Asymptotic strong Feller and weak observability inequality

For a class of non-autonomous linear SPDEs, we establish the equivalence among asymptotic regularization, weak observability, and approximate null controllability for the associated deterministic control systems.

This equivalence provides a deterministic control-theoretic characterization of stochastic smoothing and offers a systematic approach to studying SPDEs driven by spatially localized noise.

We further establish a criterion for semilinear SPDEs based on weak observability of the linearized equations.

Our approach combines methods from PDE control theory with Malliavin calculus.

As applications, we consider the stochastic Oseen equation, non-autonomous uniformly parabolic equations, and the parabolic Sine--Gordon equation, all driven by finite-dimensional, spatially localized white-in-time noise.