Abstract
We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise, with symmetric $\alpha $-stable Lévy white noise as an important special case. We identify conditions for existence of these two kinds of solutions, and, together with a new stochastic Fubini theorem, we provide conditions under which they are essentially equivalent. We apply these results to the linear stochastic heat, wave and Poisson equations driven by a symmetric $\alpha $-stable Lévy white noise.
Highlights
In this article, we consider a linear stochastic partial differential equation (SPDE) of the form Lu = X, (1.1)where L is a partial differential operator and Xis a symmetric pure jump Lévy white noise
From the random field approach to SPDEs, we have the concept of mild solution, which is a random field defined as the convolution of a fundamental solution of L with the noise
The mild solution is defined as a stochastic integral, and some conditions are needed for its existence
Summary
We consider a linear stochastic partial differential equation (SPDE) of the form. Linear SPDE driven by Lévy white noise studied, but the existence of mild solutions for various equations has been considered in [1, 3], and the approach via evolution equations is considered in [17]. A question related to (2) was studied in [8, Theorem 11] This reference gives a necessary condition on the Green’s function of the differential operator for the existence of a random field representation (see Definition 3.4) for the generalized solution to an SPDE driven by a Gaussian colored noise. The mild solution is a random field representation of the generalized solution These results are extended to the case of symmetric pure jump Lévy white noise in Section 5 (Theorem 5.2). The main results can be found in Theorems 6.6, 6.12 and 6.13
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