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.
We describe a class of explicit invariant measures for both finite and infinite dimensional Stochastic Differential Equations (SDE) driven by Lévy noise. We first discuss in details the finite dimensional case with a linear, resp. non linear, drift. In particular, we exhibit a class of such SDEs for which the invariant measures are given in explicit form, coherently in all dimensions. We then indicate how to relate them to invariant measures for SDEs on separable Hilbert spaces.
We consider the Cauchy problem for a pseudo-differential operator which has a translationinvariant and analytic symbol. For a certain set of initial conditions, a formal solution is obtained by a perturbative expansion. The series so obtained can be re-expressed in terms of generalized Feynman graphs and Feynman rules. The logarithm of the solution can then be represented by a series containing only the connected Feynman graphs. Under some conditions, it is shown that the formal solution uniquely determines the real solution by means of Borel transforms. The formalism is then applied to probabilistic Lévy distributions. Here, the Gaussian part of such a distribution is re-interpreted as a initial condition and a large diffusion expansion for Lévy densities is obtained. It is outlined how this expansion can be used in statistical problems that involve Lévy distributions.
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.
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