Two classes of stochastic Dirichlet equations which admit explicit solution formulas
Jon Gjerde · Birkhäuser Boston eBooks · 1996
In this paper we look at stochastic Dirichlet equations of the type $$Au = (\sum\limits_{i = 1}^m {ci\,\cdot\,Exp\left\{ {W_{{\phi _x}}^i} \right\}} )\diamond u - g$$ $$ u{|_{\partial }}_{D} = f $$ and $$div(Exp\left\{ {{W_{\phi x}}} \right\}\diamond u) = k\diamond u - g$$ $$ u{{|}_{\partial }}_{D} = f $$ where A is a uniformly elliptic second order differential operator and Exp $$ \left\{ {{W_{\phi x}}} \right\} $$ ,K,f and g are elements in the space (S)-1 of generalized white noise distributions. With suitable conditions on K,f and g both classes of stochastic Dirichlet equations admit unique solution formulas in the space (S)-1. These are used to give explicit solution formulas to the Scrödinger and wave equation when the boundary conditions are particularly simple.