How to determine the law of the solution to a stochastic partial differential equation driven by a Lévy space-time noise?

Abstract

We consider a stochastic partial differential equation on a lattice ∂tX=(Δ−m2)X−λXp+η, where η is a space-time Lévy noise. A perturbative (in the sense of formal power series) strong solution is given by a tree expansion, whereas the correlation functions of the solution are given by a perturbative expansion with coefficients that are represented as sums over a certain class of graphs, called Parisi-Wu graphs. The perturbative expansion of the truncated (connected) correlation functions is obtained via a linked cluster theorem as sums over connected graphs only. The moments of the stationary solution can be calculated as well. In all these solutions the cumulants of the single site distribution of the noise enter as multiplicative constants. To determine them, e.g., by comparison with an empirical correlation function, one can fit these constants (e.g., by the methods of least squares) and thereby one (approximately) determines laws of the solution and the driving noise.

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

“…Applications to various areas including physics, neurobiology, see, e.g. [2,9,11,32,33,34,51,60] and mathematical finance, see,e.g. [35] have been also provided.…”

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

“…Before we go over to describe the contents of the present paper, let us mention that our study of SPDE's with Lévy noise can also be related to the study of certain pseudo-differential equations with such noises which occur in quantum field theory and statistical mechanics (see e.g [10,11]. Also relations to certain problems in the study of statistics of processes described by Lévy noises should be mentioned [40,39].…”

Integrability and Approximability of Solutions to the Stationary Diffusion Equation with Lévy Coefficient

“…While much of the cited work employs random models based on transformed Gaussian random fields, there are effects which a Gaussian model cannot capture, particularly discontinuities and heavy-tail behavior, which nonetheless occur in applications such as flow in fractured media, anomalous diffusion and the modeling of heterogeneous materials [58,17]. It is thus of interest to consider more general stochastic models for the diffusion coefficient, and in this work we extend the Gaussian model to random fields which follow a Lévy distribution [39,5,31].…”

“…Lévy random fields have been studied in a number of contexts, including among others stochastic analysis [4], physics [2], statistics [63] and simulation [63]. For extensions that include interaction between the discrete, discontinuous particle sources of Lévy fields, see [1,31].…”