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Register a new userLimit theorems for time-dependent averages of nonlinear stochastic heat equations
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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 $mathbb{R}_+times mathbb{R}$ driven by space-time white noise with $u_0(x)=1$ for all $xin mathbb{R}$. We show that for $X_t$ (i) the weak law of large numbers holds when $lambda>lambda_1$, (ii) the strong law of large numbers holds when $lambda>lambda_2$, (iii) the central limit theorem holds when $lambda>lambda_3$, but fails when $lambda <lambda_4leq lambda_3$, (iv) the quantitative central limit theorem holds when $lambda>lambda_5$, where $lambda_i$s are positive constants depending on the moment Lyapunov exponents of $u(t, x)$.
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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.
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 $psi = mathbf{1}_{[0,1]^d}$.
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.
Due to their intrinsic link with nonlinear Fokker-Planck equations and many other applications, distribution dependent stochastic differential equations (DDSDEs for short) have been intensively investigated. In this paper we summarize some recent progresses in the study of DDSDEs, which include the correspondence of weak solutions and nonlinear Fokker-Planck equations, the well-posedness, regularity estimates, exponential ergodicity, long time large deviations, and comparison theorems.
We study the weak limits of solutions to SDEs [dX_n(t)=a_nbigl(X_n(t)bigr),dt+dW(t),] where the sequence ${a_n}$ converges in some sense to $(c_- 1mkern-4.5mumathrm{l}_{x<0}+c_+ 1mkern-4.5mumathrm{l}_{x>0})/x+gammadelta_0$. Here $delta_0$ is the Dirac delta function concentrated at zero. A limit of ${X_n}$ may be a Bessel process, a skew Bessel process, or a mixture of Bessel processes.
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