Study of a SPDE driven by a Poisson noise
Let \(U\) be a normed space. The author considers a stochastic parabolic differential equation on \(\mathbb{R}^+\times\mathbb{R}^d\times U\) driven by some time-space Poissonian noise \(\lambda^+\) \[ \begin{aligned} V(t,x)= &\;u_0(x)+\int^t_0 {1 \over 2}\Delta V(s,x)ds +\int^t_0 g\bigl(V(s,x),s,x\bigr)ds\\ & +\int^t_0 \int_{h \in U} f\bigl(V_-(s,x),s,x,h)\bigr) d\lambda^+(s,x,h).\end{aligned} \] He uses the contraction principle in order to show that this initial value problem has a unique solution if the intensity of \(\lambda^+\) is suitably bounded and if the functions \(f:\mathbb{R}\times \mathbb{R}^+\times \mathbb{R}^d\times U\) and \(g:\mathbb{R} \times\mathbb{R}^+ \times\mathbb{R}^d\) are Lipschitzian and suitably bounded. He also gives practicable conditions so that this solution is \(l\)-times continuously differentiable in space or time.
stochastic parabolic differential equation
scientific article; zbMATH DE number 1182322