Wick product and stochastic partial differential equations with Poisson measure (Q1381931)

The paper is concerned with two examples of partial differential equations with Poisson noise. Existence and uniqueness results are obtained. The first equation is a one-dimensional conservation equation of the form \[ \partial_t(u(x,t)-q((0,t]))+\partial_x(a {\textstyle\frac 12} u(x,t)^2+bu(x,t))=0 \] where \(q\) is the centered Poisson measure on \([0,\infty)\) and the products are interpreted as Wick products of generalized random variables. The other equation is a multidimensional linear stochastic differential equation \[ X_t=X_0+\int_0^t\int_E A(s,x)X_{s-}dq(s,x)+\int_0^tB_sX_sds \] where \(E\) is a compact set and the stochastic integral is understood in the Skorokhod sense.