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Register a new userSpatial asymptotics for the parabolic Anderson models with generalized time-space Gaussian noise
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Xia Chen
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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.
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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.
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}.
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 $betain(1,,dwedge 2)$.
This paper suggests a new approach to error analysis in the filtering problem for continuous time linear system driven by fractional Brownian noises. We establish existence of the large time limit of the filtering error and determine its scaling exponent with respect to the vanishing observation noise intensity. Closed form expressions are obtained in a number of important special cases.
In this note we consider the parabolic Anderson model in one dimension with time-independent fractional noise $dot{W}$ in space. We consider the case $H<frac{1}{2}$ and get existence and uniqueness of solution. In order to find the quenched asymptotics for the solution we consider its Feynman-Kac representation and explore the asymptotics of the principal eigenvalue for a random operator of the form $frac{1}{2} Delta + dot{W}$.
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