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 linke… Show more
“…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.…”
We consider stochastic differential equations with a drift term of gradient type and driven by Gaussian white noise on R d . Particular attention is given to the kernel p t , t ≥ 0 of the transition semigroup associated with the solution process.Under some rather strong regularity and growth assumptions on the coefficients, we adapt previous work by Thierry Hargé on Schrödinger operators and prove that the small time asymptotic expansion of p t , t > 0 is Borel summable.We also briefly indicate some extensions and applications.
“…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.…”
We consider stochastic differential equations with a drift term of gradient type and driven by Gaussian white noise on R d . Particular attention is given to the kernel p t , t ≥ 0 of the transition semigroup associated with the solution process.Under some rather strong regularity and growth assumptions on the coefficients, we adapt previous work by Thierry Hargé on Schrödinger operators and prove that the small time asymptotic expansion of p t , t > 0 is Borel summable.We also briefly indicate some extensions and applications.
“…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].…”
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.
“…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].…”
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confidence: 99%
“…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].…”
We investigate the stationary diffusion equation with a coefficient given by a (transformed) Lévy random field. Lévy random fields are constructed by smoothing Lévy noise fields with kernels from the Matérn class. We show that Lévy noise naturally extends Gaussian white noise within Minlos' theory of generalized random fields. Results on the distributional path spaces of Lévy noise are derived as well as the amount of smoothing to ensure such distributions become continuous paths. Given this, we derive results on the pathwise existence and measurability of solutions to the random boundary value problem (BVP). For the solutions of the BVP we prove existence of moments (in the H 1 -norm) under adequate growth conditions on the Lévy measure of the noise field. Finally, a kernel expansion of the smoothed Lévy noise fields is introduced and convergence in L n (n ≥ 1) of the solutions associated with the approximate random coefficients is proven with an explicit rate.
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