Searcharxiv⌕ Search

arXiv subjects

Jay Rosen

Publications and source records attributed to Jay Rosen.

At least 19 recordsLinked to original sources
arXiv2026

Exact moduli of continuity for the local times of Feller Brownian motions

We examine the modulus of continuity, in the spatial variable, of the local time process of Feller Brownian motion (FBM) on the half-line $[0,\infty)$. Briefly, a FBM is a strong Markov process on $[0,\infty)$ that moves like standard Brownian motion on $[0,\infty)$ up until it first encounters the state $0$. The process returns to $(0,\infty)$, either continuously (like reflecting Brownian motion) or by jumping to a (random) positive state chosen according to a specified measure. The present work is a continuation and application of our earlier work with Michael Marcus on the moduli of continuity for the local times of a Markov process built by piecing together (``rebirthing") the paths of another Markov process with finite lifetime. We first establish a general result on the resolvent and local times for a rebirthed process with a special holding state (the state $0$ for FBM). We show how our earlier approach using the Eisenbaum Isomorphism Theorem on an assemblage of excursions works out in this context. This knowledge is then used as an approximation device to obtain our main result on the exact uniform moduli of continuity for the local time of FBM on a spatial interval of the form $(0,1]$. Extensions are made, under certain conditions, to the more delicate situation of the spatial interval $[0,1]$. We also consider briefly the case of more general diffusions on $[0,\infty)$.

math.PR↗
arXiv2026

Ray-Knight theorems for the local times of rebirthed Markov processes

We prove generalizations of the first and second Ray-Knight theorems, for a large class of non-symmetric strong Markov processes. These results link the local times of the Markov process with the squares of associated Gaussian processes. This connection allows us to establish results about the exact modulus of continuity (in the spatial variable) of the local times. Our approach is different from earlier treatments which were based on associated permanental processes rather than Gaussian processes. The type of process with which we work can be described as follows. Start with a symmetric Markov process with finite lifetime; upon its death resurrect it at a place in the state space chosen at random, independent of the past. Continue in this way, resurrecting at each death, to obtain a recurrent process. The rebirthing procedure destroys the symmetry of the original process, leading to a large class of non-symmetric processes. The main results are illustrated by many examples.

math.PR↗
arXiv2024

Moduli of continuity for the local times of rebirthed Markov processes

Let $S$ a be locally compact space with a countable base. Let $\cal Y$ be a transient symmetric Borel right process with state space $S$ and continuous strictly positive $p$--potential densities $u^p(x,y)$. Local and uniform moduli of continuity are obtained for the local times of both fully and partially rebirthed versions of $\cal Y$. A fully rebirthed version of $\cal Y$ is an extension of $\cal Y$ so that instead of terminating at the end of its lifetime it is immediately ``reborn'' with a probability measure $μ$, on $S$. I.e., the process goes to the set $B\subset S$ with probability $μ(B) $, after which it continues to evolve the way $\YY$ did, being reborn with probability $μ$ each time it dies. This rebirthed version of $\cal Y$ is a recurrent Borel right process with state space $S$ and $p$-potential densities of form, \[ u^p(x,y)+h(x,y),\qquad x,y\in S,\,\, p>0, \] where $h(x,y)$ is not symmetric. The local times of the rebirthed process are given in terms of the local times of $\cal Y$ and isomorphism theorems in the spirit of Dynkin, Eisenbam and Kaspi are obtained that relate these local times to generalized chi--square processes formed by Gaussian processes with covariances $u^{q}(x,y)$ for different values of $q$. These isomorphisms allow one to obtain exact local and uniform moduli of continuity for the local times of the rebirthed process. Several explicit examples are given in which $\cal Y$ is either a modified Lévy process or a diffusion. Analogous results are obtained for partially rebirthed versions of $\cal Y$. This is obtained by starting $\cal Y$ in $S$ and when it dies returning it to $S$ with a sub-probability measure $Ξ$. (With probability $1-|Ξ|$ it is sent to a disjoint state space $S'$, where it remains.)

math.PR↗
arXiv2024

Local moduli of continuity for permanental processes that are zero at zero

Let $u(s,t)$ be a continuous potential density of a symmetric Lévy process or diffusion with state space $T$ killed at $T_{0}$, the first hitting time of $0$, or at $λ\wedge T_{0}$, where $λ$ is an independent exponential time. Let \[ f(t)=\int_{T} u(t,v)\,dμ(v), \] where $μ$ is a finite positive measure on $T$. Let $X_α=\{X_α(t),t\in T \}$ be an $α-$permanental process with kernel \[ v(s,t)=u(s,t)+f(t). \] Then when $\lim_{t\to 0}u(t,t)=0$, \[ \limsup_{t\downarrow 0}\frac{X_α(t )}{u(t,t)\log \log 1/t }\ge 1 ,\qquad \text{a.s.} \] and \[ \limsup_{t\downarrow 0}\frac{X_α(t )}{u(t,t)\log \log 1/t }\le 1+C_{u,h} ,\qquad \text{a.s.} \] where $C_{u,μ}\le |μ|$ is a constant that depends on both $u$ and $μ$, which is given explicitly, and is different in the different examples.

math.PR↗
arXiv2023

Law of the iterated logarithm for $k/2$-permanental processes and the local times of related Markov processes

Let $Y$ be a symmetric Borel right process with locally compact state space $T\subseteq R^{1}$ and potential densities $u(x,y)$ with respect to some $σ$-finite measure on $T$. Let $g$ and $f$ be finite excessive functions for $ Y$. Set $$ u_{g, f}(x,y)= u(x,y)+g(x)f(y),\qquad x,y\in T.$$ In this paper we take $Y$ to be a symmetric Lévy process, or a diffusion, that is killed at the end of an independent exponential time or the first time it hits 0. Under general smoothness conditions on $g$, $f$, $u$ and points $d\in T$, laws of the iterated logarithm are found for $X_{k/2} =\{X_{k/2}(t), t\in T \}$, a $k/2-$permanental process with kernel $ \{u_{g, f}(x,y),x,y\in T \}$, of the following form: For all integers $k\geq 1$, $$\limsup_{x \to 0}\frac{| X_{k/2}( d+x)- X_{k/2} (d)|}{ \left( 2 σ^{2}\left(x\right)\log\log 1/x\right)^{1/2}}= \left( 2 X _{k/2} (d)\right)^{1/2}, \qquad a.s. ,$$ where, $$σ^2(x)=u(d+x,d+x)+u(x,x)-2u(d+x,x).$$ Using these limit theorems and the Eisenbaum Kaspi Isomorphism Theorem, laws of the iterated logarithm are found for the local times of certain Markov processes with potential densities that have the form of $ \{u_{g, f}(x,y),x,y\in T \}$ or are slight modifications of it.

math.PR↗
arXiv2022

Tightness for Thick Points in two dimensions

Let $W_{t}$ be Brownian motion in the plane started at the origin and let $ θ$ be the first exit time of the unit disk $D_{1}$. Let \[μ_{ θ} ( x,ε) =\frac{1}{πε^{ 2} }\int_{0}^{ θ}1_{\{ B( x,ε)\}}( W_{t})\,dt,\] and set $μ^{ \ast}_{ θ} (ε)=\sup_{x\in D_{1}}μ_{ θ} ( x,ε)$. We show that \[\sqrt{μ^{\ast}_θ (ε)}-\sqrt{2/π} \left(\log ε^{-1}- \frac{1}{2}\log\log ε^{-1}\right)\] is tight.

math.PR↗
arXiv2021

Exact moduli of continuity for general chi--square processes and for permanental processes related to the Ornstein--Uhlenbeck process

Let $ \overline B=\{ \overline B_{t},t\in R^{1} \}$ be Brownian motion killed after an independent exponential time with mean $2/λ^{2}$. The process $\overline B$ has potential densities, \[ u(x,y) ={e^{-λ|y-x|}\over λ},\qquad x,y\in R^{ 1}, \] which is also the covariance of an Ornstein--Uhlenbeck process. Let $f$ be an excessive function for $\overline B$. Then, \[ {e^{-λ|y-x|}\over λ}+f(y),\qquad x,y\in R^{ 1}, \] is the kernel of an $α$-permanental process $ X_α=\{ X_α(t), t\in R^{ 1}\}$ for all $α>0$. It is shown that for all $k\ge 1$ and intervals $Δ\subseteq [0,1] $, \[ \limsup_{h\to 0}\sup_{\stackrel{|u-v|\le h }{ u,v\inΔ}} \frac{|X_{k/2} (u)-X_{k/2} (v)|}{ 2 ( |u-v| \log 1/|u-v|)^{1/2}}= \sqrt 2 \sup_{t\inΔ}X_{k/2}^{1/2}(t)\qquad a.s.\] The local modulus of continuity of $X_{k/2}$ for all $k\ge 1$ is also obtained. Local and uniform moduli of continuity are also obtained for chi--square processes which are defined by, \[ Y_{k/2}(t)=\sum_{i=1}^{k}\frac{η^2_{i}(t)}{2},\qquad t\in [0,1], \] where $η=\{η(t);t\in [0,1]\}$ is a mean zero Gaussian process and $\{η_{i};i=1,\ldots, k\}$ are independent copies of $η.$

math.PR↗
arXiv2021

Local and uniform moduli of continuity of chi--square processes

Let $η=\{η(t);t\in [0,1]\}$ be a mean zero continuous Gaussian process with covariance $U=\{U(s,t),s,t\in [ 0,1]\},$ with $U(0,0)>0$. Let $\{η_{i};i=1,\ldots, k\}$ be independent copies of $η$ and set $ Y_{k}(t)=\sum_{i=1}^{k} η^2_{i}(t), t\in [ 0,1].$ The stochastic process $Y_{k } =\{Y_{k }(t),t\in [ 0,1] \}$ is referred to as a chi--square process of order $k $ with kernel $U$. Let $ϕ(t)$ be a positive function on $[0,δ]$ for some $δ>0$. If \[\limsup_{t\to 0}\frac{ η(t)-η(0)}{ ϕ(t) }=1 \qquad a.s., \] then for all integers $k\ge 1$, \[ \limsup_{t\to 0} \frac{Y_{k }(t)-Y_{k }(0)} { ϕ(t)} = 2 Y^{1/2}_{k}(0) \qquad a.s.\] Set \[ σ^2(u,v)=E(η(u)-η(v))^2\quad\text{and}\quad \widetildeσ^2(x)=\sup_{|u-v|\le x}σ^2(u,v).\] Assume that $\inf_{t\in [0,1]}U(t,t)>0$ and, \[ \lim_{x\to0}\widetildeσ^2(x)\log 1/x =0. \] Let $φ(t)$ be a positive function on $[0,1]$. Then if \[ \lim_{h\to 0}\sup_{\stackrel{|u-v|\le h }{ u,v\inΔ}}\frac{ η(u)-η(v)}{ φ(|u-v|) }=1 \qquad a.s.\] for all intervals $Δ\subset [0,1]$, it follows that for all intervals $Δ\subset [0,1]$ and all integers $k\ge 1$, \[ \lim_{h\to 0}\sup_{\stackrel{|u-v|\le h }{ u,v\inΔ}} \frac{Y_{k }(u)-Y_{k }(v) }{ φ(|u-v|)} = 2 \sup_{u\inΔ}Y_{k }^{1/2}(u), \hspace{.2 in}a.s.\]

math.PR↗
arXiv2020

Permanental sequences that are related to a Markov chain example of Kolmogorov

Permanental sequences with non-symmetric kernels that are generalization of the potentials of a Markov chain with state space $\{0,1/2, \ldots, 1/n,\ldots\}$ that was introduced by Kolmogorov, are studied. Depending on a parameter in the kernels we obtain an exact rate of divergence of the sequence at $0$, an exact local modulus of continuity of the sequence at $0$, or a precise bounded discontinuity for the sequence at $0$.

math.PR↗
arXiv2020

Tightness for the Cover Time of the two dimensional sphere

Let $C^*_{ε,S^2}$ denote the cover time of the two dimensional sphere by a Wiener sausage of radius $ε$. We prove that $$\sqrt{C^{*}_{ε,S^2} } -\sqrt{\frac{2A_{S^2}}π}(\log ε^{-1}-\frac14\log\log ε^{-1})$$ is tight, where $A_{S^2}=4π$ denotes the Riemannian area of $S^2$.

math.PR↗
arXiv2019

Asymptotic properties of permanental sequences

Let $U=\{U_{j,k},j,k\in \overline {\mathbb N}\}$ be the potential of a transient symmetric Borel right process $X$ with state space $\overline {\mathbb N}$. For any excessive function $f=\{f_{k,k\in \overline {\mathbb N}}\}$ for $X$ , $\widetilde U=\{\widetilde U_{j,k},j,k\in\overline {\mathbb N}\}$, where \begin{equation} \widetilde U_{j,k}= U_{j,k} +f_{ k},\qquad j,k\in\overline {\mathbb N},\label{a.1} \end{equation} is the kernel of an $α$-permanental sequence $\widetilde X_α=(\widetilde X_{α, 1} ,\ldots)$ for all $α>0$. The symmetric potential $U$ is also the covariance of a mean zero Gaussian sequence $η=\{η_{j},j\in \overline {\mathbb N}\}$. Conditions are given on the potentials $U$ and excessive functions $f$ under which, \begin{equation} \limsup_{j\to \infty}\frac{ η_{j}}{( 2\,ϕ_{j})^{1/2} }=1 \quad a.s. \quad \implies \quad \limsup_{n\to \infty}\frac{\widetilde X_{α, j}}{ϕ_{j} }=1\quad a.s.,\label{a.2} \end{equation} for all $α>0$, and sequences $ϕ=\{ϕ_{j}\}$ such that $f_{j}=o(ϕ_{j})$. The function $ϕ$ is determined by $U$. Many examples are given in which $U$ is the potential of symmetric birth and death processes with and without emigration, first and higher order Gaussian autoregressive sequences and Lévy processes on $\mathbf Z$.

math.PR↗
arXiv2019

Limit law for the cover time of a random walk on a binary tree

Let $T_n$ denote the binary tree of depth $n$ augmented by an extra edge connected to its root. Let $C_n$ denote the cover time of $T_n$ by simple random walk. We prove that $\sqrt{ \mathcal{C}_{n} 2^{-(n+1) } } - m_n$ converges in distribution as $n\to \infty$, where $m_n$ is an explicit constant, and identify the limit.

math.PR↗
arXiv2018

Permanental processes with kernels that are not equivalent to a symmetric matrix

Kernels of $α$-permanental processes of the form \[ v(x,y)=u(x,y)+f(y),\qquad x,y\in S, \] in which $u(x,y)$ is symmetric, and $f$ is an excessive function for the Borel right process with potential densities $u(x,y)$, are considered. Conditions are given that determine whether $\{v(x,y);x,y\in S\}$ is symmetrizable or asymptotically symmetrizable.

math.PR↗
arXiv2017

Sample path properties of permanental processes

Let $X_α=\{X_α(t),t\in T\}$, $α>0$, be an $α$-permanental process with kernel $u(s,t)$. We show that $X^{1/2}_α$ is a subgaussian process with respect to the metric $σ(s,t)= (u(s,s)+u(t,t)-2(u(s,t)u(t,s))^{1/2})^{1/2}$. This allows us to use the vast literature on sample path properties of subgaussian processes to extend these properties to $α$-permanental processes. Local and uniform moduli of continuity are obtained as well as the behavior of the processes at infinity. Examples are given of permanental processes with kernels that are the potential density of transient Lévy processes that are not necessarily symmetric, or with kernels of the form $ \hat u(x,y)= u(x,y)+f(y)$, where $u$ is the potential density of a symmetric transient Borel right process and $f$ is an excessive function for the process.

math.PR↗
arXiv2017

Barrier estimates for a critical Galton--Watson process and the cover time of the binary tree

For the critical Galton--Watson process with geometric offspring distributions we provide sharp barrier estimates for barriers which are (small) perturbations of linear barriers. These are useful in analyzing the cover time of finite graphs in the critical regime by random walk, and the Brownian cover times of compact two dimensional manifolds. As an application of the barrier estimates, we prove that if $C_L$ denotes the cover time of the binary tree of depth $L$ by simple walk, then $\sqrt{C_L/2^{L+1}} -\sqrt{2\log 2} L+\log L/\sqrt{2\log 2}$ is tight. The latter improves results of Aldous (1991), Bramson and Zeitouni (2009) and Ding and Zeitouni (2012). In a subsequent article we use these barrier estimates to prove tightness of the Brownian cover time for the two-dimensional sphere.

math.PR↗
arXiv2015

Conditions for permanental processes to be unbounded

An $\al$-permanental process $\{X_{ t},t\in T \}$ is a stochastic process determined by a kernel $K=\{K(s,t),s,t\in T \}$, with the property that for all $t_{1},\ldots,t_{n}\in T $, $ |I+K( t_{1},\ldots,t_{n}) S|^{- \al} $ is the Laplace transform of $(X_{t_{1}},\ldots,X_{t_{n}})$, where $ K( t_{1},\ldots,t_{n})$ denotes the matrix $\{K(t_{i}, t_{j})\}_{i,j=1}^{n}$ and $S$ is the diagonal matrix with entries $s_{1},\ldots,s_{n} $. $ (X_{t_{1}},\ldots,X_{t_{n}})$ is called a permanental vector. Under the condition that $K$ is the potential density of a transient Markov process, $(X_{t_{1}},\ldots,X_{t_{n}})$ is represented as a random mixture of $n$-dimensional random variables with components that are independent gamma random variables. This representation leads to a Sudakov type inequality for the sup-norm of $(X_{t_{1}},\ldots,X_{t_{n}})$ that is used to obtain sufficient conditions for a large class of permanental processes to be unbounded almost surely. These results are used to obtain conditions for permanental processes associated with certain Lévy processes to be unbounded. Because $K$ is the potential density of a transient Markov process, for all $t_{1},\ldots,t_{n}\in T $, $A( t_{1},\ldots,t_{n}):= (K( t_{1},\ldots,t_{n}))^{-1}$ are $M$-matrices. The results in this paper are obtained by working with these $M$-matrices.

math.PR↗