Nonlinear Stochastic Heat Equation Driven by Spatially Colored Noise: Moments and Intermittency
Acta Mathematica Scientia
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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  from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al  using different techniques.
Stochastic heat equation; Moment estimates; Phase transition; Intermittency; Intermittency front; Measure-valued initial data; Moment Lyapunov exponents
Nonlinear Stochastic Heat Equation Driven by Spatially Colored Noise: Moments and Intermittency.
Acta Mathematica Scientia, 39(3),