CUBO A Mathematical Journal
Vol.15, No¯ 01, (113–117). March 2013

A Girsanov formula associated to a big order
pseudo-differential operator

Rémi Léandre

Université de Franche -Comté,

Laboratoire de Mathématiques,

25030 Besançon, France.

remi.leandre@univ-fcomte.fr

ABSTRACT

We give a quasi-invariance formula involved with a semi-group generated by a big order

elliptic pseudo-differential operator.

RESUMEN

Entregamos una fórmula de cuasi-invarianza relacionada con un semigrupo generado

por un operador seudo-diferencial eĺıptico de orden superior.

Keywords and Phrases: Pseudo-differential operators. Girsanov formula.

2010 AMS Mathematics Subject Classification: 35K41, 35S05, 60G20.



114 Rémi Léandre CUBO
15, 1 (2013)

1 Introduction

Dedicated to professor doctor N’Guérékata for his birthday

There are two basic tools in the theory of stochastic processes ([2], [6], [17]):

-) Itô formulas.

-)Quasi-invariance formulas of Girsanov type.

Roughly speaking, a semi group exp[tL] governed by a generator L whose domain is continu-

ously densely imbedded in the space of bounded continuous functions Cb(R
d) endowed with the

uniform norm is represented by a stochastic process

exp[tL]f(x) = E[f(xt(x))] (1)

if and only if the generator satisfies the maximum principle: Lf(x) ≥ 0 if the function f reaches his

maximum in x. Such semi-groups are called Markov semi-groups.

There are much more semi-groups than Markov semi-groups.

Itô formula was extended for more general partial differential equations in [7], [8], [9], [10],

[11], [15]. For an approach to Itô formula to generalized Wiener chaos, we refer to [13], [14].

Girsanov formula was extended in the framework of white noise analysis to bilaplacians in [13]

and [14].

We refer to [16] to a review.

The object of this paper is to extend the Girsanov formula to very general semi-groups gen-

erated by general pseudo-differential operators.

2 Statement and proof of the main theorem

Let (x, ξ) → a(x, ξ) a smooth function on Rd × Rd. According the terminology of [3], [4], [5] it

is called a symbol. We suppose that if |ξ| ≤ C the symbol is smooth with bounded derivatives at

each order. If |ξ| > C, we suppose that there exist a strictly positive integer m such that

sup
x∈Rd

|DkxD
k

′

ξ a(x, ξ)| ≤ C|ξ|
2m−k

′

(2)

We suppose that the symbol is elliptic:

inf
x∈Rd

|a(x, ξ)| ≥ C|ξ|2m (3)

We put by standard theory on pseudo-differential operators ([3], [4], [5])

L̂f(x) =

∫

Rd

a(x, ξ)f̂(ξ)dξ (4)



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A Girsanov formula associated to a big order pseudo-differential ... 115

where ξ → f̂(ξ) is the Fourier transform of x → f(x).He can be extended continuously on the space

of smooth functions with bounded derivatives at each order. We suppose because later we consider

Girsanov type formulas that L1 = 0.

Hypothesis: We suppose that −L is positive essentially self-adjoint on L2(Rd).

L generates a contraction semigroup exp[tL] on L2(Rd). By elliptic theory,

exp[tL]f(x) =

∫

Rd

f(y)µt(x, dy) (5)

where µt(x, dy) is a measure on R
d (But not a probability measure).

We consider an operator L1 on L2(Rd) and we suppose that it is a pseudodifferential operator

of order strictly smaller than 2m − 1 of the type (2) and (4). He can be extended continuously on

the space of smooth functions with bounded derivatives at each order. We suppose because later

we consider Girsanov type formulas that L11 = 0. We consider the pseudo-differential operator

densely defined on L2(Rd × R)

− Ltot = −L − L1
∂

∂u
+ (−1)m

∂2m

∂u2m
(6)

By elliptic theory, it generates a semi group exp[tLtot]on L2(Rd × R) (But not a contraction semi-

group due to the perturbation term L1 ∂
∂u

in the total operator Ltot). The main remark is that if

f depends only on u L1 ∂
∂u

f = 0! By elliptic theory

exp[tLtot]f(x, u) =

∫

Rd×R

f(y, v)µtott (x, u, dy, dv) (7)

where µtot is a measure on Rd × R (But not a probability measure).

We consider the operator densely defined on L2(Rd)

− Lper = −L − L1 (8)

By elliptic theory, it generates a semi-group on L2(Rd) (But not a contraction semi-group due to

the perturbation term L1). By elliptic theory, it generates a semi-group on L2(Rd) (but not a

contraction semi-group due to the perturbation term L1). By elliptic theory,

exp[tLper]f(x) =

∫

Rd

f(y)µ
per
t (x, dy) (9)

where µ
per
t (x, dy) is a measure on R

d (but not a probability measure).

We get

Theorem 2.1. (Girsanov): We have if f is continuous with compact support and if we consider

the Doleans-Dade exponential exp[u + (−1)mt] = g(u, t)

exp[tLper]f(x) = exp[tLtot][f(.)g(., t)](x, 0) (10)



116 Rémi Léandre CUBO
15, 1 (2013)

Proof:Let us begin by doing formal computations. ∂
∂u

commute with Ltot. Therefore

Ltot exp[tLtot][f(.)g(., t)](x, u) = Lexp[tLtot][f(.)g(., t)](x, u)+

L1exp[tLtot][f(.)
∂

∂v
g(., t)](x, u) + exp[tLtot][f(.)(−1)m+1

∂2m

∂v2m
g(., t)](x, u) =

A1 + A2 + A3 (11)

The term A3 is boring. This explain that we introduce exp[(−1)
mt] in the Doleans-Dade

exponential in order to remove it. Namely we consider linear semi-groups such that

exp[tLtot][f(.)g(., t)](x, u) = exp[tLtot][f(.) exp[.]](x, 0) exp[(−1)mt] (12)

Therefore A3 disappears and

∂

∂t
exp[tLtot][f(.)g(., t)](x, 0) = Lper exp[tLtot][f(.)g(., t)](x, 0) (13)

The only problem in this formal comutation is that u → exp[u] is not bounded!. But if f is with

compact support continuous

| exp[tLtot][f(.) exp[.]](x, 0)| ≤

∫

Rd×R

|f(y)| exp[v]|µtott |(x, u, dy, dv)

≤ (

∫

Rd

|f(y)|2|µt|(x, dy))
1/2(

∫

R

exp[2u]|νt|(0, dv))
1/2 (14)

In (14), νt(u, dv) represents the semi group associated to L2m = (−1)
m+1 ∂

2m

∂u2m
. By [1], this

semi-group has an heat-kernel bounded by Ct−1/4mG2m,a(
|u−v|

t1/4m
) (a > 0) where

G2m,a(u) = exp[−au
2m/2m−1] (15)

This inequality justifies the formal considerations above!

♦

Received: November 2012. Revised: February 2013.

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