ﻻ يوجد ملخص باللغة العربية
Suppose that ${u(t,, x)}_{t >0, x inmathbb{R}^d}$ is the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure $f$ which satisfies Dalangs condition. Let $boldsymbol{p}_t(x):=(2pi t)^{-d/2}exp{-|x|^2/(2t)}$ denote the standard Gaussian heat kernel on $mathbb{R}^d$. We prove that for all $t>0$, the process $U(t):={u(t,, x)/boldsymbol{p}_t(x): xin mathbb{R}^d}$ is stationary using Feynman-Kacs formula, and is ergodic under the additional condition $hat{f}{0}=0$, where $hat{f}$ is the Fourier transform of $f$. Moreover, using Malliavin-Stein method, we investigate various central limit theorems for $U(t)$ based on the quantitative analysis of $f$. In particular, when $f$ is given by Riesz kernel, i.e., $f(mathrm{d} x) = |x|^{-beta}mathrm{d} x$, we obtain a multiple phase transition for the CLT for $U(t)$ from $betain(0,,1)$ to $beta=1$ to $betain(1,,dwedge 2)$.
Let ${u(t,, x)}_{t >0, x inmathbb{R}}$ denote the solution to the parabolic Anderson model with initial condition $delta_0$ and driven by space-time white noise on $mathbb{R}_+timesmathbb{R}$, and let $bm{p}_t(x):= (2pi t)^{-1/2}exp{-x^2/(2t)}$ denot
The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has th
Partially motivated by the recent papers of Conus, Joseph and Khoshnevisan [Ann. Probab. 41 (2013) 2225-2260] and Conus et al. [Probab. Theory Related Fields 156 (2013) 483-533], this work is concerned with the precise spatial asymptotic behavior for
We study the solutions $u=u(x,t)$ to the Cauchy problem on $mathbb Z^dtimes(0,infty)$ for the parabolic equation $partial_t u=Delta u+xi u$ with initial data $u(x,0)=1_{{0}}(x)$. Here $Delta$ is the discrete Laplacian on $mathbb Z^d$ and $xi=(xi(z))_
In [1] a detailed analysis was given of the large-time asymptotics of the total mass of the solution to the parabolic Anderson model on a supercritical Galton-Watson random tree with an i.i.d. random potential whose marginal distribution is double-ex