Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient

Wensheng Wang

doi:10.3390/sym13030380

Abstract

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. On the other hand, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs and their gradient.

Highlights

The fourth order time fractional stochastic partial integro-differential equations (SPIDEs) are related to Brownian-time processes (BTPs); they form a unifying class for some different exciting processes like the iterated
SPIDE is built on the BTP [20,31,35] and extensions thereof
For the time fractional SPIDE, as in [39], we use the density of an inverse stable Lévy time Brownian motion to define their rigorous mild stochastic integral equation (SIE) formulation

Summary

Introduction

Many authors have applied fractional and higher order evolution equations as (stochastic) models in mathematical physics, fluids dynamics, turbulence and mathematical finance [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. The existence/uniqueness as well as sharp dimension-dependent L p and Hölder regularity of the linear and nonlinear noise version of Equation (1) were investigated in [30,36,37,38] These results naturally lead to the following list of motivating questions: α,γ. It was studied in [39] that the exact uniform and local moduli of continuity for the time fractional SPIDE in the time variable t and space variable x, separately. It was established in [39] that the exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for the fourth order time fractional SPIDEs and their gradient These results give the answers to spatially continuity and exact moduli of continuity of the solutions to Equation (1), and give partial answers to above questions.

Rigorous Kernel Stochastic Integral Equations Formulations

Spatial Spectral Density for Time Fractional SPIDEs and Their Gradient

Spatial Zero-One Laws for Time Fractional SPIDEs and Their Gradient

Extremes for Time Fractional SPIDEs and Their Gradient

Conclusions