Small noise asymptotic expansions for stochastic PDE’s driven by dissipative nonlinearity and Lévy noise

Abstract

We study a reaction-diffusion evolution equation perturbed by a space-time Lévy noise. The associated Kolmogorov operator is the sum of the infinitesimal generator of a C0-semigroup of strictly negative type acting in a Hilbert space and a nonlinear term which has at most polynomial growth, is non necessarily Lipschitz and is such that the whole system is dissipative.The corresponding Itô stochastic equation describes a process on a Hilbert space with dissipative nonlinear, non globally Lipschitz drift and a Lévy noise. Under smoothness assumptions on the non-linearity, asymptotics to all orders in a small parameter in front of the noise are given, with detailed estimates on the remainders.Applications to nonlinear SPDEs with a linear term in the drift given by a Laplacian in a bounded domain are included. As a particular case we provide the small noise asymptotic expansions for the SPDE equations of FitzHugh Nagumo type in neurobiology with external impulsive noise.

Explicit invariant measures for infinite dimensional SDE driven by Lévy noise with dissipative nonlinear drift I

“…See also [11] for related equations with Lévy noise. For the SPDE equations of the FitzHug-Nagumo type with Lévy noise studied in [13], existence and uniqueness of solutions was proven, as well as existence and uniqueness of invariant measures. Moreover asymptotic small noise expansion for the solutions have been established in the same paper [13].…”

“…In [13] a study was initiated concerning a class of non linear stochastic differential equations with Lévy noise and a drift term consisting of a linear unbounded space-dependent part (typically a Laplacian) and an unbounded non linear part of the dissipative type and of at most polynomial growth at infinity. This class is of particular interest since it contains the case of Fitz Hugh Nagumo equations with space dependence, on a bounded domain of R n or on bounded networks with 1-dimensional edges.…”

Borel summation of the small time expansion of some SDE’s driven by Gaussian white noise

“…(ii) One has k(t, x, y) = ψ −1 (x) u(t, x, y) ψ(y), with u(t, x, y) as in Theorem 2.5 (with ω > 0). u(t, x, y) satisfies the asymptotic expansion given by (14) and (15), and consequently k(t, x, y) has a corresponding asymptotic expansion, given by (14), (15) multiplied…”

“…The usefulness of such expansions has been pointed out very early in many contexts, including differential geometry, analysis, quantum mechanics and statistical mechanics, see, e.g. [3,4,5,6,7,8,14,15,18,19,20,21,22,23,24,30,35,41,42,43,45,46,47,49,52,59,60,61,62,63] and references therein.…”

Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena

“…see, e. g., [36], [19], [27] and references therein. Important new developments concern an analogue of the reduction theory in the presence of symmetries (well known in the deterministic case, see, e. g., work by Gaeta and coworkers, [63,64,62] and by De Vecchi,Morando,Ugolini [53,54]), see [37].…”