arXiv subjects

Xiaochuan Yang

Publications and source records attributed to Xiaochuan Yang.

27 records · Page 2Linked to original sources

arXiv2019

Restricted hypercontractivity on the Poisson space

We show that the Ornstein-Uhlenbeck semigroup associated with a general Poisson random measure is hypercontractive, whenever it is restricted to non-increasing mappings on configuration spaces. We deduce from this result some versions of Talagrand's $L^1$-$L^2$ inequality for increasing and concave mappings, and we build examples showing that such an estimate represents a strict improvement of the classical Poincaré inequality. We complement our finding with several results of independent interest, such as gradient estimates.

math.PR↗

arXiv2019

Non-integrable stable approximation by Stein's method

We develop Stein's method for $α$-stable approximation with $α\in(0,1]$, continuing the recent line of research by Xu \cite{lihu} and Chen, Nourdin and Xu \cite{C-N-X} in the case $α\in(1,2).$ The main results include an intrinsic upper bound for the error of the approximation in a variant of Wasserstein distance that involves the characterizing differential operators for stable distributions, and an application to the generalized central limit theorem. Due to the lack of first moment for the approximating sequence in the latter result, we appeal to an additional truncation procedure and investigate fine regularity properties of the solution to Stein's equation.

math.PR↗

arXiv2019

Multifractal properties of sample paths of ground state-transformed jump processes

We consider a class of Lévy-type processes with unbounded coefficients, arising as Doob $h$-transforms of Feynman-Kac type representations of non-local Schrödinger operators, where the function $h$ is chosen to be the ground state of such an operator. First, we show the existence of a càdlàg version of the so-obtained ground state-transformed processes. Next, we prove that they satisfy a related stochastic differential equation with jumps. Making use of this SDE, we then derive and prove the multifractal spectrum of local Hölder exponents of sample paths of ground state-transformed processes.

math.PR↗

arXiv2018

Uniform Hausdorff dimension result for the inverse images of stable Lévy processes

We establish a uniform Hausdorff dimension result for the inverse image sets of real-valued strictly $α$-stable Lévy processes with $1< α\le 2$. This extends a theorem of Kaufman for Brownian motion. Our method is different from that of Kaufman and depends on covering principles for Markov processes.

math.PR↗

arXiv2018

On the favorite points of symmetric Lévy processes

This paper is concerned with asymptotic behavior (at zero and at infinity) of the favorite points of Lévy processes. By exploring Molchan's idea for deriving lower tail probabilities of Gaussian processes with stationary increments, we extend the result of Marcus (2001) on the favorite points to a larger class of symmetric Lévy processes.

math.PR↗

arXiv2018

On sojourn of Brownian motion inside moving boundaries

We investigate the large scale structure of certain sojourn sets of one dimensional Brownian motion within two-sided moving boundaries. The macroscopic Hausdorff dimension, upper mass dimension and logarithmic density of these sets are computed. We also give a uniform macroscopic dimension result for the Brownian level sets.

math.PR↗

arXiv2017

Multifractality of jump diffusion processes

We study the local regularity and multifractal nature of the sample paths of jump diffusion processes, which are solutions to a class of stochastic differential equations with jumps. This article extends the recent work of Barral {\it et al.} who constructed a pure jump monotone Markov process with random multifractal spectrum. The class of processes studied here is much larger and exhibits novel features on the extreme values of the spectrum. This class includes Bass' stable-like processes and non-degenerate stable-driven SDEs.

math.PR↗

arXiv2017

Hausdorff dimension of the range and the graph of stable-like processes

We determine the Hausdorff dimension for the range of a class of pure jump Markov processes in $\mathbb{R}^d$, which turns out to be random and depends on the trajectories of these processes. The key argument is carried out through the SDE representation of these processes. The method developed here also allows to compute the Hausdorff dimension for the graph.

math.PR↗

arXiv2016

Multifractal analysis for the occupation measure of stable-like processes

In this article, we investigate the local behaviors of the occupation measure $μ$ of a class of real-valued Markov processes M, defined via a SDE. This (random) measure describes the time spent in each set A $\subset$ R by the sample paths of M. We compute the multifractal spectrum of $μ$, which turns out to be random, depending on the trajec-tory. This remarkable property is in sharp contrast with the results previously obtained on occupation measures of other processes (such as L{é}vy processes), since the multifractal spectrum is usually determinis-tic, almost surely. In addition, the shape of this multifractal spectrum is very original, reflecting the richness and variety of the local behaviors. The proof is based on new methods, which lead for instance to fine estimates on Hausdorff dimensions of certain jump configurations in Poisson point processes.

math.DS↗