Stability for the stochastic heat equation with multiplicative noise via finite-dimensional feedback

In this paper, we study the long-time behavior of a stochastic heat equation with multiplicative noise and localized control. We begin by analyzing the uncontrolled dynamics and derive explicit decay rates for both mean-square and almost sure exponential stability. These estimates show that the two notions of stability may hold under different conditions on the parameters, reflecting the interplay between the drift and the multiplicative noise. We then introduce a finite-dimensional feedback control acting on a measurable subset of positive measure, built from finitely many Fourier modes of the solution. In particular, we show that the number of controlled modes determines the decay rate and allows for arbitrarily fast stabilization in the mean-square sense. As a consequence, almost sure exponential stability is recovered via a probabilistic argument, so that both notions of stability are achieved within the same framework and with the same decay rate. As an application, we provide a new proof of controllability for the stochastic heat equation based on an iterative construction of adapted controls in feedback form, avoiding the use of the adjoint equation.