Stochastic boundary value problems: A white noise functional approach (Q1326315)

We give a program for solving stochastic boundary value problems involving functionals of (multiparameter) white noise. As an example we solve the stochastic Schrödinger equation \[ \Delta u+V \cdot u=-f \text{ in } D \subset \mathbb{R}^ d, \quad u |_{\partial D}=0, \] where \(V\) is a positive, noisy potential. We represent the potential \(V\) by a white noise functional and interpret the product of the two distribution valued processes \(V\) and \(u\) as a Wick product \(V \diamondsuit u\). Such an interpretation is in accordance with the usual interpretation of a white noise product in ordinary stochastic differential equations. The solution \(u\) will not be a generalized white noise functional but can be represented as an \(L^ 1\) functional process.