An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise

Abstract

In this paper, we develop a new general approach to the existence and uniqueness theory of infinite dimensional stochastic equations of the form

where $A(t)$ is a nonlinear monotone and demicontinuous operator from $V$ to $V^{\prime }$, coercive and with polynomial growth. Here, $V$ is a reflexive Banach space continuously and densely embedded in a Hilbert space $H$ of (generalized) functions on a domain $\mathcal{O}\subset {\mathbb{R}}^d$ and $V^{\prime }$ is the dual of $V$ in the duality induced by $H$ as pivot space. Furthermore, $W$ is a Wiener process in $H$. The new approach is based on an operatorial reformulation of the stochastic equation which is quite robust under perturbation of $A(t)$. This leads to new existence and uniqueness results of a larger class of equations with linear multiplicative noise than the one treatable by the known approaches. In addition, we obtain regularity results for the solutions with respect to both the time and spatial variable which are sharper than the classical ones. New applications include stochastic partial differential equations, as e.g. stochastic transport equations.