Variations of the solution to a fourth order time-fractional stochastic partial integro-differential equation

Abstract

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study the realized power variations for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by the space-time white noise in one-to-three dimensional spaces, in time, have infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. We use the underlying explicit kernels and spectral/harmonic analysis, yielding temporal central limit theorems for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of variations of general Gaussian processes and stochastic heat equation driven by the space-time white noise. On the other hand, it builds on and complements Allouba’s earlier works on time fractional SPIDEs and their gradient.