Moments, Intermittency and Growth Indices for the Nonlinear Fractional Stochastic Heat Equation

TL;DR

This paper analyzes the nonlinear fractional stochastic heat equation driven by space-time white noise, establishing moment bounds, growth indices, and demonstrating how fractional operators alter the solution's behavior compared to classical models.

Contribution

It provides new uniform moment bounds, refines intermittency results, and introduces exponential growth indices to describe peak propagation in fractional stochastic heat equations.

Findings

Established existence and uniqueness of solutions.

Derived uniform bounds on all moments for the equation.

Showed that fractional operators significantly change solution behavior.

Abstract

We study the nonlinear fractional stochastic heat equation in the spatial domain $\mathbb{R}$ driven by space-time white noise. The initial condition is taken to be a measure on $\mathbb{R}$, such as the Dirac delta function, but this measure may also have non-compact support. Existence and uniqueness, as well as upper and lower bounds on all $p$-th moments $(p\ge 2)$, are obtained. These bounds are uniform in the spatial variable, which answers an open problem mentioned in Conus and Khoshnevisan [9]. We improve the weak intermittency statement by Foondun and Khoshnevisan [14], and we show that the growth indices (of linear type) introduced in [9] are infinite. We introduce the notion of "growth indices of exponential type" in order to characterize the manner in which high peaks propagate away from the origin, and we show that the presence of a fractional differential operator leads to significantly different behavior compared with the standard stochastic heat equation.