id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-77	Penney, Richard C.; Urban, Roman	Poisson measures on semi-direct products of infinite-dimensional Hilbert spaces	2022	15	.pdf	application/pdf	5211	356	82	Then from (4.1) for 0 < s < t,∫ t 0 EaLσt−sβ v(s, x, σt−s) = Ea ∫ t 0 Lσt−sβ Uσ(0, s)f(x, σt−s) = ∫ t 0 Ea ∫ `2 ∞∑ j=1 e2λj(σt−s)βj∂ 2 xjv(x+ y)N2[β]Aσ(s,0)(dy) ds. Since ∣∣Ea ∫ t 0 e2λj(σt−s) ds ∣∣ = ∣∣Ea ∫ t 0 e2λj(σu) du ∣∣ = ∣∣Ea ∫ t 0 e2(σu)j du ∣∣ = ∣∣Ea ∫ t 0 e2(bu)j−αjt du ∣∣ ≤ ∣∣∣∣Ea ∫ t 0 e2(bu)j du ∣∣∣∣ = Ct we obtain (since β ∈ `1) that |Laβv(s, x, a)| ≤ ∣∣∣‖v‖2∞ ∞∑ j=1 βjEa ∫ t 0 e2λj(σt−s) ds ∣∣∣ ≤ ‖v‖2∞Ct ∣∣ ∞∑ j=1 βj ∣∣ ≤ Ct‖v‖∞‖β‖`1 .	cache/ejde-77.pdf	txt/ejde-77.txt