  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.
