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

We study the stochastic time-fractional stochastic heat equation 0.1 ∂ ^α/∂ t^αY(t,x)=λ Y(t,x)+σ W(t,x); (t,x)∈ (0,∞ )×ℝ^d, where d∈ℕ={1,2,...} and ∂ ^α/∂ t^α is the Caputo derivative of order α∈ (0,2) , and λ >0 and σ∈ℝ are given constants. Here denotes the Laplacian operator, W ( t , x ) is time-space white noise, defined by 0.2 W(t,x)=∂/∂ t∂ ^dB(t,x)/∂ x_1...∂ x_d, B(t,x)=B(t,x,ω ); t≥ 0, x ∈ℝ^d, ω∈ being time-space Brownian motion with probability law ℙ . We consider the equation ( 0.1 ) in the sense of distribution, and we find an explicit expression for the 𝒮' -valued solution Y ( t , x ), where 𝒮' is the space of tempered distributions. Following the terminology of Y. Hu [ 11 ], we say that the solution is mild if Y(t,x) ∈ L^2(ℙ) for all t , x . It is well-known that in the classical case with α = 1 , the solution is mild if and only if the space dimension d=1 . We prove that if α∈ (1,2) the solution is mild if d=1 or d=2 . If α < 1 we prove that the solution is not mild for any d .