937 103 13 framework framework NOUN cana-937 103 14 for for ADP cana-937 103 15 modeling model VERB cana-937 103 16 systems system NOUN cana-937 103 17 subject subject ADJ cana-937 103 18 to to ADP cana-937 103 19 random random ADJ cana-937 103 20 fluctuations fluctuation NOUN cana-937 103 21 , , PUNCT cana-937 103 22 bridging bridge VERB cana-937 103 23 the the DET cana-937 103 24 gap gap NOUN cana-937 103 25 between between ADP cana-937 103 26 deterministic deterministic ADJ cana-937 103 27 theory theory NOUN cana-937 103 28 and and CCONJ cana-937 103 29 stochastic stochastic ADJ cana-937 103 30 observations observation NOUN cana-937 103 31 . . PUNCT cana-937 104 1 throughout throughout ADP cana-937 104 2 history history NOUN cana-937 104 3 , , PUNCT cana-937 104 4 eminent eminent ADJ cana-937 104 5 physicists physicist NOUN cana-937 104 6 such such ADJ cana-937 104 7 as as ADP cana-937 104 8 albert albert PROPN cana-937 104 9 einstein einstein PROPN cana-937 104 10 , , PUNCT cana-937 104 11 norbert norbert PROPN cana-937 104 12 wiener wiener PROPN cana-937 104 13 , , PUNCT cana-937 104 14 and and CCONJ cana-937 104 15 paul paul PROPN cana-937 104 16 langevin langevin PROPN cana-937 104 17 have have AUX cana-937 104 18 made make VERB cana-937 104 19 pioneering pioneering ADJ cana-937 104 20 contributions contribution NOUN cana-937 104 21 to to ADP cana-937 104 22 the the DET cana-937 104 23 development development NOUN cana-937 104 24 of of ADP cana-937 104 25 stochastic stochastic ADJ cana-937 104 26 processes process NOUN cana-937 104 27 and and CCONJ cana-937 104 28 their their PRON cana-937 104 29 applications application NOUN cana-937 104 30 in in ADP cana-937 104 31 physics physics NOUN cana-937 104 32 . . PUNCT cana-937 105 1 their their PRON cana-937 105 2 groundbreaking groundbreake VERB cana-937 105 3 work work NOUN cana-937 105 4 laid lay VERB cana-937 105 5 the the DET cana-937 105 6 foundation foundation NOUN cana-937 105 7 for for ADP cana-937 105 8 the the DET cana-937 105 9 modern modern ADJ cana-937 105 10 understanding understanding NOUN cana-937 105 11 of of ADP cana-937 105 12 stochastic stochastic ADJ cana-937 105 13 dynamics dynamic NOUN cana-937 105 14 , , PUNCT cana-937 105 15 inspiring inspire VERB cana-937 105 16 generations generation NOUN cana-937 105 17 of of ADP cana-937 105 18 researchers researcher NOUN cana-937 105 19 to to PART cana-937 105 20 explore explore VERB cana-937 105 21 the the DET cana-937 105 22 rich rich ADJ cana-937 105 23 interplay interplay NOUN cana-937 105 24 between between ADP cana-937 105 25 randomness randomness NOUN cana-937 105 26 and and CCONJ cana-937 105 27 determinism determinism NOUN cana-937 105 28 in in ADP cana-937 105 29 physical physical ADJ cana-937 105 30 systems system NOUN cana-937 105 31 . . PUNCT cana-937 106 1 communications communication NOUN cana-937 106 2 on on ADP cana-937 106 3 applied apply VERB cana-937 106 4 nonlinear nonlinear ADJ cana-937 106 5 analysis analysis NOUN cana-937 106 6 issn issn NOUN cana-937 106 7 : : PUNCT cana-937 106 8 1074 1074 NUM cana-937 106 9 - - PUNCT cana-937 106 10 133x 133x NUM cana-937 106 11 vol vol NOUN cana-937 106 12 31 31 NUM cana-937 106 13 no no NOUN cana-937 106 14 . . PUNCT cana-937 107 1 4s 4s NUM cana-937 107 2 ( ( PUNCT cana-937 107 3 2024 2024 NUM cana-937 107 4 ) ) PUNCT cana-937 107 5 439 439 NUM cana-937 107 6 https://internationalpubls.com https://internationalpubls.com X cana-937 107 7 from from ADP cana-937 107 8 statistical statistical ADJ cana-937 107 9 mechanics mechanic NOUN cana-937 107 10 to to ADP cana-937 107 11 quantum quantum ADJ cana-937 107 12 mechanics mechanic NOUN cana-937 107 13 and and CCONJ cana-937 107 14 condensed condense VERB cana-937 107 15 matter matter NOUN cana-937 107 16 physics physic NOUN cana-937 107 17 , , PUNCT cana-937 107 18 sdes sde NOUN cana-937 107 19 find find VERB cana-937 107 20 applications application NOUN cana-937 107 21 across across ADP cana-937 107 22 a a DET cana-937 107 23 wide wide ADJ cana-937 107 24 range range NOUN cana-937 107 25 of of ADP cana-937 107 26 phenomena phenomenon NOUN cana-937 107 27 , , PUNCT cana-937 107 28 including include VERB cana-937 107 29 diffusion diffusion NOUN cana-937 107 30 processes process NOUN cana-937 107 31 , , PUNCT cana-937 107 32 phase phase NOUN cana-937 107 33 transitions transition NOUN cana-937 107 34 , , PUNCT cana-937 107 35 and and CCONJ cana-937 107 36 quantum quantum NOUN cana-937 107 37 stochastic stochastic NOUN cana-937 107 38 processes process NOUN cana-937 107 39 . . PUNCT cana-937 108 1 by by ADP cana-937 108 2 incorporating incorporate VERB cana-937 108 3 stochastic stochastic ADJ cana-937 108 4 effects effect NOUN cana-937 108 5 into into ADP cana-937 108 6 theoretical theoretical ADJ cana-937 108 7 models model NOUN cana-937 108 8 , , PUNCT cana-937 108 9 physicists physicist NOUN cana-937 108 10 can can AUX cana-937 108 11 gain gain VERB cana-937 108 12 insights insight NOUN cana-937 108 13 into into ADP cana-937 108 14 the the DET cana-937 108 15 emergent emergent ADJ cana-937 108 16 behavior behavior NOUN cana-937 108 17 of of ADP cana-937 108 18 complex complex ADJ cana-937 108 19 systems system NOUN cana-937 108 20 and and CCONJ cana-937 108 21 make make VERB cana-937 108 22 predictions prediction NOUN cana-937 108 23 about about ADP cana-937 108 24 their their PRON cana-937 108 25 behavior behavior NOUN cana-937 108 26 under under ADP cana-937 108 27 different different ADJ cana-937 108 28 conditions condition NOUN cana-937 108 29 . . PUNCT cana-937 109 1 numerical numerical ADJ cana-937 109 2 solutions solution NOUN cana-937 109 3 of of ADP cana-937 109 4 sdes sde NOUN cana-937 109 5 play play VERB cana-937 109 6 a a DET cana-937 109 7 crucial crucial ADJ cana-937 109 8 role role NOUN cana-937 109 9 in in ADP cana-937 109 10 simulating simulate VERB cana-937 109 11 the the DET cana-937 109 12 dynamics dynamic NOUN cana-937 109 13 of of ADP cana-937 109 14 stochastic stochastic ADJ cana-937 109 15 systems system NOUN cana-937 109 16 and and CCONJ cana-937 109 17 validating validate VERB cana-937 109 18 theoretical theoretical ADJ cana-937 109 19 models model NOUN cana-937 109 20 . . PUNCT cana-937 110 1 techniques technique NOUN cana-937 110 2 such such ADJ cana-937 110 3 as as ADP cana-937 110 4 the the DET cana-937 110 5 euler euler NOUN cana-937 110 6 - - PUNCT cana-937 110 7 maruyama maruyama NOUN cana-937 110 8 method method NOUN cana-937 110 9 and and CCONJ cana-937 110 10 stochastic stochastic ADJ cana-937 110 11 rungekutta rungekutta NOUN cana-937 110 12 methods method NOUN cana-937 110 13 enable enable VERB cana-937 110 14 researchers researcher NOUN cana-937 110 15 to to PART cana-937 110 16 approximate approximate ADJ cana-937 110 17 solutions solution NOUN cana-937 110 18 of of ADP cana-937 110 19 sdes sde NOUN cana-937 110 20 and and CCONJ cana-937 110 21 analyze analyze VERB cana-937 110 22 the the DET cana-937 110 23 behavior behavior NOUN cana-937 110 24 of of ADP cana-937 110 25 stochastic stochastic ADJ cana-937 110 26 systems system NOUN cana-937 110 27 in in ADP cana-937 110 28 a a DET cana-937 110 29 computationally computationally ADV cana-937 110 30 efficient efficient ADJ cana-937 110 31 manner manner NOUN cana-937 110 32 . . PUNCT cana-937 111 1 through through ADP cana-937 111 2 case case NOUN cana-937 111 3 studies study NOUN cana-937 111 4 such such ADJ cana-937 111 5 as as ADP cana-937 111 6 brownian brownian ADJ cana-937 111 7 motion motion NOUN cana-937 111 8 and and CCONJ cana-937 111 9 the the DET cana-937 111 10 quantum quantum ADJ cana-937 111 11 harmonic harmonic ADJ cana-937 111 12 oscillator oscillator NOUN cana-937 111 13 , , PUNCT cana-937 111 14 we we PRON cana-937 111 15 have have AUX cana-937 111 16 demonstrated demonstrate VERB cana-937 111 17 the the DET cana-937 111 18 practical practical ADJ cana-937 111 19 relevance relevance NOUN cana-937 111 20 of of ADP cana-937 111 21 sdes sde NOUN cana-937 111 22 in in ADP cana-937 111 23 physics physics NOUN cana-937 111 24 , , PUNCT cana-937 111 25 illustrating illustrate VERB cana-937 111 26 how how SCONJ cana-937 111 27 they they PRON cana-937 111 28 enable enable VERB cana-937 111 29 researchers researcher NOUN cana-937 111 30 to to PART cana-937 111 31 model model VERB cana-937 111 32 and and CCONJ cana-937 111 33 analyze analyze VERB cana-937 111 34 real real ADJ cana-937 111 35 - - PUNCT cana-937 111 36 world world NOUN cana-937 111 37 phenomena phenomenon NOUN cana-937 111 38 with with ADP cana-937 111 39 stochastic stochastic ADJ cana-937 111 40 components component NOUN cana-937 111 41 . . PUNCT cana-937 112 1 by by ADP cana-937 112 2 combining combine VERB cana-937 112 3 theoretical theoretical ADJ cana-937 112 4 insights insight NOUN cana-937 112 5 with with ADP cana-937 112 6 numerical numerical ADJ cana-937 112 7 simulations simulation NOUN cana-937 112 8 and and CCONJ cana-937 112 9 experimental experimental ADJ cana-937 112 10 observations observation NOUN cana-937 112 11 , , PUNCT cana-937 112 12 physicists physicist NOUN cana-937 112 13 can can AUX cana-937 112 14 deepen deepen VERB cana-937 112 15 our our PRON cana-937 112 16 understanding understanding NOUN cana-937 112 17 of of ADP cana-937 112 18 the the DET cana-937 112 19 natural natural ADJ cana-937 112 20 world world NOUN cana-937 112 21 and and CCONJ cana-937 112 22 pave pave VERB cana-937 112 23 the the DET cana-937 112 24 way way NOUN cana-937 112 25 for for ADP cana-937 112 26 future future ADJ cana-937 112 27 advancements advancement NOUN cana-937 112 28 in in ADP cana-937 112 29 science science NOUN cana-937 112 30 and and CCONJ cana-937 112 31 technology technology NOUN cana-937 112 32 . . PUNCT cana-937 113 1 in in ADP cana-937 113 2 conclusion conclusion NOUN cana-937 113 3 , , PUNCT cana-937 113 4 stochastic stochastic ADJ cana-937 113 5 differential differential ADJ cana-937 113 6 equations equation NOUN cana-937 113 7 serve serve VERB cana-937 113 8 as as ADP cana-937 113 9 indispensable indispensable ADJ cana-937 113 10 tools tool NOUN cana-937 113 11 in in ADP cana-937 113 12 the the DET cana-937 113 13 physicist physicist NOUN cana-937 113 14 's 's PART cana-937 113 15 toolkit toolkit NOUN cana-937 113 16 , , PUNCT cana-937 113 17 offering offer VERB cana-937 113 18 a a DET cana-937 113 19 powerful powerful ADJ cana-937 113 20 framework framework NOUN cana-937 113 21 for for ADP cana-937 113 22 modeling modeling NOUN cana-937 113 23 and and CCONJ cana-937 113 24 understanding understand VERB cana-937 113 25 the the DET cana-937 113 26 stochastic stochastic ADJ cana-937 113 27 behavior behavior NOUN cana-937 113 28 of of ADP cana-937 113 29 physical physical ADJ cana-937 113 30 systems system NOUN cana-937 113 31 . . PUNCT cana-937 114 1 as as SCONJ cana-937 114 2 we we PRON cana-937 114 3 continue continue VERB cana-937 114 4 to to PART cana-937 114 5 explore explore VERB cana-937 114 6 the the DET cana-937 114 7 frontiers frontier NOUN cana-937 114 8 of of ADP cana-937 114 9 stochastic stochastic NOUN cana-937 114 10 processes process NOUN cana-937 114 11 and and CCONJ cana-937 114 12 their their PRON cana-937 114 13 applications application NOUN cana-937 114 14 , , PUNCT cana-937 114 15 we we PRON cana-937 114 16 embark embark VERB cana-937 114 17 on on ADP cana-937 114 18 a a DET cana-937 114 19 journey journey NOUN cana-937 114 20 of of ADP cana-937 114 21 discovery discovery NOUN cana-937 114 22 and and CCONJ cana-937 114 23 innovation innovation NOUN cana-937 114 24 , , PUNCT cana-937 114 25 unraveling unravel VERB cana-937 114 26 the the DET cana-937 114 27 mysteries mystery NOUN cana-937 114 28 of of ADP cana-937 114 29 the the DET cana-937 114 30 universe universe ADJ cana-937 114 31 one one NUM cana-937 114 32 equation equation NOUN cana-937 114 33 at at ADP cana-937 114 34 a a DET cana-937 114 35 time time NOUN cana-937 114 36 . . PUNCT cana-937 115 1 thank thank VERB cana-937 115 2 you you PRON cana-937 115 3 for for ADP cana-937 115 4 reading read VERB cana-937 115 5 this this DET cana-937 115 6 paper paper NOUN cana-937 115 7 , , PUNCT cana-937 115 8 and and CCONJ cana-937 115 9 may may AUX cana-937 115 10 it it PRON cana-937 115 11 inspire inspire VERB cana-937 115 12 further further ADJ cana-937 115 13 research research NOUN cana-937 115 14 and and CCONJ cana-937 115 15 exploration exploration NOUN cana-937 115 16 in in ADP cana-937 115 17 the the DET cana-937 115 18 fascinating fascinating ADJ cana-937 115 19 field field NOUN cana-937 115 20 of of ADP cana-937 115 21 stochastic stochastic ADJ cana-937 115 22 differential differential ADJ cana-937 115 23 equations equation NOUN cana-937 115 24 in in ADP cana-937 115 25 physics physics PROPN cana-937 115 26 . . PUNCT cana-937 116 1 references reference NOUN cana-937 116 2 : : PUNCT cana-937 117 1 [ [ X cana-937 117 2 1 1 NUM cana-937 117 3 ] ] X cana-937 117 4 gardiner gardiner NOUN cana-937 117 5 , , PUNCT cana-937 117 6 c. c. PROPN cana-937 117 7 w. w. PROPN cana-937 117 8 ( ( PUNCT cana-937 117 9 2009 2009 NUM cana-937 117 10 ) ) PUNCT cana-937 117 11 . . PUNCT cana-937 118 1 stochastic stochastic ADJ cana-937 118 2 methods method NOUN cana-937 118 3 : : PUNCT cana-937 118 4 a a DET cana-937 118 5 handbook handbook NOUN cana-937 118 6 for for ADP cana-937 118 7 the the DET cana-937 118 8 natural natural ADJ cana-937 118 9 and and CCONJ cana-937 118 10 social social ADJ cana-937 118 11 sciences science NOUN cana-937 118 12 . . PUNCT cana-937 119 1 [ [ X cana-937 119 2 2 2 NUM cana-937 119 3 ] ] X cana-937 119 4 kloeden kloeden PROPN cana-937 119 5 , , PUNCT cana-937 119 6 p. p. PROPN cana-937 119 7 e. e. PROPN cana-937 119 8 , , PUNCT cana-937 119 9 & & CCONJ cana-937 119 10 platen platen PROPN cana-937 119 11 , , PUNCT cana-937 119 12 e. e. PROPN cana-937 119 13 ( ( PUNCT cana-937 119 14 1992 1992 NUM cana-937 119 15 ) ) PUNCT cana-937 119 16 . . PUNCT cana-937 120 1 numerical numerical ADJ cana-937 120 2 solution solution NOUN cana-937 120 3 of of ADP cana-937 120 4 stochastic stochastic ADJ cana-937 120 5 differential differential ADJ cana-937 120 6 equations equation NOUN cana-937 120 7 . . PUNCT cana-937 121 1 [ [ X cana-937 121 2 3 3 NUM cana-937 121 3 ] ] X cana-937 121 4 risken risken VERB cana-937 121 5 , , PUNCT cana-937 121 6 h. h. PROPN cana-937 121 7 ( ( PUNCT cana-937 121 8 1989 1989 NUM cana-937 121 9 ) ) PUNCT cana-937 121 10 . . PUNCT cana-937 122 1 the the DET cana-937 122 2 fokker fokker NOUN cana-937 122 3 - - PUNCT cana-937 122 4 planck planck NOUN cana-937 122 5 equation equation NOUN cana-937 122 6 : : PUNCT cana-937 122 7 methods method NOUN cana-937 122 8 of of ADP cana-937 122 9 solution solution NOUN cana-937 122 10 and and CCONJ cana-937 122 11 applications application NOUN cana-937 122 12 . . PUNCT cana-937 123 1 [ [ X cana-937 123 2 4 4 NUM cana-937 123 3 ] ] X cana-937 123 4 ΓΈksendal ΓΈksendal NOUN cana-937 123 5 , , PUNCT cana-937 123 6 b. b. PROPN cana-937 123 7 ( ( PUNCT cana-937 123 8 2013 2013 NUM cana-937 123 9 ) ) PUNCT cana-937 123 10 . . PUNCT cana-937 124 1 stochastic stochastic ADJ cana-937 124 2 differential differential ADJ cana-937 124 3 equations equation NOUN cana-937 124 4 : : PUNCT cana-937 124 5 an an DET cana-937 124 6 introduction introduction NOUN cana-937 124 7 with with ADP cana-937 124 8 applications application NOUN cana-937 124 9 . . PUNCT cana-937 125 1 [ [ X cana-937 125 2 5 5 NUM cana-937 125 3 ] ] SYM cana-937 125 4 karatzas karatzas PROPN cana-937 125 5 , , PUNCT cana-937 125 6 i. i. PROPN cana-937 125 7 , , PUNCT cana-937 125 8 & & CCONJ cana-937 125 9 shreve shreve PROPN cana-937 125 10 , , PUNCT cana-937 125 11 s. s. PROPN cana-937 125 12 e. e. PROPN cana-937 125 13 ( ( PUNCT cana-937 125 14 1991 1991 NUM cana-937 125 15 ) ) PUNCT cana-937 125 16 . . PUNCT cana-937 126 1 brownian brownian ADJ cana-937 126 2 motion motion NOUN cana-937 126 3 and and CCONJ cana-937 126 4 stochastic stochastic ADJ cana-937 126 5 calculus calculus NOUN cana-937 126 6 ( ( PUNCT cana-937 126 7 vol vol NOUN cana-937 126 8 . . NOUN cana-937 126 9 113 113 NUM cana-937 126 10 ) ) PUNCT cana-937 126 11 . . PUNCT cana-937 127 1 springer springer PROPN cana-937 127 2 science science PROPN cana-937 127 3 & & CCONJ cana-937 127 4 business business NOUN cana-937 127 5 media medium NOUN cana-937 127 6 . . PUNCT cana-937 128 1 [ [ X cana-937 128 2 6 6 NUM cana-937 128 3 ] ] X cana-937 128 4 gardiner gardiner NOUN cana-937 128 5 , , PUNCT cana-937 128 6 c. c. PROPN cana-937 128 7 w. w. PROPN cana-937 128 8 , , PUNCT cana-937 128 9 & & CCONJ cana-937 128 10 zoller zoller PROPN cana-937 128 11 , , PUNCT cana-937 128 12 p. p. NOUN cana-937 128 13 ( ( PUNCT cana-937 128 14 2004 2004 NUM cana-937 128 15 ) ) PUNCT cana-937 128 16 . . PUNCT cana-937 129 1 quantum quantum NOUN cana-937 129 2 noise noise NOUN cana-937 129 3 : : PUNCT cana-937 129 4 a a DET cana-937 129 5 handbook handbook NOUN cana-937 129 6 of of ADP cana-937 129 7 markovian markovian PROPN cana-937 129 8 and and CCONJ cana-937 129 9 non non ADJ cana-937 129 10 - - ADJ cana-937 129 11 markovian markovian ADJ cana-937 129 12 quantum quantum NOUN cana-937 129 13 stochastic stochastic NOUN cana-937 129 14 methods method NOUN cana-937 129 15 with with ADP cana-937 129 16 applications application NOUN cana-937 129 17 to to ADP cana-937 129 18 quantum quantum ADJ cana-937 129 19 optics optic NOUN cana-937 129 20 . . PUNCT cana-937 130 1 springer springer NOUN cana-937 130 2 science science PROPN cana-937 130 3 & & CCONJ cana-937 130 4 business business NOUN cana-937 130 5 media medium NOUN cana-937 130 6 . . PUNCT cana-937 131 1 [ [ X cana-937 131 2 7 7 X cana-937 131 3 ] ] X cana-937 131 4 van van PROPN cana-937 131 5 kampen kampen PROPN cana-937 131 6 , , PUNCT cana-937 131 7 n. n. PROPN cana-937 131 8 g. g. PROPN cana-937 131 9 ( ( PUNCT cana-937 131 10 2007 2007 NUM cana-937 131 11 ) ) PUNCT cana-937 131 12 . . PUNCT cana-937 132 1 stochastic stochastic ADJ cana-937 132 2 processes process NOUN cana-937 132 3 in in ADP cana-937 132 4 physics physics NOUN cana-937 132 5 and and CCONJ cana-937 132 6 chemistry chemistry NOUN cana-937 132 7 . . PUNCT cana-937 133 1 elsevier elsevier NOUN cana-937 133 2 . . PUNCT cana-937 134 1 [ [ X cana-937 134 2 8 8 NUM cana-937 134 3 ] ] X cana-937 134 4 risken risken VERB cana-937 134 5 , , PUNCT cana-937 134 6 h. h. PROPN cana-937 134 7 ( ( PUNCT cana-937 134 8 1996 1996 NUM cana-937 134 9 ) ) PUNCT cana-937 134 10 . . PUNCT cana-937 135 1 the the DET cana-937 135 2 fokker fokker NOUN cana-937 135 3 - - PUNCT cana-937 135 4 planck planck NOUN cana-937 135 5 equation equation NOUN cana-937 135 6 : : PUNCT cana-937 135 7 methods method NOUN cana-937 135 8 of of ADP cana-937 135 9 solution solution NOUN cana-937 135 10 and and CCONJ cana-937 135 11 applications application NOUN cana-937 135 12 ( ( PUNCT cana-937 135 13 vol vol NOUN cana-937 135 14 . . PROPN cana-937 135 15 18 18 NUM cana-937 135 16 ) ) PUNCT cana-937 135 17 . . PUNCT cana-937 136 1 springer springer PROPN cana-937 136 2 science science PROPN cana-937 136 3 & & CCONJ cana-937 136 4 business business NOUN cana-937 136 5 media medium NOUN cana-937 136 6 . . PUNCT cana-937 137 1 [ [ X cana-937 137 2 9 9 NUM cana-937 137 3 ] ] SYM cana-937 137 4 kloeden kloeden PROPN cana-937 137 5 , , PUNCT cana-937 137 6 p. p. PROPN cana-937 137 7 e. e. PROPN cana-937 137 8 , , PUNCT cana-937 137 9 & & CCONJ cana-937 137 10 platen platen PROPN cana-937 137 11 , , PUNCT cana-937 137 12 e. e. PROPN cana-937 137 13 ( ( PUNCT cana-937 137 14 1999 1999 NUM cana-937 137 15 ) ) PUNCT cana-937 137 16 . . PUNCT cana-937 138 1 numerical numerical ADJ cana-937 138 2 solution solution NOUN cana-937 138 3 of of ADP cana-937 138 4 stochastic stochastic ADJ cana-937 138 5 differential differential ADJ cana-937 138 6 equations equation NOUN cana-937 138 7 ( ( PUNCT cana-937 138 8 vol vol NOUN cana-937 138 9 . . PROPN cana-937 138 10 23 23 NUM cana-937 138 11 ) ) PUNCT cana-937 138 12 . . PUNCT cana-937 139 1 springer springer PROPN cana-937 139 2 science science PROPN cana-937 139 3 & & CCONJ cana-937 139 4 business business NOUN cana-937 139 5 media medium NOUN cana-937 139 6 . . PUNCT cana-937 140 1 [ [ X cana-937 140 2 10 10 NUM cana-937 140 3 ] ] X cana-937 140 4 hΓ€nggi hΓ€nggi NOUN cana-937 140 5 , , PUNCT cana-937 140 6 p. p. NOUN cana-937 140 7 , , PUNCT cana-937 140 8 talkner talkner NOUN cana-937 140 9 , , PUNCT cana-937 140 10 p. p. NOUN cana-937 140 11 , , PUNCT cana-937 140 12 & & CCONJ cana-937 140 13 borkovec borkovec NOUN cana-937 140 14 , , PUNCT cana-937 140 15 m. m. NOUN cana-937 140 16 ( ( PUNCT cana-937 140 17 1990 1990 NUM cana-937 140 18 ) ) PUNCT cana-937 140 19 . . PUNCT cana-937 141 1 reaction reaction NOUN cana-937 141 2 - - PUNCT cana-937 141 3 rate rate NOUN cana-937 141 4 theory theory NOUN cana-937 141 5 : : PUNCT cana-937 141 6 fifty fifty NUM cana-937 141 7 years year NOUN cana-937 141 8 after after ADP cana-937 141 9 kramers kramer NOUN cana-937 141 10 . . PUNCT cana-937 142 1 reviews review NOUN cana-937 142 2 of of ADP cana-937 142 3 modern modern ADJ cana-937 142 4 physics physic NOUN cana-937 142 5 , , PUNCT cana-937 142 6 62(2 62(2 NOUN cana-937 142 7 ) ) PUNCT cana-937 142 8 , , PUNCT cana-937 142 9 251 251 NUM cana-937 142 10 . . PUNCT cana-937 143 1 [ [ X cana-937 143 2 11 11 NUM cana-937 143 3 ] ] X cana-937 143 4 hughes hughes PROPN cana-937 143 5 , , PUNCT cana-937 143 6 b. b. PROPN cana-937 143 7 d. d. PROPN cana-937 143 8 ( ( PUNCT cana-937 143 9 1995 1995 NUM cana-937 143 10 ) ) PUNCT cana-937 143 11 . . PUNCT cana-937 144 1 random random ADJ cana-937 144 2 walks walk NOUN cana-937 144 3 and and CCONJ cana-937 144 4 random random ADJ cana-937 144 5 environments environment NOUN cana-937 144 6 ( ( PUNCT cana-937 144 7 vol vol NOUN cana-937 144 8 . . NOUN cana-937 144 9 1 1 NUM cana-937 144 10 ) ) PUNCT cana-937 144 11 . . PUNCT cana-937 145 1 oxford oxford PROPN cana-937 145 2 university university PROPN cana-937 145 3 press press NOUN cana-937 145 4 . . PUNCT cana-937 146 1 [ [ X cana-937 146 2 12 12 NUM cana-937 146 3 ] ] PUNCT cana-937 146 4 haenggi haenggi NOUN cana-937 146 5 , , PUNCT cana-937 146 6 p. p. NOUN cana-937 146 7 ( ( PUNCT cana-937 146 8 2018 2018 NUM cana-937 146 9 ) ) PUNCT cana-937 146 10 . . PUNCT cana-937 147 1 stochastic stochastic ADJ cana-937 147 2 processes process NOUN cana-937 147 3 in in ADP cana-937 147 4 physics physics NOUN cana-937 147 5 , , PUNCT cana-937 147 6 chemistry chemistry NOUN cana-937 147 7 , , PUNCT cana-937 147 8 and and CCONJ cana-937 147 9 biology biology NOUN cana-937 147 10 . . PUNCT cana-937 148 1 springer springer NOUN cana-937 148 2 . . PUNCT cana-937 149 1 [ [ X cana-937 149 2 13 13 NUM cana-937 149 3 ] ] X cana-937 149 4 engel engel PROPN cana-937 149 5 , , PUNCT cana-937 149 6 a. a. PROPN cana-937 149 7 , , PUNCT cana-937 149 8 & & CCONJ cana-937 149 9 schmid schmid PROPN cana-937 149 10 , , PUNCT cana-937 149 11 g. g. PROPN cana-937 149 12 ( ( PUNCT cana-937 149 13 2001 2001 NUM cana-937 149 14 ) ) PUNCT cana-937 149 15 . . PUNCT cana-937 150 1 stochastic stochastic ADJ cana-937 150 2 dynamics dynamic NOUN cana-937 150 3 of of ADP cana-937 150 4 molecular molecular ADJ cana-937 150 5 systems system NOUN cana-937 150 6 . . PUNCT cana-937 151 1 chemical chemical NOUN cana-937 151 2 physics physics PROPN cana-937 151 3 , , PUNCT cana-937 151 4 266(1 266(1 NUM cana-937 151 5 - - SYM cana-937 151 6 3 3 NUM cana-937 151 7 ) ) PUNCT cana-937 151 8 , , PUNCT cana-937 151 9 139 139 NUM cana-937 151 10 - - SYM cana-937 151 11 151 151 NUM cana-937 151 12 . . PUNCT cana-937 152 1 [ [ X cana-937 152 2 14 14 NUM cana-937 152 3 ] ] X cana-937 152 4 bao bao PROPN cana-937 152 5 , , PUNCT cana-937 152 6 f. f. PROPN cana-937 152 7 , , PUNCT cana-937 152 8 cao cao PROPN cana-937 152 9 , , PUNCT cana-937 152 10 j. j. PROPN cana-937 152 11 , , PUNCT cana-937 152 12 & & CCONJ cana-937 152 13 zhang zhang PROPN cana-937 152 14 , , PUNCT cana-937 152 15 q. q. PROPN cana-937 152 16 ( ( PUNCT cana-937 152 17 2018 2018 NUM cana-937 152 18 ) ) PUNCT cana-937 152 19 . . PUNCT cana-937 153 1 stochastic stochastic ADJ cana-937 153 2 differential differential ADJ cana-937 153 3 equations equation NOUN cana-937 153 4 in in ADP cana-937 153 5 finance finance NOUN cana-937 153 6 and and CCONJ cana-937 153 7 engineering engineering NOUN cana-937 153 8 : : PUNCT cana-937 153 9 a a DET cana-937 153 10 unified unified ADJ cana-937 153 11 treatment treatment NOUN cana-937 153 12 with with ADP cana-937 153 13 matlab matlab PROPN cana-937 153 14 examples example NOUN cana-937 153 15 . . PUNCT cana-937 154 1 springer springer NOUN cana-937 154 2 . . PUNCT