具耗散阻尼的随机波动方程的多项式混合
Polynomial mixing of a stochastic wave equation with dissipative damping
AI总结：
本文研究受加性高斯噪声扰动的半线性波动方程，证明在非线性耗散阻尼作用下其混合速率至少为任意阶多项式，方法结合Lyapunov条件、转移半群压缩性与d-小集。
摘要：
我们研究一类半线性波动方程的长时间统计行为，该方程刻画悬浮于连续介质中的粒子在受到加性高斯噪声随机扰动时的运动。与解已知具有几何遍历性的非线性反应情形相比，我们发现在非线性耗散阻尼的影响下，混合速率至少为任意阶多项式。这一结论依赖于Lyapunov条件、马尔可夫转移半群的压缩性质以及$d$--小集概念的组合运用。
英文摘要：
We study the long time statistics of a class of semi--linear wave equations modeling the motions of a particle suspended in continuous media while being subjected to random perturbations via an additive Gaussian noise. By comparison with the nonlinear reaction settings, of which the solutions are known to possess geometric ergodicity, we find that, under the impact of nonlinear dissipative damping, the mixing rate is at least polynomial of any order. This relies on a combination of Lyapunov conditions, the contracting property of the Markov transition semigroup as well as the notion of $d$--small sets.
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