Almost Sure Asymptotic Stability of Parabolic SPDEs with Small Multiplicative Noise

Abstract: A better understanding of the instability margin will eventually optimize the operational range for safety-critical industries. In this paper, we investigate the almost-sure exponential asymptotic stability of the trivial solution of a parabolic semilinear stochastic partial differential equation (SPDE) driven by multiplicative noise near the deterministic Hopf bifurcation point. We show the existence and uniqueness of the invariant measure under appropriate assumptions, and approximate the exponential growth rate via asymptotic expansion, given that the strength of the noise is small. This approximate quantity can readily serve as a robust indicator of the change of almost-sure stability.