论文标题
随机微分方程的固定措施和退化阻尼
Stationary measures for stochastic differential equations with degenerate damping
论文作者
论文摘要
各种物理现象涉及能量从弱阻尼模式的非线性转移，该模式受到外部强迫的影响，这些模式更严重。在这项工作中，我们在$ \ mathbb r^n $中使用（有限维）随机微分方程探索了这一点，并带有二次，保守的非线性$ b（x，x）$和线性抑制术语$ -ax $，它以$ \ nathrm {ker} a \ nebestysys $ neqebtysys $ ne \ neemet $。我们研究了足够的条件，以推断出相关的马尔可夫半群的固定度量的存在。如果$ a $是完整的排名，但这种措施的存在是简单的，但是否则，能量可能会积累在$ \ mathrm {ker} a $中，并导致几乎没有界限的轨迹，从而使存在固定措施的存在是不可能的。我们基于$ \ Mathrm {ker}社区的轨迹的轨迹估计的相对简单和一般的足够条件，并且可以进行许多估算的示例。
A variety of physical phenomena involve the nonlinear transfer of energy from weakly damped modes subjected to external forcing to other modes which are more heavily damped. In this work we explore this in (finite-dimensional) stochastic differential equations in $\mathbb R^n$ with a quadratic, conservative nonlinearity $B(x,x)$ and a linear damping term $-Ax$ which is degenerate in the sense that $\mathrm{ker} A \neq \emptyset$. We investigate sufficient conditions to deduce the existence of a stationary measure for the associated Markov semigroups. Existence of such measures is straightforward if $A$ is full rank, but otherwise, energy could potentially accumulate in $\mathrm{ker} A$ and lead to almost-surely unbounded trajectories, making the existence of stationary measures impossible. We give a relatively simple and general sufficient condition based on time-averaged coercivity estimates along trajectories in neighborhoods of $\mathrm{ker} A$ and many examples where such estimates can be made.
