Transportation-cost inequalities for invariant measures of stochastic reaction-diffusion equations driven by space-time white noise

In this paper, we establish transportation-cost inequalities for invariant measures of stochastic reaction-diffusion equations driven by multiplicative space-time white noise.

In particular, the transportation-cost inequalities are established with respect to the $L^1$ metric, the $L^2$ metric and the uniform metric under different dissipativity strengths of the drift.

We first obtain a global-in-time and spatially uniform moment estimate for stochastic convolutions driven by space-time white noise, which is of independent interest.

We then establish Lipschitz continuity of the solutions to the associated stochastic controlled equations with respect to the controls with a time-independent Lipschitz constant.

Applying this Lipschitz continuity estimate, we finally prove transportation-cost inequalities for the invariant measures of stochastic reaction-diffusion equations.