Markov limits of steady states of the KPZ equation on an interval
Wlodek Bryc, Alexey Kuznetsov
备注：
This is an expanded version of the paper with some additional details and post-publication corrections: corrected a typo in the published version of Theorems 2.6 and 2.8; corrected the proof and the statement of formula (1.14)
发表信息：
ALEA, Lat. Am. J. Probab. Math. Stat. 19, 1329-1351 (2022)
本文基于 Corwin 和 Knizel 的研究，他们证明了区间上 KPZ 方程平稳测度的存在性，并通过 Laplace 变换公式对其进行了刻画。Bryc、Kuznetsov、Wang 和 Wesolowski 利用某些 Markov 核的 Doob 变换，找到了平稳测度的概率描述；几乎同时，Barraquand 和 Le Doussal 的工作中出现了另一种将平稳测度与 Brownian 运动的指数泛函联系起来的描述。我们的第一个主要结果阐明并证明了这些平稳测度的两种概率描述的等价性。然后，我们利用 Markov 描述对 Barraquand 和 Le Doussal 中声称的一些结果给出了严格证明。我们分析了有限区间上 KPZ 方程的平稳测度在大尺度下的行为。我们研究了 G. Barraquand 和 P. Le Doussal 最近得到的 KPZ 方程稳态的极限中，哪些可以在对参数范围施加额外限制的条件下，由空间变量中的 Markov 过程来表示。
英文摘要：
This paper builds upon the research of Corwin and Knizel who proved the existence of stationary measures for the KPZ equation on an interval and characterized them through a Laplace transform formula. Bryc, Kuznetsov, Wang and Wesolowski found a probabilistic description of the stationary measures in terms of a Doob transform of some Markov kernels; essentially at the same time, another description connecting the stationary measures to the exponential functionals of the Brownian motion appeared in work of Barraquand and Le Doussal.
Our first main result clarifies and proves the equivalence of the two probabilistic description of these stationary measures. We then use the Markovian description to give rigorous proofs of some of the results claimed in Barraquand and Le Doussal. We analyze how the stationary measures of the KPZ equation on finite interval behave at large scale. We investigate which of the limits of the steady states of the KPZ equation obtained recently by G. Barraquand and P. Le Doussal can be represented by Markov processes in spatial variable under an additional restriction on the range of parameters.
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