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Annals of Mathematics, II. Series Vol. 151, No. 3, pp. 877960 (2000) 

Invariant measures for Burgers equation with stochastic forcingWeinan E, K. Khanin, A. Mazel and Ya. SinaiReview from Zentralblatt MATH: The authors study the following Burgers equation $${\partial u\over\partial t}+{\partial\over\partial x} \Biggl({u^2 \over 2}\Biggr)= \varepsilon {\partial^2u\over\partial x^2}+ f(x,t),$$ where $f(x,t)= {\partial F\over\partial x} (x, t)$ is a random forcing function, which is periodic in $x$ with period $1$, and with white noise in $t$. The general form for the potentials of such forces is given by $$F(x,t)= \sum^\infty_{k=1} F_k(x)\dot B_k(t),$$ where the $\{B_k(t), t\in(\infty,\infty)\}$'s are independent standard Wiener processes defined on a probability space $(\Omega,{\cal F},{\cal P})$ and the $F_k$'s are periodic with period $1$. The authors assume for some $r\ge 3$ $$f_k(x)= F_k'(x)\in \bbfC^r(S^1),\quad \f_k\_{\bbfC^r}\le {C\over k^2},$$ where $S^1$ denotes the unit circle, and $C$ a generic constant. Without loss of generality, the authors assume that for all $k$: $\int^1_0 F_k(x) dx= 0$. They denote the elements in the proabability space $\Omega$ by $\omega= (\dot B_1(\cdot),\dot B_2(\cdot),\dots)$. Except in Section 8, where they study the convergence as $\varepsilon\to 0$, the authors restrict their attention to the case when $\varepsilon= 0$: $${\partial u\over\partial t}+{\partial\over\partial x} \Biggl({u^2\over 2}\Biggr)= {\partial F\over\partial x} (x,t).\tag 1$$ Besides establishing existence and uniqueness of an invariant measure for the Markov process corresponding to (1) the authors give a detailed description of the structure and regularity properties for the solutions that live on the support of this measure. Reviewed by Stanislaw Wedrychowicz Keywords: Burgers equation; random forcing function; Wiener processes; probability space; existence; uniqueness; invariant measure; Markov process Classification (MSC2000): 35R60 35Q53 37A50 35B10 60J25 Full text of the article:
Electronic fulltext finalized on: 8 Sep 2001. This page was last modified: 22 Jan 2002.
© 2001 Johns Hopkins University Press
