Stochastic partial differential equations driven by multi-parameter white noise of Lévy processes

Abstract

Abstract. We give a short introduction to the white noise theory for multiparameter Lévy processes and its application to stochastic partial differential equations driven by such processes. Examples include temperature distribution with a Lévy white noise heat source, and heat propagation with a multiplicative Lévy white noise heat source. Introduction. The white noise theory was originally developed by T. Hida for Brownian motion {B(t)} t≥0. See e.g. [7] and [8] and the references therein. The main idea was that a rigorous mathematical foundation for the time derivative of B(t),

On extended stochastic integrals with respect to Lévy processes

“…Finally, the domains of integrals (24) and (29) coincide because both these domains are given by the condition: the result of integration is an element of (L 2 ), see (25) and (30).…”

“…In the case of the "Nualart-Schoutens's CRP" one can use term by term integration of a NualartSchoutens decomposition for an integrand with respect to a random measure corresponding to L. In the case of the "Lytvynov's CRP" one can construct the extended stochastic integral as in the Meixner case [17] (see also [18]): with use of a "special symmetrization" for kernels from the Lytvynov decomposition, or as the conjugated operator to the Hida stochastic derivative. The reader can find more information about extended stochastic integrals with respect to Lévy processes in, e.g., [3,21,10,8,11,25,9], for a general information about stochastic integration on infinite-dimensional spaces see, e.g., [1].…”

WICK-TYPE STOCHASTIC KdV EQUATION BASED ON LÉVY WHITE NOISE

“…On the other hand, an extension of Gaussian white noise analysis to non-Gaussian white noise analysis was established in [21], and developed further in [22,23]. Based on this extension, Løkka et al [24] and Øksendal [25] developed a white noise framework for the study of SPDEs driven by a d-parameter Lévy white noise, which is in fact a non-Gaussian white noise. Recently, Hyder and Zakarya [20] have developed a non-Gaussian Wick calculus based on the theory of hypercomplex systems ( ) ( ).…”

“…Recently, Hyder and Zakarya [20] have developed a non-Gaussian Wick calculus based on the theory of hypercomplex systems ( ) ( ). The main objective of this paper is to use the theory introduced in [24][25][26] [1,27], the motion of long, unidirectional, weakly nonlinear water waves on a channel can be described by Equation (1.2). Moreover, Wang and Wang [28] gave the exact solutions of Equation (1.1) by using the homogeneous balance principle.…”

Existence and Exponential Mixing of Infinite White $\alpha$-Stable Systems with Unbounded Interactions

“…The SPDEs driven by Lévy noises were intensively studied in the past several decades ( [24], [3], [25], [28], [7], [5], [22], [21], · · · ). The noises can be Wiener( [11], [12]) Poisson ( [5]), α-stable types ( [27], [33]) and so on.…”