Title

Nonlinear Stochastic Heat Equation Driven by Spatially Colored Noise: Moments and Intermittency

Document Type

Article

Publication Date

5-30-2019

Publication Title

Acta Mathematica Scientia

Volume

39

Issue

3

First page number:

645

Last page number:

668

Abstract

In this article, we study the nonlinear stochastic heat equation in the spatial domain ℝdsubject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies Dalang’s condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. With these moment bounds, we first establish some necessary and sufficient conditions for the phase transition of the moment Lyapunov exponents, which extends the classical results from the stochastic heat equation on ℤd to that on ℝd. Then, we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan [9] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al [17] using different techniques.

Keywords

Stochastic heat equation; Moment estimates; Phase transition; Intermittency; Intermittency front; Measure-valued initial data; Moment Lyapunov exponents

Disciplines

Applied Mathematics

Language

English

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