论文标题
非线性随机热方程的不变措施，没有漂移项
Invariant measures for the nonlinear stochastic heat equation with no drift term
论文作者
论文摘要
This paper deals with the long term behavior of the solution to the nonlinear stochastic heat equation $\partial u /\partial t - \frac{1}{2}Δu = b(u)\dot{W}$, where $b$ is assumed to be a globally Lipschitz continuous function and the noise $\dot{W}$ is a centered and spatially homogeneous Gaussian noise那是白色的。使用[9，10]中获得的力矩公式，我们确定了初始数据，相关度量和权重函数$ρ$的一组条件，这将共同保证在加权空间中存在不变的度量$ l^2_至（\ MATHBB {r}^d）$。特别是，我们的结果包括从Dirac Delta度量开始的抛物线Anderson模型（即，$ b（u）=λu$）的情况。
This paper deals with the long term behavior of the solution to the nonlinear stochastic heat equation $\partial u /\partial t - \frac{1}{2}Δu = b(u)\dot{W}$, where $b$ is assumed to be a globally Lipschitz continuous function and the noise $\dot{W}$ is a centered and spatially homogeneous Gaussian noise that is white in time. Using the moment formulas obtained in [9, 10], we identify a set of conditions on the initial data, the correlation measure and the weight function $ρ$, which will together guarantee the existence of an invariant measure in the weighted space $L^2_ρ(\mathbb{R}^d)$. In particular, our result includes the parabolic Anderson model (i.e., the case when $b(u) = λu$) starting from the Dirac delta measure.
