Equivalence of laws and null controllability for SPDEs driven by a fractional Brownian motion

TL;DR

This paper establishes conditions under which the probability laws of solutions to certain SPDEs driven by fractional Brownian motion are equivalent, linking this to null controllability of associated deterministic control problems.

Contribution

It provides a comprehensive characterization of law equivalence for SPDEs driven by Gaussian noise, including fractional Brownian motion, and connects this to controllability properties.

Findings

Necessary and sufficient conditions for law equivalence are identified.

Law equivalence is linked to null controllability of the deterministic control problem.

Results apply to semilinear reaction-diffusion equations driven by fractional Brownian motion.

Abstract

We obtain necessary and sufficient conditions for equivalence of law for linear stochastic evolution equations driven by a general Gaussian noise by identifying the suitable space of controls for the corresponding deterministic control problem. This result is applied to semilinear (reaction-diffusion) equations driven by a fractional Brownian motion. We establish the equivalence of continuous dependence of laws of solutions to semilinear equations on the initial datum in the topology of pointwise convergence of measures and null controllability for the corresponding deterministic control problem.