Phase transitions arising in stochastic ergodic control associated with viscous Hamilton-Jacobi equations with bounded inward drift
Phase transitions arising in stochastic ergodic control associated with viscous Hamilton-Jacobi equations with bounded inward drift
复制标题
与具有有限向内漂移的粘性 Hamilton-Jacobi 方程相关的随机遍历控制中出现的相变
DOI：
10.1007/s42985-021-00072-0
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Ichihara Naoyuki
中科院分区：
文献类型：
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作者：
Chasseigne Emmanuel;Ichihara Naoyuki;Ichihara Naoyuki
本文研究一类含真实的参数的随机遍历控制问题中的相变现象。我们发现，最优扩散的大时间行为的变化急剧的一些临界值附近。具体地说，最优扩散是经常性的，而它是短暂的。我们还研究了最优扩散的大时间行为，结果与前两种情况不同，并且更微妙。我们的证明是基于李雅普诺夫方法给出的分析标准的复发性和瞬态的扩散。其关键在于分析相应的粘性Hamilton-Jacobi方程的有界内漂的解。特别是，粘性Hamilton-Jacobi方程的解的梯度估计的改进版本起着重要的作用。
This paper is concerned with certain phase transition phenomena arising in a family of stochastic ergodic control problems having real parameter. We show that the large time behavior of the optimal diffusion changes drastically in the vicinity of some critical value. Specifically, the optimal diffusion is recurrent for, while it is transient for. We also investigate the large time behavior of the optimal diffusion forwhich turns out to be different from the previous two cases and more subtle. Our proof is based on the Lyapunov method giving analytical criteria for recurrence and transience of diffusions. The key lies in the analysis of solutions to the associated viscous Hamilton–Jacobi equation with bounded inward drift. In particular, a refined version of the gradient estimate for solutions to viscous Hamilton–Jacobi equations plays a substantial role.