A splitting algorithm for stochastic partial differential equations driven by linear multiplicative noise

Abstract

It is discussed the convergence of a Douglas–Rachford type splitting algorithm for the infinite dimensional stochastic differential equation $$\begin{aligned} dX+A(t)(X)dt=X\,dW \text{ in } (0,T); X(0)=x, \end{aligned}$$ where $$A(t):V\rightarrow V'$$ is a nonlinear, monotone, coercive and demicontinuous operator with sublinear growth and V is a real Hilbert with the dual $$V'$$ . V is densely and continuously embedded in the Hilbert space H and W is an H-valued Wiener process. The general case of a maximal monotone operators $$A(t):H\rightarrow H$$ is also investigated.