Intermittency properties for a large class of stochastic PDEs driven by fractional space-time noises

TL;DR

This paper investigates the intermittency properties of a broad class of stochastic PDEs driven by fractional Gaussian noises, introducing a novel Feynman diagram approach to establish matching upper and lower moment bounds.

Contribution

It develops a new Feynman diagram formula for moments and applies it to various stochastic PDEs, including those with fractional operators, filling a gap in lower bound estimates.

Findings

Established matching upper and lower moment bounds for multiple stochastic PDEs.

Extended intermittency analysis to equations with fractional Laplacians and fractional in time and space.

Provided a framework applicable to stochastic heat, wave, and diffusion equations with Gaussian noises.

Abstract

In this paper, we study intermittency properties for various stochastic PDEs with varieties of space time Gaussian noises via matching upper and lower moment bounds of the solution. Due to the absence of the powerful Feynman Kac formula, the lower moment bounds have been missing for many interesting equations except for the stochastic heat equation. This work introduces and explores the Feynman diagram formula for the moments of the solution and the small ball nondegeneracy for the Green's function to obtain the lower bounds for all moments which match the upper moment bounds. Our upper and lower moments are valid for various interesting equations, including stochastic heat equations, stochastic wave equations, stochastic heat equations with fractional Laplcians, and stochastic diffusions which are both fractional in time and in space.