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Michael B. Marcus

Publications and source records attributed to Michael B. Marcus.

At least 19 recordsLinked to original sources
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↗
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↗
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↗
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↗
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↗
arXiv2015

Permanental fields, loop soups and continuous additive functionals

A permanental field, $ψ=\{ψ(ν),ν\in {\mathcal{V}}\}$, is a particular stochastic process indexed by a space of measures on a set $S$. It is determined by a kernel $u(x,y)$, $x,y\in S$, that need not be symmetric and is allowed to be infinite on the diagonal. We show that these fields exist when $u(x,y)$ is a potential density of a transient Markov process $X$ in $S$. A permanental field $ψ$ can be realized as the limit of a renormalized sum of continuous additive functionals determined by a loop soup of $X$, which we carefully construct. A Dynkin-type isomorphism theorem is obtained that relates $ψ$ to continuous additive functionals of $X$ (continuous in $t$), $L=\{L_t^ν,(ν,t)\in {\mathcal{V}}\times R_+\}$. Sufficient conditions are obtained for the continuity of $L$ on ${\mathcal{V}}\times R_+$. The metric on ${\mathcal{V}}$ is given by a proper norm.

math.PR↗
arXiv2013

Intersection local times, loop soups and permanental Wick powers

Several stochastic processes related to transient Lévy processes with potential densities $u(x,y)=u(y-x)$, that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures $V$ endowed with a metric $d$. Sufficient conditions are obtained for the continuity of these processes on $(V,d)$. The processes include $n$-fold self-intersection local times of transient Lévy processes and permanental chaoses, which are `loop soup $n$-fold self-intersection local times' constructed from the loop soup of the Lévy process. Loop soups are also used to define permanental Wick powers, which generalizes standard Wick powers, a class of $n$-th order Gaussian chaoses. Dynkin type isomorphism theorems are obtained that relate the various processes. Poisson chaos processes are defined and permanental Wick powers are shown to have a Poisson chaos decomposition. Additional properties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.

math.PR↗
arXiv2013

A sufficient condition for the continuity of permanental processes with applications to local times of Markov processes

We provide a sufficient condition for the continuity of real valued permanental processes. When applied to the subclass of permanental processes which consists of squares of Gaussian processes, we obtain the sufficient condition for continuity which is also known to be necessary. Using an isomorphism theorem of Eisenbaum and Kaspi which relates Markov local times and permanental processes, we obtain a general sufficient condition for the joint continuity of local times.

math.PR↗
arXiv2011

Permanental Vectors

A permanental vector is a generalization of a vector with components that are squares of the components of a Gaussian vector, in the sense that the matrix that appears in the Laplace transform of the vector of Gaussian squares is not required to be either symmetric or positive definite. In addition the power of the determinant in the Laplace transform of the vector of Gaussian squares, which is -1/2, is allowed to be any number less than zero. It was not at all clear what vectors are permanental vectors. In this paper we characterize all permanental vectors in $R^{3}_{+}$ and give applications to permanental vectors in $R^{n}_{+}$ and to the study of permanental processes.

math.PR↗
arXiv2010

Permanental Processes

This is a survey of results about permanental processes, real valued positive processes which are a generalization of squares of Gaussian processes. In a certain sense the symmetric positive definite function that determines a Gaussian process is replaced by a function that is not necessarily symmetric nor positive definite, but that nevertheless determines a stochastic process. This is a new avenue of research with very many open problems.

math.PR↗
arXiv2009

An almost sure limit theorem for Wick powers of Gaussian differences quotients

Let G={G(x), x\in R_+}, G(0)=0, be a mean zero Gaussian process with $E(G(x)-G(y))^2=σ^2(x-y) $. Let $ ρ(x)= \frac12{d^{2}\over dx^2}σ^2(x)$, $x\ne 0 $. When $ρ^{k}$ is integrable at zero and satisfies some additional regularity conditions, \[ \lim_{h\downarrow 0} \int :(\frac{G(x+h)-G(x)}{h})^{k}:g(x) dx=:(G') ^{k}:(g){.3 in}a.s. \] for all $g\in B_{0}(R^{+})$, the set of bounded Lebesgue measurable functions on $R_+$ with compact support. Here $G'$ is a generalized derivative of $G$ and $:(\cd)^{k}:$ is the $k$--th order Wick power.

math.PR↗
arXiv2009

A CLT for the $L^{2}$ norm of increments of local times of Lévy processes as time goes to infinity

Let $X=\{X_{t},t\in R_{+}\}$ be a symmetric Lévy process with local time $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$. When the Lévy exponent $ψ(\la)$ is regularly varying at zero with index $1<β\leq 2$, and satisfies some additional regularity conditions, \begin{eqnarray*} && {\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- E(\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx)\over t\sqrt{ψ^{-1}(1/t)}}\label{r5.0tweaksabs} && \stackrel{\mathcal{L}}{\Longrightarrow}(8c_{ψ,1 })^{1/2}(\int_{-\infty}^{\finfty} (L_{β,1}^{x})^{2} dx)^{1/2} η\end{eqnarray*} as $t\rar\infty$, where $L_{\bb,1}=\{L^{x}_{β, 1} ; x \in R^{1} \}$ denotes the local time, at time 1, of a symmetric stable process with index $β$, $η$ is a normal random variable with mean zero and variance one that is independent of $L_{β,1}$, and $c_{ψ,1}$ is a known constant that depends on $ψ$.

math.PR↗