无对称Dynkin同构

Dynkin同构定理与随机heat equation

Dynkin's isomorphism theorem and the stochastic heat equation

考虑随机热方程$\partial_t u = \sL u + \dot{W}$，其中$\sL$是对偶中[Borel右]马尔可夫过程的生成元。我们表明，该解决方案是局部相互绝对连续的光滑扰动的高斯过程相关联的，通过Dynkin的同构定理，到本地时间的复制对称过程，对应于$\sL$ .在$\sL$的情况下，是一个发电机的Lévy过程$\R^d$，我们的结果给出了概率的解释，最近的研究结果Foondun等人。

On $H^{2|2}$ Isomorphism theorems and reinforced loop soup

We show that supersymmetric (susy) hyperbolic isomorphism theorems that relate Vertex Reinforced Jump Processes and $H^{2|2}$ field, introduced in [2] and [3], are annealed version of isomorphism theorems relating Markov processes and Gaussian free field, with the help of a Bayes formula that relates susy hyperbolic field to susy free field. On the other hand, we also prove a BFS-Dynkin's isomorphism theorem for reinforced loop soup. Moreover, we provide yet another proof of BFS-Dynkin's isomorphism for VRJP a la Feynman-Kac.

Dynkin isomorphism and Mermin--Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process

We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma models whose target space is a hyperbolic space $\mathbb{H}^n$ or its supersymmetric counterpart $\mathbb{H}^{2|2}$. These results are analogues of well-known relations between the Gaussian free field and the local times of simple random walk. The second ingredient is a Mermin--Wagner theorem for these sigma models. This result is of intrinsic interest for the s…

Isomorphisms of $β$-Dyson's Brownian motion with Brownian local time

We show that the Brydges-Fröhlich-Spencer-Dynkin and the Le Jan's isomorphisms between the Gaussian free fields and the occupation times of symmetric Markov processes generalize to the $β$-Dyson's Brownian motion. For $β\in\{1,2,4\}$ this is a consequence of the Gaussian case, however the relation holds for general $β$. We further raise the question whether there is an analogue of $β$-Dyson's Brownian motion on general electrical networks, interpolating and extrapolating the fields of eigenvalues in matrix-valued Gaussian free fields. In the case $n=2$ we give a simple construction.

同构定理讲义

Lectures on Isomorphism Theorems

这些笔记来自我2013年5月在马赛的一系列讲座。我被邀请介绍同构定理，起源于Dynkin，连接马尔可夫当地时间和高斯过程。这是一个领域，我曾在一段时间前，甚至写了一本书，但后来转移到其他事情。然而，同构定理已成为兴趣再次，既因为新的应用程序的研究覆盖时间的图和高斯领域，并因为新的同构定理的非对称马尔可夫过程及其连接与循环汤和泊松过程。因此，我觉得时机已经成熟，可以对这个话题进行新的介绍。在这些笔记中，我试图保持讲座的非正式气氛，并经常简单地让读者参考我的书和其他来源的细节。