## Invariant measures for Burgers equation with stochastic forcing.(English)Zbl 0972.35196

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 $$(\Omega,{\mathcal F},{\mathcal P})$$ and the $$F_k$$’s are periodic with period $$1$$. The authors assume for some $$r\geq 3$$ $f_k(x)= F_k'(x)\in \mathbb{C}^r(S^1),\quad \|f_k\|_{\mathbb{C}^r}\leq {C\over k^2},$ 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 probability 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.

### MSC:

 35R60 PDEs with randomness, stochastic partial differential equations 35Q53 KdV equations (Korteweg-de Vries equations) 37A50 Dynamical systems and their relations with probability theory and stochastic processes 35B10 Periodic solutions to PDEs 60J25 Continuous-time Markov processes on general state spaces
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