We investigate the asymptotic behaviors of the solution $u(t,\cdot)$ to a stochastic heat equation with a periodic, gradient-type nonlinear term. We extend the central limit theorem for finite-dimensional diffusions presented in \cite[\S9.1]{KLO12} to infinite-dimensional settings. Due to our results, as $t\rightarrow\infty$, $\frac{1}{\sqrt{t}}u(t,\cdot)$ converges weakly to a centered Gaussian variable whose covariance operator is described through Poisson's equations. Different from the finite-dimensional case, the fluctuation in space vanishes in the limit distribution. Furthermore, we verify the tightness and present an invariance principle for $\{\epsilon u(\epsilon^{-2}t,\cdot)\}_{t \in[0,T]}$ as $\epsilon \downarrow 0$.

An invariance principle for stochastic heat equations with periodic coefficients

Xu L.
2018-01-01

Abstract

We investigate the asymptotic behaviors of the solution $u(t,\cdot)$ to a stochastic heat equation with a periodic, gradient-type nonlinear term. We extend the central limit theorem for finite-dimensional diffusions presented in \cite[\S9.1]{KLO12} to infinite-dimensional settings. Due to our results, as $t\rightarrow\infty$, $\frac{1}{\sqrt{t}}u(t,\cdot)$ converges weakly to a centered Gaussian variable whose covariance operator is described through Poisson's equations. Different from the finite-dimensional case, the fluctuation in space vanishes in the limit distribution. Furthermore, we verify the tightness and present an invariance principle for $\{\epsilon u(\epsilon^{-2}t,\cdot)\}_{t \in[0,T]}$ as $\epsilon \downarrow 0$.
2018
stochastic heat equations, central limit theorem, invariance principle
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/31104
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