30,673 research outputs found
On the intermittency front of stochastic heat equation driven by colored noises
We study the propagation of high peaks (intermittency front) of the solution
to a stochastic heat equation driven by multiplicative centered Gaussian noise
in . The noise is assumed to have a general homogeneous
covariance in both time and space, and the solution is interpreted in the
senses of the Wick product. We give some estimates for the upper and lower
bounds of the propagation speed, based on a moment formula of the solution.
When the space covariance is given by a Riesz kernel, we give more precise
bounds for the propagation speed
Large time asymptotics for the parabolic Anderson model driven by spatially correlated noise
In this paper we study the linear stochastic heat equation, also known as
parabolic Anderson model, in multidimension driven by a Gaussian noise which is
white in time and it has a correlated spatial covariance. Examples of such
covariance include the Riesz kernel in any dimension and the covariance of the
fractional Brownian motion with Hurst parameter in
dimension one. First we establish the existence of a unique mild solution and
we derive a Feynman-Kac formula for its moments using a family of independent
Brownian bridges and assuming a general integrability condition on the initial
data. In the second part of the paper we compute Lyapunov exponents, lower and
upper exponential growth indices in terms of a variational quantity. The last
part of the paper is devoted to study the phase transition property of the
Anderson model
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