Parabolic SPDEs driven by Poisson white noise
Let \((\Omega ,\mathcal F,(\mathcal F_{t}),P)\) be a complete probability space with a filtration satisfying the usual conditions and \((U,\mathcal B,\nu)\) a \(\sigma \)-finite measure space. Denote by \(p\) the Poisson random measure on \(\mathbb R_{+}\times U\) and set \(q([0,t],A,\omega)=p([0,t],A,\omega)-t\nu (A)\), \(t\geq 0\), \(A\in \mathcal B\), \(\nu (A)<\infty \), \(\omega \in \Omega \). A semilinear heat equation driven by a Poisson white noise \[ \frac {\partial u}{\partial t} = \frac 12\frac {\partial ^{2}u} {\partial x^{2}} + f(t,x,u) + \int_{U}g(t,x,u,y)\eta_{t} ( dy,\omega), \quad u(0,\cdot) = u_{0}, \tag{1} \] is studied, where \(f:\mathbb R_{+}\times \mathbb R\times \mathbb R\to \mathbb R\) and \(g:\mathbb R_{+}\times \mathbb R\times \mathbb R\times U\to \mathbb R\) are measurable functions and \(\eta_{t}\) is the Poisson white noise, defined heuristically as the Radon-Nikodym derivative of \(q\) with respect to the Lebesgue measure on \(\mathbb R_{+}\). Assume that the functions \(f(t,x,\cdot)\) and \(\int_{U} g(t,x,\cdot ,u) d\nu (y)\) obey Lipschitz and linear growth type conditions. Existence and uniqueness of a mild solution to (1) are established for any \(u_{0}\in L^{2}(\mathbb R)\): there exists a unique adapted random field \(\{u(t,x)\}_{t\geq 0, x\in \mathbb R}\) such that \(t\mapsto u(t,x)\) is a càdlàg \(L^{2}(\Omega)\)-valued process, \(u(t,\cdot)\) is continuous almost surely, and \[ \begin{aligned} u(t,x) = &\int_{\mathbb R} G(t,x-z)u_{0}(z) dz + \int ^{t}_{0}\int_{\mathbb R} G(t-s,x-z)f(s,z,u(s,z)) dz ds \\ &+ \int ^{t+}_{0}\int_{U}\int_{\mathbb R} G(t-s,x-z) g(s,z,u(s,z),y) dz q( ds, dy), \end{aligned} \] \(G\) denoting the fundamental solution to the heat equation on \(\mathbb R\).
scientific article; zbMATH DE number 1364454
scientific article