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Yimin Xiao

Publications and source records attributed to Yimin Xiao.

At least 73 records · Page 4Linked to original sources
arXiv2016

On the Macroscopic Fractal Geometry of Some Random Sets

This paper is concerned mainly with the macroscopic fractal behavior of various random sets that arise in modern and classical probability theory. Among other things, it is shown here that the macroscopic behavior of Boolean coverage processes is analogous to the microscopic structure of the Mandelbrot fractal percolation. Other, more technically challenging, results of this paper include: (i) The computation of the macroscopic dimension of the graph of a large family of Lévy processes; and (ii) The determination of the macroscopic monofractality of the extreme values of symmetric stable processes. As a consequence of (i), it will be shown that the macroscopic fractal dimension of the graph of Brownian motion differs from its microscopic fractal dimension. Thus, there can be no scaling argument that allows one to deduce the macroscopic geometry from the microscopic. Item (ii) extends the recent work of Khoshnevisan, Kim, and Xiao \cite{KKX} on the extreme values of Brownian motion, using a different method.

math.PR↗
arXiv2016

The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments

Let $X=\{X(t),t\in {\mathbb{R}}^N\}$ be a centered Gaussian random field with stationary increments and $X(0)=0$. For any compact rectangle $T\subset {\mathbb{R}}^N$ and $u\in {\mathbb{R}}$, denote by $A_u=\{t\in T:X(t)\geq u\}$ the excursion set. Under $X(\cdot)\in C^2({\mathbb{R}}^N)$ and certain regularity conditions, the mean Euler characteristic of $A_u$, denoted by ${\mathbb{E}}\{φ(A_u)\}$, is derived. By applying the Rice method, it is shown that, as $u\to\infty$, the excursion probability ${\mathbb{P}}\{\sup_{t\in T}X(t)\geq u\}$ can be approximated by ${\mathbb{E}}\{φ(A_u)\}$ such that the error is exponentially smaller than ${\mathbb{E}}\{φ(A_u)\}$. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.

math.PR↗
arXiv2016

L-Kuramoto-Sivashinsky SPDEs vs. time-fractional SPIDEs: exact continuity and gradient moduli, 1/2-derivative criticality, and laws

We establish exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for (1) the fourth order L-Kuramoto-Sivashinsky (L-KS) SPDEs and for (2) the time-fractional stochastic partial integro-differential equations (SPIDEs), driven by space-time white noise in one-to-three dimensional space. Both classes were introduced---with Brownian-time-type kernel formulations---by Allouba in a series of articles starting in 2006, where he presented class (2) in its rigorous stochastic integral equations form. He proved existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above two classes of equations in $d=1,2,3$. We show that both classes are $(1/2)^-$ Hölder continuously differentiable in space when $d=1$, and we give the exact uniform and local moduli of continuity for the gradient in both cases. This is unprecedented for SPDEs driven by space-time white noise. Our results on exact moduli show that the half-derivative SPIDE is a critical case. It signals the onset of rougher modulus regularity in space than both time-fractional SPIDEs with time-derivatives of order $<1/2$ and L-KS SPDEs. This is despite the fact that they all have identical spatial Hölder regularity, as shown earlier by Allouba. Moreover, we show that the temporal laws governing (1) and (2) are fundamentally different. We relate L-KS SPDEs to the Houdré-Villa bifractional Brownian motion, yielding a Chung-type law of the iterated logarithm for these SPDEs. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on delicate sample path properties of Gaussian random fields. On the other hand, it builds on and complements Allouba's earlier works on (1) and (2). Similar regularity results hold for Allen-Cahn nonlinear versions of (1) and (2) on compacts via change of measure.

math.AP↗
arXiv2016

Strong Local Nondeterminism and Exact Modulus of Continuity for Spherical Gaussian Fields

In this paper, we are concerned with sample path properties of isotropic spherical Gaussian fields on $§^2$. In particular, we establish the property of strong local nondeterminism of an isotropic spherical Gaussian field based on the high-frequency behaviour of its angular power spectrum; we then exploit this result to establish an exact uniform modulus of continuity for its sample paths. We also discuss the range of values of the spectral index for which the sample functions exhibit fractal or smooth behaviour.

math.PR↗
arXiv2016

Excursion probability of Gaussian random fields on sphere

Let $X=\{X(x): x\in\mathbb{S}^N\}$ be a real-valued, centered Gaussian random field indexed on the $N$-dimensional unit sphere $\mathbb{S}^N$. Approximations to the excursion probability ${\mathbb{P}}\{\sup_{x\in\mathbb{S}^N}X(x)\ge u\}$, as $u\to\infty$, are obtained for two cases: (i) $X$ is locally isotropic and its sample functions are non-smooth and; (ii) $X$ is isotropic and its sample functions are twice differentiable. For case (i), the excursion probability can be studied by applying the results in Piterbarg (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), Mikhaleva and Piterbarg (Theory Probab. Appl. 41 (1997) 367--379) and Chan and Lai (Ann. Probab. 34 (2006) 80--121). It is shown that the asymptotics of ${\mathbb{P}}\{\sup_{x\in\mathbb {S}^N}X(x)\ge u\}$ is similar to Pickands' approximation on the Euclidean space which involves Pickands' constant. For case (ii), we apply the expected Euler characteristic method to obtain a more precise approximation such that the error is super-exponentially small.

math.PR↗
arXiv2015

Tail Asymptotics for the Extremes of Bivariate Gaussian Random Fields

Let $\{X(t)= (X_1(t),X_2(t))^T,\ t \in \mathbb{R}^N\}$ be an $\mathbb{R}^2$-valued continuous locally stationary Gaussian random field with $\mathbb{E}[X(t)]=\mathbf{0}$. For any compact sets $A_1, A_2 \subset \mathbb{R}^N$, precise asymptotic behavior of the excursion probability \[ \mathbb{P}\bigg(\max_{s\in A_1} X_1(s)>u,\, \max_{t\in A_2} X_2(t)>u\bigg),\ \ \text{ as }\ u \rightarrow \infty \] is investigated by applying the double sum method. The explicit results depend not only on the smoothness parameters of the coordinate fields $X_1$ and $X_2$, but also on their maximum correlation $ρ$.

math.PR↗
arXiv2015

Exact moduli of continuity for operator-scaling Gaussian random fields

Let $X=\{X(t),t\in\mathrm{R}^N\}$ be a centered real-valued operator-scaling Gaussian random field with stationary increments, introduced by Biermé, Meerschaert and Scheffler (Stochastic Process. Appl. 117 (2007) 312-332). We prove that $X$ satisfies a form of strong local nondeterminism and establish its exact uniform and local moduli of continuity. The main results are expressed in terms of the quasi-metric $τ_E$ associated with the scaling exponent of $X$. Examples are provided to illustrate the subtle changes of the regularity properties.

math.ST↗
arXiv2015

Polarity of points for Gaussian random fields

We show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients, such as the stochastic heat equation or wave equation with space-time white noise, or colored noise in spatial dimensions $k\geq 1$. Our approach builds on a delicate covering argument developed by M. Talagrand for the study of fractional Brownian motion, and uses a harmonizable representation of the solutions of these stochastic pde's.

math.PR↗
arXiv2015

Smoothness of Local Times and Self-Intersection Local Times of Gaussian Random Fields

This paper is concerned with the smoothness (in the sense of Meyer-Watanabe) of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.

math.PR↗
arXiv2015

Intermittency and multifractality: A case study via parabolic stochastic PDEs

Let $ξ$ denote space-time white noise, and consider the following stochastic partial differential equations: (i) $\dot{u}=\frac{1}{2} u" + uξ$, started identically at one; and (ii) $\dot{Z}=\frac12 Z" + ξ$, started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in different universality classes. We prove that the tall peaks of both systems are multifractals in a natural large-scale sense. Some of this work is extended to also establish the multifractal behavior of the peaks of stochastic PDEs on $\mathbf{R}_+\times\mathbf{R}^d$ with $d\ge 2$. G. Lawler has asked us if intermittency is the same as multifractality. The present work gives a negative answer to this question. As a byproduct of our methods, we prove also that the peaks of the Brownian motion form a large-scale monofractal, whereas the peaks of the Ornstein--Uhlenbeck process on $\mathbf{R}$ are multifractal. Throughout, we make extensive use of the macroscopic fractal theory of M.T. Barlow and S. J. Taylor (1989, 1992). We expand on aspects of the Barlow-Taylor theory, as well.

math.PR↗
arXiv2015

Sample Paths of the Solution to the Fractional-colored Stochastic Heat Equation

Let u = {u(t, x), t $\in$ [0, T ], x $\in$ R d } be the solution to the linear stochastic heat equation driven by a fractional noise in time with correlated spatial structure. We study various path properties of the process u with respect to the time and space variable, respectively. In particular, we derive their exact uniform and local moduli of continuity and Chung-type laws of the iterated logarithm.

math.PR↗
arXiv2015

Brownian motion and thermal capacity

Let $W$ denote $d$-dimensional Brownian motion. We find an explicit formula for the essential supremum of Hausdorff dimension of $W(E)\cap F$, where $E\subset(0,\infty)$ and $F\subset \mathbf {R}^d$ are arbitrary nonrandom compact sets. Our formula is related intimately to the thermal capacity of Watson [Proc. Lond. Math. Soc. (3) 37 (1978) 342-362]. We prove also that when $d\ge2$, our formula can be described in terms of the Hausdorff dimension of $E\times F$, where $E\times F$ is viewed as a subspace of space time.

math.PR↗
arXiv2014

Weak existence of a solution to a differential equation driven by a very rough fBm

We prove that if $f:\mathbb{R}\to\mathbb{R}$ is Lipschitz continuous, then for every $H\in(0,1/4]$ there exists a probability space on which we can construct a fractional Brownian motion $X$ with Hurst parameter $H$, together with a process $Y$ that: (i) is Hölder-continuous with Hölder exponent $γ$ for any $γ\in(0,H)$; and (ii) solves the differential equation $dY_t = f(Y_t) dX_t$. More significantly, we describe the law of the stochastic process $Y$ in terms of the solution to a non-linear stochastic partial differential equation.

math.PR↗
arXiv2013

Generalized dimensions of images of measures under Gaussian processes

We show that for certain Gaussian random processes and fields X:R^N to R^d, D_q(mu_X) = min{d, D_q(mu)/alpha} a.s. for an index alpha which depends on Holder properties and strong local nondeterminism of X, where q>1, where D_q denotes generalized q-dimension and where mu_X is the image of the measure mu under X. In particular this holds for index-alpha fractional Brownian motion, for fractional Riesz-Bessel motions and for certain infinity scale fractional Brownian motions.

math.PR↗
arXiv2012

Critical Brownian sheet does not have double points

We derive a decoupling formula for the Brownian sheet which has the following ready consequence: An $N$-parameter Brownian sheet in $\mathbf{R}^d$ has double points if and only if $d<4N$. In particular, in the critical case where $d=4N$, the Brownian sheet does not have double points. This answers an old problem in the folklore of the subject. We also discuss some of the geometric consequences of the mentioned decoupling, and establish a partial result concerning $k$-multiple points in the critical case $k(d-2N)=d$.

math.PR↗
arXiv2012

Discrete Fractal Dimensions of the Ranges of Random Walks in $\Z^d$ Associate with Random Conductances

Let X= {X_t, t \ge 0} be a continuous time random walk in an environment of i.i.d. random conductances {μ_e \in [1, \infty), e \in E_d}, where E_d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice Z^d and d\ge 3. Let R = {x \in Z^d: X_t = x for some t \ge 0} be the range of X. It is proved that, for almost every realization of the environment, dim_H (R) = dim_P (R) = 2 almost surely, where dim_H and dim_P denote respectively the discrete Hausdorff and packing dimension. Furthermore, given any set A \subseteq Z^d, a criterion for A to be hit by X_t for arbitrarily large t>0 is given in terms of dim_H(A). Similar results for Bouchoud's trap model in Z^d (d \ge 3) are also proven.

math.PR↗
arXiv2011

Spectral conditions for strong local nondeterminism and exact Hausdorff measure of ranges of Gaussian random fields

Let $X= \{X(t), t \in \R^N\}$ be a Gaussian random field with values in $\R^d$ defined by \[ X(t) = \big(X_1(t),..., X_d(t)\big),\qquad t \in \R^N, \] where $X_1, ..., X_d$ are independent copies of a real-valued, centered, anisotropic Gaussian random field $X_0$ which has stationary increments and the property of strong local nondeterminism. In this paper we determine the exact Hausdorff measure function for the range $X([0, 1]^N)$. We also provide a sufficient condition for a Gaussian random field with stationary increments to be strongly locally nondeterministic. This condition is given in terms of the spectral measures of the Gaussian random fields which may contain either an absolutely continuous or discrete part. This result strengthens and extends significantly the related theorems of Berman (1973, 1988), Pitt (1978) and Xiao (2007, 2009), and will have wider applicability beyond the scope of the present paper.

math.PR↗