The stochastic heat equation with fractional time and fractional time–space white noise

Abstract

We give an introduction to the time-fractional stochastic heat equation driven by 1+d-parameter fractional time–space white noise, in the following two cases:(i) With additive fractional time–space white noise(ii) With multiplicative time–space Brownian noiseThe fractional time derivative is interpreted as the Caputo derivative of order α∈(0,2) and we assume that the Hurst coefficient H=(H0,H1,H2,…,Hd) of the time–space fractional white noise is in (12,1)1+d.We find an explicit expression for the unique solution in the sense of distribution of the equation in the additive noise case (i).In the multiplicative case (ii) we show that there is a unique solution in the Hida space (S)∗ of stochastic distributions and we show that the solution coincides with the solution of an associated fractional stochastic Volterra equation. Then we give an explicit expression for the solution of this Volterra equation. See Appendix below or Di Nunno et al. (2008), Holden et al. (2010) and Øksendal (2023) for more information about Hida spaces and white noise theory.A solution Y(t,x) is called mild if E[Y2(t,x)]<∞ for all t,x. For both the additive noise case and the multiplicative noise case we show that if α≥1 then the solution is mild if d=1 or d=2, while if α<1 the solution is not mild for any d.This paper is partly a survey paper, explaining the concepts and methods behind the results. It is also partly a research paper, in the sense that some results are new, to the best of our knowledge.