ABSTRACT

In this paper we establish the existence of an invariant measure for a stochastic reaction–diffusion equation of the type $du=(\mathrm{\Delta }u+f(x,u))\mathrm{d}t+\sigma (x,u)\mathrm{d}W(x,t)$, where f and σ are nonlinear maps and W is an infinite dimensional Q-–Wiener process. Our emphasis is on unbounded domain ${\mathbb{R}}^d$. Under a very mild dissipation assumption, we show the existence of a solution which is bounded in probability using a general Ito's formula. In addition, we investigate a type of equation for which bounded solutions may be obtained as a limit of an iteration scheme. Together with the compactness property of the heat equation semigroup, stochastic continuity and Feller property of the transition semigroup, the existence of such solution implies the existence of an invariant measure which is an important step in establishing the ergodic behaviour of the underlying physical system.

摘要

在本文中，我们建立了类型为随机反应-扩散方程的不变测度的存在 $dü=（\mathrm{\Delta }ü+F（X，ü））\mathrm{d}Ť+\sigma （X，ü）\mathrm{d}\mathrm{w\; ^}（X，Ť）$，其中f和σ是非线性映射，W是无限维Q-Wiener过程。我们的重点是无限领域${\mathbb{[R}}^d$。在一个非常温和的耗散假设下，我们使用通用的Ito公式证明存在一个概率受限的解。此外，我们研究了一类方程，可以为其求出有界解作为迭代方案的极限。连同热方程半群的紧致性质，过渡半群的随机连续性和Feller性质一起，这种解决方案的存在意味着存在不变量度，这是建立基础物理系统的遍历行为的重要步骤。