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Register a new userSpatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition
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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}.
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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 $betain(1,,dwedge 2)$.
Let ${u(t,,x)}_{tge 0, xin mathbb{R}^d}$ denote the solution of a $d$-dimensional nonlinear stochastic heat equation that is driven by a Gaussian noise, white in time with a homogeneous spatial covariance that is a finite Borel measure $f$ and satisfies Dalangs condition. We prove two general functional central limit theorems for occupation fields of the form $N^{-d} int_{mathbb{R}^d} g(u(t,,x)) psi(x/N), mathrm{d} x$ as $Nrightarrow infty$, where $g$ runs over the class of Lipschitz functions on $mathbb{R}^d$ and $psiin L^2(mathbb{R}^d)$. The proof uses Poincare-type inequalities, Malliavin calculus, compactness arguments, and Paul Levys classical characterization of Brownian motion as the only mean zero, continuous Levy process. Our result generalizes central limit theorems of Huang et al cite{HuangNualartViitasaari2018,HuangNualartViitasaariZheng2019} valid when $g(u)=u$ and $psi = mathbf{1}_{[0,1]^d}$.
An urn contains balls of d colors. At each time, a ball is drawn and then replaced together with a random number of balls of the same color. Assuming that some colors are dominated by others, we prove central limit theorems. Some statistical applications are discussed.
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.
The approximation of integral type functionals is studied for discrete observations of a continuous It^o semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved for $L^2$-Sobolev functions with fractional smoothness. An explicit $L^2$-lower bound shows that already lower order quadrature rules, such as the trapezoidal rule and the classical Riemann estimator, are rate optimal, but only the trapezoidal rule is efficient, achieving the minimal asymptotic variance.
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