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 ( C ) for solutions of nonlinear stochastic partial differential equation of fractional order in both space and time variables with deterministic and bounded initial conditions: ∂ t β u ( t , x ) + ( − Δ ) α / 2 u ( t , x ) = I t γ σ ( u ( t , x ) ) W ̇ ( t , x ) in ( 0 , ∞ ) × R d , where α > 0 , β ∈ ( 0 , 2 ] , γ ≥ 0 , ∂ t β is the Caputo fractional derivative, − ( − Δ ) α / 2 is the fractional/power of Laplacian, I t γ is the Riemann–Liouville integral operator, W ̇ ( t , x ) is a space–time white noise, and σ : R → 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.