The fractional stochastic heat equation driven by time-space white noise

Abstract

AbstractWe study the stochastic time-fractional stochastic heat equation$$\begin{aligned} \frac{\partial ^{\alpha }}{\partial t^{\alpha }}Y(t,x)=\lambda \varDelta Y(t,x)+\sigma W(t,x);\; (t,x)\in (0,\infty )\times \mathbb {R}^{d}, \end{aligned}$$∂α∂tαY(t,x)=λΔY(t,x)+σW(t,x);(t,x)∈(0,∞)×Rd,where$$d\in \mathbb {N}=\{1,2,...\}$$d∈N={1,2,...}and$$\frac{\partial ^{\alpha }}{\partial t^{\alpha }}$$∂α∂tαis the Caputo derivative of order$$\alpha \in (0,2)$$α∈(0,2), and$$\lambda >0$$λ>0and$$\sigma \in \mathbb {R}$$σ∈Rare given constants. Here$$\varDelta $$Δdenotes the Laplacian operator,W(t, x) is time-space white noise, defined by$$\begin{aligned} W(t,x)=\frac{\partial }{\partial t}\frac{\partial ^{d}B(t,x)}{\partial x_{1}...\partial x_{d}}, \end{aligned}$$W(t,x)=∂∂t∂dB(t,x)∂x1...∂xd,$$B(t,x)=B(t,x,\omega ); t\ge 0, x \in \mathbb {R}^d, \omega \in \varOmega $$B(t,x)=B(t,x,ω);t≥0,x∈Rd,ω∈Ωbeing time-space Brownian motion with probability law$$\mathbb {P}$$P. We consider the equation (0.1) in the sense of distribution, and we find an explicit expression for the$$\mathcal {S}'$$S′-valued solutionY(t, x), where$$\mathcal {S}'$$S′is the space of tempered distributions. Following the terminology of Y. Hu [11], we say that the solution ismildif$$Y(t,x) \in L^2(\mathbb {P})$$Y(t,x)∈L2(P)for allt, x. It is well-known that in the classical case with$$\alpha = 1$$α=1, the solution is mild if and only if the space dimension$$d=1$$d=1. We prove that if$$\alpha \in (1,2)$$α∈(1,2)the solution is mild if$$d=1$$d=1or$$d=2$$d=2. If$$\alpha < 1$$α<1we prove that the solution is not mild for anyd.