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Register a new userSpatial stationarity, ergodicity and CLT for parabolic Anderson model with delta initial condition in dimension $dgeq 1$
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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 $betain(1,,dwedge 2)$.
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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 $bm{p}_t(x):= (2pi t)^{-1/2}exp{-x^2/(2t)}$ denote the standard Gaussian heat kernel on the line. We use a non-trivial adaptation of the methods in our companion papers cite{CKNP,CKNP_b} in order to prove that the random field $xmapsto u(t,,x)/bm{p}_t(x)$ is ergodic for every $t >0$. And we establish an associated quantitative central limit theorem following the approach based on the Malliavin-Stein method introduced in Huang, Nualart, and Viitasaari cite{HNV2018}.
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 the covariance structure of a fractional Brownian motion with Hurst parameter greater than 1/4 and less than 1/2 in the space variable.
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 the parabolic Anderson equation [cases{displaystyle {frac{partial u}{partial t}}(t,x)={frac{1}{2}}Delta u(t,x)+V(t,x)u(t,x),cr u(0,x)=u_0(x),}] where the homogeneous generalized Gaussian noise $V(t,x)$ is, among other forms, white or fractional white in time and space. Associated with the Cole-Hopf solution to the KPZ equation, in particular, the precise asymptotic form [lim_{Rtoinfty}(log R)^{-2/3}logmax_{|x|le R}u(t,x)={frac{3}{4}}root 3of {frac{2t}{3}}qquad a.s.] is obtained for the parabolic Anderson model $partial_tu={frac{1}{2}}partial_{xx}^2u+dot{W}u$ with the $(1+1)$-white noise $dot{W}(t,x)$. In addition, some links between time and space asymptotics for the parabolic Anderson equation are also pursued.
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 $xi=(xi(z))_{zinmathbb Z^d}$ is an i.i.d. random field with doubly-exponential upper tails. We prove that, for large $t$ and with large probability, a majority of the total mass $U(t):=sum_x u(x,t)$ of the solution resides in a bounded neighborhood of a site $Z_t$ that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian $Delta+xi$ and the distance to the origin. The processes $tmapsto Z_t$ and $t mapsto tfrac1t log U(t)$ are shown to converge in distribution under suitable scaling of space and time. Aging results for $Z_t$, as well as for the solution to the parabolic problem, are also established. The proof uses the characterization of eigenvalue order statistics for $Delta+xi$ in large sets recently proved by the first two authors.
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-exponential. Under the assumption that the degree distribution has bounded support, two terms in the asymptotic expansion were identified under the quenched law, i.e., conditional on the realisation of the random tree and the random potential. The second term contains a variational formula indicating that the solution concentrates on a subtree with minimal degree according to a computable profile. The present paper extends the analysis to degree distributions with unbounded support. We identify the weakest condition on the tail of the degree distribution under which the arguments in [1] can be pushed through. To do so we need to control the occurrence of large degrees uniformly in large subtrees of the Galton-Watson tree.
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