Penalising symmetric stable Lévy paths

Excursions away from a regular point for one-dimensional symmetric Levy processes without Gaussian part

The characteristic measure of excursions away from a regular point is studied for a class of symmetric Lévy processes without Gaussian part. It is proved that the harmonic transform of the killed process enjoys Feller property. The result is applied to prove extremeness of the excursion measure and to prove several sample path behaviors of the excursion and the $ h $-path processes.

Explicit LDP for a slowed RW driven by a symmetric exclusion process

We consider a random walk (RW) driven by a simple symmetric exclusion process (SSE). Rescaling the RW and the SSE in such a way that a joint hydrodynamic limit theorem holds we prove a joint path large deviation principle. The corresponding large deviation rate function can be split into two components, the rate function of the SSE and the one of the RW given the path of the SSE. Such components have different structures (Gaussian and Poissoinian, respectively) and to overcome this difficulty we make use of the theory of Orlicz spaces. In particular, the component of the rate function correspo…

Laws of the iterated logarithm for symmetric jump processes

Based on two-sided heat kernel estimates for a class of symmetric jump processes on metric measure spaces, the laws of the iterated logarithm (LILs) for sample paths, local times and ranges are established. In particular, the LILs are obtained for $β$-stable-like processes on $α$-sets with $β>0$.

由布朗运动和对称稳定过程驱动的一维随机微分方程的逐路稳定性

Pathwise stability for one-dimensional SDEs driven by Brownian motion and a symmetric stable process

该研究针对由布朗运动和对称α-稳定过程驱动的一维SDE，通过同步耦合建立了定量逐路稳定性估计，明确了不同Hölder指数下的误差率，还得到了含可预测强迫误差和时间一致尾界的估计。

Intersection conductance and canonical alternating paths, methods for general finite Markov chains

We extend the conductance and canonical paths methods to the setting of general finite Markov chains, including non-reversible non-lazy walks. The new path method is used to show that a known bound for mixing time of a lazy walk on a Cayley graph with symmetric generating set also applies to the non-lazy non-symmetric case, often even when there is no holding probability.

简单对称排斥过程的路径空间高阶高斯近似

Higher-Order Gaussian Approximations on Path Space for the Simple Symmetric Exclusion Process

该研究针对简单对称排斥过程的非平衡密度涨落，构造了含高阶格点信息的修正Ornstein-Uhlenbeck过程，推导了路径空间高阶高斯近似的误差界，改进了中心极限速率。

具有 delta 势的对称 Levy 过程生成元的极限定理

Limit theorems for the generator of a symmetric Levy process with the delta potential

本文研究具有局部时的一维对称 Levy 过程，构造带 delta 势的生成元自伴扩张并推广 Feynman-Kac 公式，同时建立路径吸引分布族及其弱收敛极限定理。

The Construction of Two Kinds of Bijections in Simple Random Walk Paths

It is known that for the 2n-step symmetric simple random walk on Z, two events have the same probability if and only if their sets of paths have the same cardinality. In this article, we construct two kinds of bijections between sets of paths with the same cardinality. The construction is natural and simple. It can be easily realized through programming. More importantly, this construction opens a door to prove that two events in the 2n-step symmetric simple random walk on Z have the same probability and some further related results.

Hitting distributions of alpha-stable processes via path censoring and self-similarity

In this paper we return to the problem of Blumenthal-Getoor-Ray, published in 1961, which gave the law of the position of first entry of a symmetric alpha-stable process into the unit ball. Specifically, we are interested in establishing the same law, but now for a one dimensional alpha-stable process which enjoys two-sided jumps, and which is not necessarily symmetric. Our method is modern in the sense that we appeal to the relationship between alpha-stable processes and certain positive self-similar Markov processes. However there are two notable additional innovations. First, we make use of…

On Number of Turns in Reduced Random Lattice Paths

We consider the tree-reduced path of symmetric random walk on $\ZZ^{d}$. It is interesting to ask about the number of turns $T_n$ in the reduced path after $n$ steps. This question arises from inverting signature for lattice paths. We show that, when $n$ is large, the mean and variance of $T_n$ have the same order as $n$, while the second order terms are O(1). We then use these estimates to obtain limit theorems for $T_n$. Similar results hold for any other finite patterns as well.

A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential

Additive white noise may significantly increase the response of bistable systems to a periodic driving signal. We consider two classes of double-well potentials, symmetric and asymmetric, modulated periodically in time with period $1/\eps$, where $\eps$ is a moderately (not exponentially) small parameter. We show that the response of the system changes drastically when the noise intensity $σ$ crosses a threshold value. Below the threshold, paths are concentrated near one potential well, and have an exponentially small probability to jump to the other well. Above the threshold, transitions betw…