Yimin Xiao

• Ph.D.
• MSU Foundation Professor at Michigan State University

Let $(X_t)_{t \geqslant 0}$ be the solution of the stochastic differential equation \[\mathrm{d} X_t = b(X_t) \,\mathrm{d} t+A\,\mathrm{d} Z_t, \quad X_{0}=x,\] where $b\,:\, \mathbb{R}^d \rightarrow \mathbb{R}^d$ is a Lipschitz-continuous function, $A \in \mathbb{R}^{d \times d}$ is a positive-definite matrix, $(Z_t)_{t\geqslant 0}$ is a d -dimens...

We establish a sample path moderate deviation principle for the integrated shot noise process with Poisson arrivals and non-stationary noises. As in Pang and Taqqu (2019), we assume that the noise is conditionally independent given the arrival times, and the distribution of each noise depends on its arrival time. As applications, we derive moderate...

Let $W= \{W(t): t \in \mathbb{R}_+^N \}$ be an $(N, d)$-Brownian sheet and let $E \subset (0, \infty)^N$ and $F \subset \mathbb{R}^d$ be compact sets. We prove a necessary and sufficient condition for $W(E)$ to intersect $F$ with positive probability and determine the essential supremum of the Hausdorff dimension of the intersection set $W(E)\cap F...

Consider the nonlinear stochastic heat equation $$ \frac{\partial u (t,x)}{\partial t}=\frac{\partial^2 u (t,x)}{\partial x^2}+ \sigma(u (t,x))\dot{W}(t,x),\quad t> 0,\, x\in \mathbb{R}, $$ where $\dot W$ is a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in(\frac 14,\frac 12)$...

Let $X:=\{X(t)\}_{t\ge0}$ be a generalized fractional Brownian motion: $$ \{X(t)\}_{t\ge0}\overset{d}{=}\left\{ \int_{\mathbb R} \left((t-u)_+^{\alpha}-(-u)_+^{\alpha} \right) |u|^{-\gamma/2} B(du) \right\}_{t\ge0}, $$ with parameters $\gamma \in (0, 1)$ and $\alpha\in \left(-1/2+ \gamma/2, \, 1/2+ \gamma/2 \right)$. This is a self-similar Gaussian...

We consider the following stochastic space-time fractional diffusion equation with vanishing initial condition:$$ \partial^{\beta} u(t, x)=- \left(-\Delta\right)^{\alpha / 2} u(t, x)+ I_{0+}^{\gamma}\left[\dot{W}(t, x)\right],\quad t\in[0,T],\: x \in \mathbb{R}^d,$$ where $\alpha>0$, $\beta\in(0,2)$, $\gamma\in[0,1)$, $\left(-\Delta\right)^{\alpha/...

Non-Gaussian Harmonizable Fractional Stable Motion (HFSM) is a natural and important extension of the well-known Fractional Brownian Motion to the framework of heavy-tailed stable distributions. It was introduced several decades ago; however its properties are far from being completely understood. In our present paper we determine the optimal power...

Consider the stochastic partial differential equation ∂ ∂ t u t ( x ) = − ( − Δ ) α 2 u t ( x ) + b ( u t ( x ) ) + σ ( u t ( x ) ) F ˙ ( t , x ) , t ≥ 0 , x ∈ R d , \begin{equation*} \frac {\partial }{\partial t}u_t(\boldsymbol {x})= -(-\Delta )^{\frac {\alpha }{2}}u_t(\boldsymbol {x}) +b\left (u_t(\boldsymbol {x})\right )+\sigma \left (u_t(\bolds...

We study the local times of anisotropic Gaussian random fields satisfying strong local nondeterminism with respect to an anisotropic metric. We derive moment estimates and prove local and global H\"{o}lder conditions for the local times. As a consequence, we deduce sample path properties that are related to Chung's law of the iterated logarithm and...

We study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in Rd${\mathbb {R}}^{d}$, d≥1$d\ge 1$, with respect to subordinate killed Brownian motions via α/2$\alpha /2$‐stable subordinators and establish the asymptotic behavior of the spectral heat content as t→0$t\rightarrow 0$ for the full range...

Let $u(t, x) = (u_1(t, x), \dots, u_d(t, x))$ be the solution to the systems of nonlinear stochastic heat equations \[ \begin{split} \frac{\partial}{\partial t} u(t, x) &= \frac{\partial^2}{\partial x^2} u(t, x) + \sigma(u(t, x)) \dot{W}(t, x),\\ u(0, x) &= u_0(x), \end{split} \] where $t \ge 0$, $x \in \mathbb{R}$, $\dot{W}(t, x) = (\dot{W}_1(t, x...

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 $\math...

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

Let \((X,\mathscr {B}, \mu ,T,d)\) be a measure-preserving dynamical system with exponentially mixing property, and let \(\mu \) 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...

Let X:={X(t)}t≥0 be a generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019):{X(t)}t⩾0=d{∫R((t−u)+α−(−u)+α)|u|−γB(du)}t⩾0, where γ∈[0,1/2) and α∈(−12+γ,12+γ) are constants. Ichiba, Pang and Taqqu (2021) and Wang and Xiao (2021) have studied some asymptotic and regularity properties of GFBM X. However, the problem of estab...

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≤2 under the condition that the innovations are centered if 1<α≤2 and are symmetric if α=1. We e...

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...

The generalized fractional Brownian motion (GFBM) X:={X(t)}t≥0 with parameters γ∈[0,1) and α∈-12+γ2,12+γ2 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-γ2+12∈(0,1). When γ=0, X is the ordinary fractional Brownian motion. When γ∈(0,1), GFBM X does n...

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 unifor...

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 $\alpha/2$-stable subordinators and establish the asymptotic behavior of the spectral heat content as $t\to 0$ for the full range of $\alpha\in (0,2)$....

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

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.

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 $0<\alpha\leq2$ under the condition that the innovations are centered if $1<\alpha\leq2$ and are sy...

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 as...

We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) with $\alpha\in(0,2]$, to include both linear multifractional Brownian sheets ($\alpha=2$) and linear multifractional stable sheets ($\alpha<2$). The purpose of the present paper is to study the existence and joint continuity of the local times of LMSS. The ma...

Let $(X,\mathscr{B}, \mu,T,d)$ be a measure-preserving dynamical system with exponentially mixing property, and let $\mu$ 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 inte...

Let $X:=\{X(t)\}_{t\ge0}$ be a generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019): $$ \big\{X(t)\big\}_{t\ge0}\overset{d}{=}\left\{ \int_{\mathbb R} \left((t-u)_+^{\alpha}-(-u)_+^{\alpha} \right) |u|^{-\gamma} B(du) \right\}_{t\ge0}, $$ with parameters $\gamma \in (0, 1/2)$ and $\alpha\in \left(-\frac12+ \gamma , \, \...

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], \mathc...

The generalized fractional Brownian motion (GFBM) $X:=\{X(t)\}_{t\ge0}$ with parameters $\gamma \in [0, 1)$ and $\alpha\in \left(-\frac12+\frac{\gamma}{2}, \, \frac12+\frac{\gamma}{2} \right)$ 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 = \alpha...

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 (C R Acad Sci Paris Sér I Math 300:281–282, 1985) for Brownian motion and that of Song et al. (Electron Commun Probab 23:10, 2018) for \(\alpha...

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.

We determine the Hausdorff dimension of the set of k-multiple points for a symmetric operator semistable Lévy process \(X=\{X(t), t\in {\mathbb {R}}_+\}\) in terms of the eigenvalues of its stability exponent. We also give a necessary and sufficient condition for the existence of k-multiple points. Our results extend to all \(k\ge 2\) the recent wo...

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)+\sigma(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...

In this article, we study the following stochastic heat equation ∂tu(t,x)=Lu(t,x)+Ḃ, u(0,x)=0, 0≤t≤T, x∈Rd, where L is the generator of a Lévy process X in Rd, B is a fractional-colored Gaussian noise with Hurst index H∈(12,1) in the time variable and spatial covariance function f which is the Fourier transform of a tempered measure μ. After establ...

This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen’s functional laws of the iterated logarithm at zero and infinity respectively. The sets of limit points of those Gaussian random fields are obtained. The main results are applied t...

We study the existence and propagation of singularities of the solution to a one-dimensional stochastic wave equation (SWE) driven by an additive Gaussian noise that is white in time and colored in space. Our approach is based on general methods for Gaussian processes and the relation between the solution of SWE and the fractional Brownian sheet.

This paper is concerned with asymptotic behavior (at zero and at infinity) of the favorite points of L\'evy 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 L\'evy processes.

In this paper, we consider block thresholding wavelet estimators of spatial regression functions on stationary Gaussian random fields observed over a rectangular domain indexed with \({{\mathbb {Z}}}^2\), whose covariance function is assumed to satisfy some weak condition. We investigate their asymptotic rates of convergence under the mean integrat...

For \(0 < H \le 1/2\), let \(\mathbf {B}^H = \{ \mathbf {B}^H(t);\; t\in \mathbb R^N_+ \}\) be the Gaussian random field obtained from the set-indexed fractional Brownian motion restricted to the rectangles of \(\mathbb R^N_+\). We prove that \(\mathbf {B}^H\) is tangent to a multiparameter fBm which is isotropic in the \(l^1\)-norm and we determin...

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 a...

Strong laws of large numbers are established for random fields with weak or strong dependence. These limit theorems are applicable to random fields with heavy-tailed distributions including fractional stable random fields. The conditions for SLLN are described in terms of the p-th moments of the partial sums of the random fields, which are convenie...

We consider the linear stochastic wave equation driven by a Gaussian noise. We show that the solution satisfies a certain form of strong local nondeterminism and we use this property to derive the exact uniform modulus of continuity for the solution.

We prove uniform Hausdorff and packing dimension results for the inverse images of a large class of real-valued symmetric L\'evy 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 $\alpha$-stable L\'evy processes with $1<\alpha<2$. Along the way, we a...

Let ZH={ZH(t),t∈RN} be a real-valued N-parameter harmonizable fractional stable sheet with index H=(H1,…,HN)∈(0,1)N. We establish a random wavelet series expansion for ZH which is almost surely convergent in all the Hölder spaces Cγ([−M,M]N), where M>0 and γ∈(0,min{H1,…,HN}) are arbitrary. One of the main ingredients for proving the latter result i...

Let $Z^H= \{Z^H(t), t \in \R^N\}$ be a real-valued $N$-parameter harmonizable fractional stable sheet with index $H = (H_1, \ldots, H_N) \in (0, 1)^N$. We establish a random wavelet series expansion for $Z^H$ which is almost surely convergent in all the H\"older spaces $C^\gamma ([-M,M]^N)$, where $M>0$ and $\gamma\in (0, \min\{H_1,\ldots, H_N\})$...

We establish the exact moduli of non-differentiability of fractional Brownian motion. As an application of the result, we prove that the uniform Hölder condition for the maximum local times of fractional Brownian motion obtained in Xiao (1997) is optimal.

We study the Cramér type moderate deviation for partial sums of random fields by applying the conjugate method. The results are applicable to the partial sums of linear random fields with short or long memory and to nonparametric regression with random field errors.

We study the Cram\'er type moderate deviation for partial sums of random fields by applying the conjugate method. The results are applicable to the partial sums of linear random fields with short or long memory and to nonparametric regression with random field errors.

We study the linear stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere S ² . The existence and uniqueness of its solution in certain Sobolev space is investigated and sample path regularity properties are established. In particular, the exact uniform modulus of continuity of the solution...

This paper studies the problem of equivalence of Gaussian measures induced by Gaussian random fields (GRFs) with stationary increments and proves a sufficient condition for the equivalence in terms of the behavior of the spectral measures at infinity. The main results extend those of Stein (2004), Van Zanten (2007, 2008) and are applicable to a ric...

Strong laws of large numbers are established for random fields with weak or strong dependence. These limit theorems are applicable to random fields with heavy-tailed distributions including fractional stable random fields. The conditions for SLLN are described in terms of the $p$-th moments of the partial sums of the random fields, which are conven...

In this paper, we study the following stochastic heat equation \[ \partial_tu=\mathcal{L} u(t,x)+\dot{B},\quad u(0,x)=0,\quad 0\le t\le T,\quad x\in\mathbb{R}d, \] where $\mathcal{L}$ is the generator of a L\'evy process $X$ taking value in $\mathbb{R}^d$, $B$ is a fractional-colored Gaussian noise with Hurst index $H\in\left(\frac12,\,1\right)$ fo...

We study the stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere $\mathbb{S}^{2}$. The existence and uniqueness of its solution in certain Sobolev space is investigated and sample path regularity properties are established. In particular, the exact uniform modulus of continuity of the sol...

This paper investigate the local times and modulus of nondifferentiability of the spherical Gaussian random fields. We extend the methods for studying the local times of Gaussian to the spherical setting. The new main ingredient is the property of strong local nondeterminism established recently in Lan et al (2018).

This paper studies the problem of equivalence of Gaussian measures induced by Gaussian random fields (GRFs) with stationary increments and proves a sufficient condition for the equivalence in terms of the behavior of the spectral measures at infinity. The main results extend those of Stein (2004), Van Zanten (2007, 2008) and are applicable to a ric...

It is generally argued that the solution to a stochastic PDE with multiplicative noise---such as $\dot{u}=\frac12 u"+u\xi$, where $\xi$ denotes space-time white noise---routinely produces exceptionally-large peaks that are "macroscopically multifractal." See, for example, Gibbon and Doering (2005), Gibbon and Titi (2005), and Zimmermann et al (2000...

This paper proves sharp bounds on the tails of the Lévy exponent of an operator semistable law on Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb R^d}$$\end{...

We determine the Hausdorff dimension of $k$-multiple points for a symmetric operator semistable L\'evy process $X=\{X(t), t\in\mathbb{R}_+\}$ in terms of the eigenvalues of its stability exponent. We also give a necessary and sufficient condition for the existence of $k$-multiple points. Our results extend to all $k\geq2$ the recent work [23], wher...

We establish a uniform Hausdorff dimension result for the inverse image sets of real-valued strictly α-stable Lévy processes with 1 < α ≤ 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.

We establish a uniform Hausdorff dimension result for the inverse image sets of real-valued strictly $\alpha$-stable L\'evy processes with $1< \alpha\le 2$. 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.

We establish a uniform Hausdorff dimension result for the inverse image sets of real-valued strictly α-stable Lévy processes with 1 < α ≤ 2. This extends a theorem of Kaufman [11] for Brownian motion. Our method is different from that of [11] and depends on covering principles for Markov processes.

This paper is concerned with asymptotic behavior (at zero and at infinity) of the favorite points of L\'evy 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 L\'evy processes.

Based on the seminal works of Rosinski (1995, 2000) and Samorodnitsky (2004a), we solve an open problem mentioned in a paper of Xiao (2010) and provide sharp upper bound on the rate of growth of maximal moments for many stationary symmetric stable random fields. We also investigate the relationship between this rate of growth and the path regularit...

In this work, we solve an open problem mentioned in Xiao (2010) and provide sharp bounds on the rate of growth of maximal moments for stationary symmetric stable random fields using structure theorem of finitely generated abelian groups and ergodic theory of quasi-invariant group actions. We also investigate the relationship between this rate of gr...

By extending the methods in Peligrad et al. (2014a, b), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for studying nonparametric regression with random field errors and strong limit theorems.

By extending the methods in Peligrad et al. (2014a, b), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for studying nonparametric regression with random field errors and strong limit theorems.

Let $B = \left\{ B\left( x\right),\, x\in \mathbb{S}^{2}\right\} $ be the fractional Brownian motion indexed by the unit sphere $\mathbb{S}^{2}$ with index $0<H\leq \frac{1}{2}$, introduced by Istas \cite{IstasECP05}. We establish optimal estimates for its angular power spectrum $\{d_\ell, \ell = 0, 1, 2, \ldots\}$, and then exploit its high-freque...

Let $B = \left\{ B\left( x\right),\, x\in \mathbb{S}^{2}\right\} $ be the fractional Brownian motion indexed by the unit sphere $\mathbb{S}^{2}$ with index $0<H\leq \frac{1}{2}$, introduced by Istas \cite{IstasECP05}. We establish optimal estimates for its angular power spectrum $\{d_\ell, \ell = 0, 1, 2, \ldots\}$, and then exploit its high-freque...

In this paper we prove uniform Hausdorff and packing dimension results for the images of a large family of Markov processes. The main tools are the two covering principles of Xiao (second author). As applications, uniform Hausdorff and packing dimension results for certain classes of L\'evy processes, stable jump diffusion and non-symmetric stable-...

In this paper we prove uniform Hausdorff and packing dimension results for the images of a large family of Markov processes. The main tools are the two covering principles of Xiao (second author). As applications, uniform Hausdorff and packing dimension results for certain classes of L\'evy processes, stable jump diffusion and non-symmetric stable-...

It is generally argued that the solution to a stochastic PDE with multiplicative noise---such as $\dot{u}=\frac12 u"+u\xi$, where $\xi$ denotes space-time white noise---routinely produces exceptionally-large peaks that are "macroscopically multifractal." See, for example, Gibbon and Doering (2005), Gibbon and Titi (2005), and Zimmermann et al (2000...

We determine the Hausdorff dimension of the set of double points for a symmetric operator stable L\'evy process in terms of the eigenvalues of its stability exponent.

We provide sharp lower and upper bounds for the Hausdorff dimension of the intersection of a typical random covering set with a fixed analytic set both in Ahlfors regular metric spaces and in the $d$-dimensional torus. In metric spaces, we consider covering sets generated by balls and, in the torus, we deal with general analytic generating sets.

For modeling multiple variables, the fractal indices of the components of the underlying multivariate process play a key role in characterizing the dependence structure among the components and statistical properties of the multivariate process. In this paper, under the infill asymptotics framework, we establish joint asymptotic results for the inc...

Multivariate (or vector-valued) processes are important for modeling multiple variables. The fractal indices of the components of the underlying multivariate process play a key role in characterizing the dependence structures and statistical properties of the multivariate process. In this paper, under the infill asymptotics framework, we establish...

This paper proves sharp bounds on the tails of the L\'evy exponent of an operator semistable law on $\mathbb R^d$. These bounds are then applied to explicitly compute the Hausdorff and packing dimensions of the range, graph, and other random sets describing the sample paths of the corresponding operator semi-selfsimilar L\'evy processes. The proofs...

We consider wavelet-based nonlinear estimators, which are constructed by using the thresholding of the empirical wavelet coefficients, for the mean regression functions with strong mixing errors and investigate their asymptotic rates of convergence. We show that these estimators achieve nearly optimal convergence rates within a logarithmic term ove...

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 c...

In this paper, we are concerned with sample path properties of isotropic spherical Gaussian fields on $\S^2$. 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...

In this paper, we are concerned with sample path properties of isotropic spherical Gaussian fields on $\S^2$. 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...

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---w...

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---w...

We provide sharp lower and upper bounds for the Hausdorff dimension of the intersection of a typical random covering set with a fixed analytic set both in Ahlfors regular metric spaces and in the $d$-dimensional torus. In metric spaces, we consider covering sets generated by balls and, in the torus, we deal with general analytic generating sets.

By applying a Fourier analytic argument, we prove that, for every α∈(0,2), the N-parameter harmonizable fractional α-stable field (HFαSF) is locally nondeterministic. When 0<α<1, this solves an open problem in Nolan (1989). Also, it allows us to establish the joint continuity of the local times of an (N,d)-HFαSF for an arbitrary α∈(0,2), and to obt...

Let be a Gaussian random field with values in and let be a Borel set. We determine the packing dimension of the image set in terms of the packing dimension profiles in the canonical metric of , which are extensions of the packing dimension profiles of Falconer and Howroyd (1997) and the box-counting dimension profiles of Howroyd (2001).

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 s...

Let $X=\{X(t),t\in\mathrm{R}^N\}$ be a centered real-valued operator-scaling Gaussian random field with stationary increments, introduced by Bierm\'{e}, 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 conti...

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...

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...

We determine the Hausdorff dimension of the set of double points for a symmetric operator stable Lévy process X = {X(t), t ∈ R+} in terms of the eigenvalues of its stability exponent.

We determine the Hausdorff dimension of the set of double points for a symmetric operator stable Lévy process in terms of the eigenvalues of its stability exponent.