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Yimin Xiao

Publications and source records attributed to Yimin Xiao.

At least 37 records · Page 2Linked to original sources
arXiv2023

Polarity of points for systems of nonlinear stochastic heat equations in the critical dimension

Let $u(t, x) = (u_1(t, x), \dots, u_d(t, x))$ be the solution to the systems of nonlinear stochastic heat equations \[ \begin{split} \frac{\partial}{\partial t} u(t, x) &= \frac{\partial^2}{\partial x^2} u(t, x) + σ(u(t, x)) \dot{W}(t, x),\\ u(0, x) &= u_0(x), \end{split} \] where $t \ge 0$, $x \in \mathbb{R}$, $\dot{W}(t, x) = (\dot{W}_1(t, x), \dots, \dot{W}_d(t, x))$ is a vector of $d$ independent space-time white noises, and $σ: \mathbb{R}^d \to \mathbb{R}^{d\times d}$ is a matrix-valued function. We say that a subset $S$ of $\mathbb{R}^d$ is polar for $\{u(t, x), t \ge 0, x \in \mathbb{R}\}$ if \[ \mathbb{P}\{u(t,x) \in S \text{ for some } t>0 \text{ and } x\in\mathbb{R} \}=0. \] The main result of this paper shows that, in the critical dimension $d=6$, all points in $\mathbb{R}^d$ are polar for $\{u(t, x), t \ge 0, x \in \mathbb{R}\}$. This solves an open problem of Dalang, Khoshnevisan and Nualart (2009, 2013) and Dalang, Mueller and Xiao (2021). We also provide a sufficient condition for a subset $S$ of $\mathbb{R}^d$ to be polar.

math.PR↗
arXiv2023

The expected Euler characteristic approximation to excursion probabilities of Gaussian vector fields

Let $\{(X(t), Y(s)): t\in T, s\in S\}$ be an $\mathbb{R}^2$-valued, centered, unit-variance smooth Gaussian vector field, where $T$ and $S$ are compact rectangles in $\mathbb{R}^N$. It is shown that, as $u\to \infty$, the joint excursion probability $\mathbb{P} \{\sup_{t\in T} X(t) \geq u, \sup_{s\in S} Y(s) \geq u \}$ can be approximated by $\mathbb{E}\{χ(A_u)\}$, the expected Euler characteristic of the excursion set $A_u=\{(t,s)\in T\times S: X(t) \ge u, Y(s) \ge u\}$, such that the error is super-exponentially small. This verifies the expected Euler characteristic heuristic (cf. Taylor, Takemura and Alder (2005), Alder and Taylor (2007)) for a large class of smooth Gaussian vector fields.

math.PR↗
arXiv2022

Linear Multifractional Stable Sheets in the Broad Sense: Existence and Joint Continuity of Local Times

We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) with $α\in(0,2]$, to include both linear multifractional Brownian sheets ($α=2$) and linear multifractional stable sheets ($α<2$). The purpose of the present paper is to study the existence and joint continuity of the local times of LMSS, and also the local Hölder condition of the local times in the set variable. Among the main results of this paper, Theorem 2.4 provides a sufficient and necessary condition for the existence of local times of LMSS; Theorem 3.1 shows a sufficient condition for the joint continuity of local times; and Theorem 4.1 proves a sharp local Hölder condition for the local times in the set variable. All these theorems improve significantly the existing results for the local times of multifractional Brownian sheets and linear multifractional stable sheets in the literature.

math.PR↗
arXiv2022

Limit theorems for linear random fields with innovations in the domain of attraction of a stable law

In this paper we study the convergence in distribution and the local limit theorem for the partial sums of linear random fields with i.i.d. innovations that have infinite second moment and belong to the domain of attraction of a stable law with index $0<α\leq2$ under the condition that the innovations are centered if $1<α\leq2$ and are symmetric if $α=1$. We establish these two types of limit theorems as long as the linear random fields are well-defined, the coefficients are either absolutely summable or not absolutely summable.

math.PR↗
arXiv2022

Hitting probabilities of Gaussian random fields and collision of eigenvalues of random matrices

Let $X= \{X(t), t \in \mathbb R^N\}$ be a centered Gaussian random field with values in $\mathbb R^d$ satisfying certain conditions and let $F \subset \mathbb R^d$ be a Borel set. In our main theorem, we provide a sufficient condition for $F$ to be polar for $X$, i.e. $\mathbb P \big( X(t) \in F \hbox{ for some } t \in \mathbb R^N \big) = 0$, which improves significantly the main result in Dalang et al [7], where the case of $F$ being a singleton was considered. We provide a variety of examples of Gaussian random field for which our result is applicable. Moreover, by using our main theorem, we solve a problem on the existence of collisions of the eigenvalues of random matrices with Gaussian random field entries that was left open in Jaramillo and Nualart [14] and Song et al [21].

math.PR↗
arXiv2021

Strong Local Nondeterminism and Exact Modulus of Continuity for Isotropic Gaussian Random Fields on Compact Two-Point Homogeneous Spaces

This paper is concerned with sample path properties of isotropic Gaussian fields on compact two-point homogeneous spaces. In particular, we establish the property of strong local nondeterminism of an isotropic Gaussian field based on the high-frequency behavior of its angular power spectrum, and then exploit this result to establish an exact uniform modulus of continuity for its sample paths.

math.PR↗
arXiv2021

On the Intersection of Dynamical Covering Sets with Fractals

Let $(X,\mathscr{B}, μ,T,d)$ be a measure-preserving dynamical system with exponentially mixing property, and let $μ$ be an Ahlfors $s$-regular probability measure. The dynamical covering problem concerns the set $E(x)$ of points which are covered by the orbits of $x\in X$ infinitely many times. We prove that the Hausdorff dimension of the intersection of $E(x)$ and any regular fractal $G$ equals $\dim_{\rm H}G+α-s$, where $α=\dim_{\rm H}E(x)$ $μ$--a.e. Moreover, we obtain the packing dimension of $E(x)\cap G$ and an estimate for $\dim_{\rm H}(E(x)\cap G)$ for any analytic set $G$.

math.DS↗
arXiv2021

Spectral heat content on a class of fractal sets for subordinate killed Brownian motions

We study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in $\mathbb{R}^{d}$, $d\geq 1$, with respect to subordinate killed Brownian motions via $α/2$-stable subordinators and establish the asymptotic behavior of the spectral heat content as $t\to 0$ for the full range of $α\in (0,2)$. Our main theorems show that these asymptotic behaviors depend on whether the sequence of logarithms of the coefficients of the similitudes is arithmetic when $α\in [d-\b,2)$, where $\b$ is the interior Minkowski dimension of the boundary of the open set. The main tools for proving the theorems are the previous results on the spectral heat content for Brownian motions and the renewal theorem.

math.PR↗
arXiv2021

Chung-type law of the iterated logarithm and exact moduli of continuity for a class of anisotropic Gaussian random fields

We establish a Chung-type law of the iterated logarithm and the exact local and uniform moduli of continuity for a large class of anisotropic Gaussian random fields with a harmonizable-type integral representation and the property of strong local nondeterminism. Compared with the existing results in the literature, our results do not require the assumption of stationary increments and provide more precise upper and lower bounds for the limiting constants. The results are applicable to the solutions of a class of linear stochastic partial differential equations driven by a fractional-colored Gaussian noise, including the stochastic heat equation.

math.PR↗
arXiv2021

Exact uniform modulus of continuity and Chung's LIL for the generalized fractional Brownian motion

The generalized fractional Brownian motion (GFBM) $X:=\{X(t)\}_{t\ge0}$ with parameters $γ\in [0, 1)$ and $α\in \left(-\frac12+\fracγ{2}, \, \frac12+\fracγ{2} \right)$ is a centered Gaussian $H$-self-similar process introduced by Pang and Taqqu (2019) as the scaling limit of power-law shot noise processes, where $H = α-\fracγ{2}+\frac12 \in(0,1)$. When $γ= 0$, $X$ is the ordinary fractional Brownian motion. For $γ\in (0, 1)$, GFBM $X$ does not have stationary increments, and its sample path properties such as Hölder continuity, path differentiability/non-differentiability, and the functional law of the iterated logarithm have been investigated recently by Ichiba, Pang and Taqqu (2020). They mainly focused on sample path properties that are described in terms of the self-similarity index $H$ (e.g., LILs at the origin and at infinity). In this paper, we further study the sample path properties of GFBM $X$ and establish the exact uniform modulus of continuity, small ball probabilities, and Chung's law of iterated logarithm. Our results show that the local regularity properties of GFBM $X$ away from the origin are determined by the index $α+\frac1 2$, instead of the self-similarity index $H$. This is in contrast with the properties of ordinary fractional Brownian motion whose local and asymptotic properties are determined by the single index $H$.

math.PR↗
arXiv2021

Propagation of Singularities for the Stochastic Wave Equation

We study the existence and propagation of singularities of the solution to a one-dimensional linear stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. Our approach is based on a simultaneous law of the iterated logarithm and general methods for Gaussian processes.

math.PR↗
arXiv2021

Lower functions and Chung's LILs of the generalized fractional Brownian motion

Let $X:=\{X(t)\}_{t\ge0}$ be a generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019): $$ \big\{X(t)\big\}_{t\ge0}\overset{d}{=}\left\{ \int_{\mathbb R} \left((t-u)_+^α-(-u)_+^α \right) |u|^{-γ} B(du) \right\}_{t\ge0}, $$ with parameters $γ\in (0, 1/2)$ and $α\in \left(-\frac12+ γ, \, \frac12+ γ\right)$. Continuing the studies of sample path properties of GFBM $X$ in Ichiba, Pang and Taqqu (2021) and Wang and Xiao (2021), we establish integral criteria for the lower functions of $X$ at $t=0$ and at infinity by modifying the arguments of Talagrand (1996). As a consequence of the integral criteria, we derive the Chung-type laws of the iterated logarithm of $X$ at the $t=0$ and at infinity, respectively. This solves a problem in Wang and Xiao (2021).

math.PR↗
arXiv2021

On eigenvalues of the Brownian sheet matrix

We derive a system of stochastic partial differential equations satisfied by the eigenvalues of the symmetric matrix whose entries are the Brownian sheets. We prove that the sequence $\left\{L_{d}(s,t), (s,t)\in[0,S]\times [0,T]\right\}_{d\in\mathbb N}$ of empirical spectral measures of the rescaled matrices is tight on $C([0,S]\times [0,T], \mathcal P(\mathbb R))$ and hence is convergent as $d$ goes to infinity by Wigner's semicircle law. We also obtain PDEs which are satisfied by the high-dimensional limiting measure.

math.PR↗
arXiv2021

Multiple Points of Gaussian Random Fields

This paper is concerned with the existence of multiple points of Gaussian random fields. Under the framework of Dalang et al. (2017), we prove that, for a wide class of Gaussian random fields, multiple points do not exist in critical dimensions. The result is applicable to fractional Brownian sheets and the solutions of systems of stochastic heat and wave equations.

math.PR↗
arXiv2020

On collision of multiple eigenvalues for matrix-valued Gaussian processes

For real symmetric and complex Hermitian Gaussian processes whose values are $d\times d$ matrices, we characterize the conditions under which the probability that at least $k$ eigenvalues collide is positive for $2\le k\le d$, and we obtain the Hausdorff dimension of the set of collision times.

math.PR↗
arXiv2020

Polarity of almost all points for systems of non-linear stochastic heat equations in the critical dimension

We study vector-valued solutions $u(t,x)\in\mathbb{R}^d$ to systems of nonlinear stochastic heat equations with multiplicative noise: \begin{equation*} \frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)+σ(u(t,x))\dot{W}(t,x). \end{equation*} Here $t\geq 0$, $x\in\mathbb{R}$ and $\dot{W}(t,x)$ is an $\mathbb{R}^d$-valued space-time white noise. We say that a point $z\in\mathbb{R}^d$ is polar if \begin{equation*} P\{u(t,x)=z\text{ for some $t>0$ and $x\in\mathbb{R}$}\}=0. \end{equation*} We show that in the critical dimension $d=6$, almost all points in $\mathbb{R}^d$ are polar.

math.PR↗
arXiv2019

Uniform dimension results for the inverse images of symmetric Lévy processes

We prove uniform Hausdorff and packing dimension results for the inverse images of a large class of real-valued symmetric Lévy processes. Our main result for the Hausdorff dimension extends that of Kaufman (1985) for Brownian motion and that of Song, Xiao, and Yang (2018) for $α$-stable Lévy processes with $1<α<2$. Along the way, we also prove an upper bound for the uniform modulus of continuity of the local times of these processes.

math.PR↗