Transportation cost-information inequality for non-linear time-fractional stochastic heat equation driven by space–time white noise

We establish transportation cost-information inequalities $T_2\left(C\right)$ for solutions of nonlinear stochastic partial differential equation of fractional order in both space and time variables with deterministic and bounded initial conditions: ${\partial }_t^{\beta }u(t,x)+{(-\Delta )}^{\alpha /2}u(t,x)=I_t^{\gamma }\left(\sigma \left(u(t,x)\right)\dot{W}(t,x)\right)\text{in}(0,\infty )\times R^d,$where $\alpha >0$, $\beta \in (0,2]$, $\gamma \ge 0$, ${\partial }_t^{\beta }$ is the Caputo fractional derivative, $-{(-\Delta )}^{\alpha /2}$ is the fractional/power of Laplacian, $I_t^{\gamma }$ is the Riemann–Liouville integral operator, $\dot{W}(t,x)$ is a space–time white noise, and $\sigma :R\to R$ is a bounded and Lipschitz function. Since the space variable is defined on the unbounded domain $R^d$, the inequalities are proved under a weighted $L^2$-norm in the spatial domain.