In this paper, we established a quadratic transportation cost inequality under the uniform/maximum norm for solutions of stochastic heat equations driven by multiplicative space-time white noise. The proof is based on a new inequality we obtained for the moments of the stochastic convolution with respect to space-time white noise, which is of independent interest. The solutions of such stochastic partial differential equations are typically not semimartingales on the state space.
"Talagrand concentration inequalities for stochastic heat-type equations under uniform distance." Electron. J. Probab. 24 1 - 15, 2019. https://doi.org/10.1214/19-EJP388