Wong-Zakai approximations of stochastic evolution equations

Abstract

Abstract. Theorems on weak convergence of the laws of the Wong-Zakai approximations for evolution equationare proved. The operator A in the equation generates an analytic semigroup of linear operators on a Hilbert space H. The tightness of the approximating sequence is established using the stochastic factorization formula. Applications to strongly damped wave and plate equations as well as to stochastic invariance are discussed.

Weak approximation of a fractional SDE

“…This kind of result, usually known as diffusion approximation, has been thoroughly studied in the literature (see e.g. [17,32,33]), since it also shows that equations like (1) may emerge as the limit of a noisy equation driven by a fast oscillating function. The diffusion approximation program has also been taken up in the fBm case by Marty in [23], with some random wave problems in mind, but only in the cases where H > 1/2 or the dimension d of the fBm is 1.…”

Non-random Invariant Sets for Some Systems of Parabolic Stochastic Partial Differential Equations

“…In the finitedimensional case this kind of approximation theorems is well-known (see, e.g., Refs. [9,14,32,33,51,53,54] and the literature cited there). There is a substantial number of publications devoted to Wong-Zakaï type approximations of infinite-dimensional stochastic equations (see, e.g., Refs.…”

“…[8,19,34,50] and their references), and it is a well established fact that such invariant sets along with the validity of comparison principles play an important role in the analysis of the qualitative properties of solutions (see, for instance, Refs. [7,38,40,51,59] and their references), the analysis of comparison principles has been essentially limited to single parabolic stochastic partial differential equations (see, for instance, Refs. There are comparatively fewer related results regarding stochastic partial differential equations; in fact, whereas there exist several papers devoted to the proof of the so-called stochastic invariance for certain stochastic evolution equations (see, for instance, Refs.…”

Wong–Zakai Type Approximation of SPDEs of Lévy Noise

“…Twardowska [18][19][20][21][22][23] considered certain cases of SPDEs driven by Wiener noise. One should also mention the papers of Brzeźniak and Carroll [3], Brzeźniak, Capinski and Flandoli [2], Brzeźniak and Flandoli [4], Gyöngy [7], Gyöngy and Pröhle [8], Gyöngy and Shmatkov [9], Nowak [14] and the preprint of Tessitore and Zabczyk [17]. Proppe [16] has extended the result in finite dimension to stochastic differential equations driven by Poisson random measures.…”