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

Abstract

We study the stochastic time-fractional stochastic heat equation 0.1∂α∂tαY(t,x)=λΔY(t,x)+σW(t,x);(t,x)∈(0,∞)×Rd,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\\partial ^{\\alpha }}{\\partial t^{\\alpha }}Y(t,x)=\\lambda \\varDelta Y(t,x)+\\sigma W(t,x);\\; (t,x)\\in (0,\\infty )\ imes \\mathbb {R}^{d}, \\end{aligned}$$\\end{document}where din mathbb {N}={1,2,...} and frac{partial ^{alpha }}{partial t^{alpha }} is the Caputo derivative of order alpha in (0,2), and lambda >0 and sigma in mathbb {R} are given constants. Here varDelta denotes the Laplacian operator, W(t, x) is time-space white noise, defined by 0.2W(t,x)=∂∂t∂dB(t,x)∂x1...∂xd,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} W(t,x)=\\frac{\\partial }{\\partial t}\\frac{\\partial ^{d}B(t,x)}{\\partial x_{1}...\\partial x_{d}}, \\end{aligned}$$\\end{document}B(t,x)=B(t,x,omega ); tge 0, x in mathbb {R}^d, omega in varOmega being time-space Brownian motion with probability law mathbb {P}. We consider the equation (0.1) in the sense of distribution, and we find an explicit expression for the mathcal {S}'-valued solution Y(t, x), where mathcal {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) in L^2(mathbb {P}) for all t, x. It is well-known that in the classical case with alpha = 1, the solution is mild if and only if the space dimension d=1. We prove that if alpha in (1,2) the solution is mild if d=1 or d=2. If alpha < 1 we prove that the solution is not mild for any d.