Transportation cost inequalities for stochastic reaction diffusion equations on the whole line $\mathbb{R}$
Yue Li, Shijie Shang, Tusheng Zhang
TL;DR
The paper addresses quadratic transportation cost inequalities for the laws of solutions to stochastic reaction–diffusion equations on the real line driven by space–time white noise. It develops new moment bounds for stochastic convolution in weighted tempered spaces $L^2_{tem}$ and $C_{tem}$, and then uses a Girsanov change of measure to couple the laws and Gronwall-type estimates to bound the Wasserstein distance by relative entropy. The main contributions include proving $W_2(Q,P^{\mu})^2\le c\,H(Q|P^{\mu})$ for the deterministic-initial and random-initial settings on $L^2_{tem}$ and $C_{tem}$ path spaces, and extending these results to random initial data via Lipschitz continuity of the solution map. The work provides a rigorous transport-inequality framework for SPDEs on unbounded domains, with explicit constants depending on the model data, heat-kernel bounds, and weight parameters. This yields concentration-of-measure and large-deviation insights for SPDEs in unbounded spatial domains, broadening the applicability of Talagrand-type inequalities in stochastic PDEs.
Abstract
In this paper, we established quadratic transportation cost inequalities for solutions of stochastic reaction diffusion equations driven by multiplicative space-time white noise on the whole line $\mathbb{R}$. Since the space variable is defined on the unbounded domain $\mathbb{R}$, the inequalities are proved under a weighted $L^2$-norm and a weighted uniform metric in the so called $L^2_{tem}$, $C_{tem}$ spaces. The new moments estimates of the stochastic convolution with respect to space-time white noise play an important role. In addition, the transportation cost inequalities are also obtained for the stochastic reaction diffusion equations with random initial values.