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Suppose that ${u(t,, x)}_{t >0, x inmathbb{R}^d}$ is the solution to a $d$-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalangs condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form $N^{-d} int_{[0,N]^d} g(u(t,,x)), mathrm{d} x$, as $Nrightarrowinfty$, where $g$ is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Steins method for normal approximations. Our results include a central limit theorem for the {it Hopf-Cole} solution to KPZ equation. We also establish a functional central limit theorem for these spatial averages.
We obtain explicit error bounds for the $d$-dimensional normal approximation on hyperrectangles for a random vector that has a Stein kernel, or admits an exchangeable pair coupling, or is a non-linear statistic of independent random variables or a su
We study limit theorems for time-dependent averages of the form $X_t:=frac{1}{2L(t)}int_{-L(t)}^{L(t)} u(t, x) , dx$, as $tto infty$, where $L(t)=exp(lambda t)$ and $u(t, x)$ is the solution to a stochastic heat equation on $mathbb{R}_+times mathbb{R
We combine Steins method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by P
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 satisf
We define a multi-group version of the mean-field or Curie-Weiss spin model. For this model, we show how, analogously to the classical (single-group) model, the three temperature regimes are defined. Then we use the method of moments to determine for