A white noise approach to stochastic Neumann boundary-value problems (Q5937962)

The aim of the paper is to illustrate the application of white noise calculus to stochastic partial differential equations (SPDEs) by studying the stochastic boundary-value problem of Neumann type: \[ LU(x)-c(x)U(x)=0 \text{ for } x \in {\mathcal D}, \qquad {\gamma (x)}U(x)=-W(x) \text{ for } x\in {\partial}{\mathcal D}, \] where \({\mathcal D}\subset {\mathbb R}^d\) is a given bounded \(C^2\) domain, \(L\) is a differential operator of second order and \(W(x)\) is white noise. The authors look for the solutions \(u(x, \cdot)\) as a stochastic distribution (in \(\omega\)) for all \(x\). Such an approach fits well with the white noise theory and permits to avoid some specific difficulties. The authors prove that, under additional conditions, the stochastic boundary-value problem mentioned above has a unique solution. The solution is presented via the pair \((x_t, {\xi}_t)\), that is in turn the solution of the Skorokhod SDE of the form \[ dx_t=b(x_t) dt+ {\sigma}(x_t) d{\beta}_t + {\gamma}(x_t) d{\xi}_t. \] Here \({\beta}(t)\) is \(d\)-dimensional Brownian motion, \({\xi}_t\) is called local time of \(x_t\) on \({\partial}{\mathcal D}\) and other coefficients have some physical sense.