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Jay Rosen

Publications and source records attributed to Jay Rosen.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2014

Lectures on Isomorphism Theorems

These notes originated in a series of lectures I gave in Marseille in May, 2013. I was invited to give an introduction to the isomorphism theorems, originating with Dynkin, which connect Markov local times and Gaussian processes. This is an area I had worked on some time ago, and even written a book about, but had then moved on to other things. However, isomorphism theorems have become of interest once again, both because of new applications to the study of cover times of graphs and Gaussian fields, and because of new isomorphism theorems for non-symmetric Markov processes and their connection with loop soups and Poisson processes. Thus I felt the time was ripe for a new introduction to this topic. In these notes I have tried to preserve the informal atmosphere of the lectures, and often simply refer the reader to my book and other sources for details.

math.PR↗
arXiv2014

Perturbation of the loop measure

The loop measure is associated with a Markov generator. We compute the variation of the loop measure induced by an in nitesimal variation of the generator a ecting the killing rates or the jumping rates.

math.PR↗
arXiv2014

Markovian loop soups: permanental processes and isomorphism theorems

We construct loop soups for general Markov processes without transition densities and show that the associated permanental process is equal in distribution to the loop soup local time. This is used to establish isomorphism theorems connecting the local time of the original process with the associated permanental process. Further properties of the loop measure are studied.

math.PR↗
arXiv2014

Intersection local times for interlacements

We define renormalized intersection local times for random interlacements of Lévy processes in R^{d} and prove an isomorphism theorem relating renormalized intersection local times with associated Wick polynomials.

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↗
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

A CLT for the third integrated moment of Brownian local time increments

Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of Brownian motion. Our main result is to show that for each fixed $t$ $${\int (L^{x+h}_t- L^x_t)^3 dx-12h\int (L^{x+h}_t - L^x_t)L^x_t dx-24h^{2}t\over h^2} \stackrel{\mathcal{L}}{\Longrightarrow}\sqrt{192}(\int (L^x_t)^3dx)^{1/2}η$$ as $h\to 0$, where $η$ is a normal random variable with mean zero and variance one that is independent of $L^{x}_{t}$. This generalizes our previous result for the second moment. We also explain why our approach will not work for higher moments

math.PR↗
arXiv2009

Large deviations and renormalization for Riesz potentials of stable intersection measures

We study the object formally defined as γ\big([0,t]^{2}\big)=\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-σ} dr ds-E\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-σ} dr ds, where $X_{t}$ is the symmetric stable processes of index $0<β\le 2$ in $R^{d}$. When $β\leσ<\displaystyle\min \Big\{{3\over 2}β, d\Big\}$, this has to be defined as a limit, in the spirit of renormalized self-intersection local time. We obtain results about the large deviations and laws of the iterated logarithm for $γ$. This is applied to obtain results about stable processes in random potentials.

math.PR↗
arXiv2009

Asymptotic expansions for functions of the increments of certain Gaussian processes

Let $G=\{G(x),x\ge 0\}$ be a mean zero Gaussian process with stationary increments and set $σ^2(|x-y|)= E(G(x)-G(y))^2$. Let $f$ be a function with $Ef^{2}(η)<\ff$, where $η=N(0,1)$. When $σ^2$ is regularly varying at zero and \[ \lim_{h\to 0}{h^2\over σ^2(h)}= 0\qquad {and}\qquad \lim_{h\to 0}{σ^2(h)\over h}= 0 \quad {but} \quad ({d^{2}\over ds^2}σ^2(s))^{j_0} \] is locally integrable for some integer $j_0\ge 1$, and satisfies some additional regularity conditions, \bea && \int_a^bf(\frac{G(x+h)-G(x)}{σ(h)}) dx \label{abst}\nn &&\qquad = \sum_{j=0}^{j_0} (h/σ(h))^{j} {E(H_{j}(η) f(η))\over\sqrt {j!}} :(G')^{j}:(I_{[a,b]}) +o({h\overσ(h)})^{j_0}\nn \eea in $L^2$. Here $H_j$ is the $j$-th Hermite polynomial. Also $:(G')^{j}:(I_{[a,b]})$ is a $j $-th order Wick power Gaussian chaos constructed from the Gaussian field $ G'(g) $, with covariance \[ E(G'(g)G'(\wt g)) = \int \int ρ(x-y)g(x)\wt g(y) dx dy\label{3.7bqs}, \] where $ ρ(s)={1/2}{d^{2}\over ds^2}σ^2(s)$.

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↗
arXiv2009

A CLT for the $L^{2}$ moduli of continuity of local times of Levy processes

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 infinity with index $1<β\leq 2$ and satisfies some additional regularity conditions && \sqrt{hψ^{2}(1/h)} \lc \int (L^{x+h}_{1}- L^{x}_{1})^{2} dx- E(\int (L^{x+h}_{1}- L^{x}_{1})^{2} dx)\rc\nn && {1 in} \stackrel{\mathcal{L}}{\Longrightarrow} (8c_{β,1})^{1/2} η(\int (L_{1}^{x})^{2} dx)^{1/2} \nn, as $h\rar 0$, where $η$ is a normal random variable with mean zero and variance one that is independent of $L^{x}_{t}$, and $c_{β,1}$ is a known constant.

math.PR↗
arXiv2009

A CLT for the L^{2} modulus of continuity of Brownian local time

Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of Brownian motion and \[ α_{t}:=\int_{-\infty}^{\infty} (L^{x}_{t})^{2} dx . \] Let $η=N(0,1)$ be independent of $α_{t}$. For each fixed $t$ \[ {\int_{-\infty}^{\infty} (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}} \stackrel{\mathcal{L}}{\to}({64 \over 3})^{1/2}\sqrt{α_{t}} η, \] as $h\rar 0$. Equivalently \[ {\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- 4t\over t^{3/4}} \stackrel{\mathcal{L}}{\to}({64 \over 3} )^{1/2}\sqrt{α_{1}} η, \] as $t\rar\infty$.

math.PR↗
arXiv2008

Existence of a critical point for the infinite divisibility of squares of Gaussian vectors in $R^{2}$ with non--zero mean

Let $G=(G_{1},G_{2})$ be a Gaussian vector in $R^{2}$ with $EG_{1}G_{2}\neq 0$. Let $c_{1},c_{2}\in R^{1}$. A necessary and sufficient condition for $G=((G_{1}+c_{1}α)^{2},(G_{2}+c_{2}α)^{2})$ to be infinitely divisible for all $α\in R^{1}$ is that \[ \Ga_{i,i}\geq \frac{c_{i}}{c_{j}}\Ga_{i,j}>0\qquad\forall 1\le i\ne j\le 2.\] In this paper we show that when this does not hold there exists an $0<α_{0}<\ff $ such that $G=((G_{1}+c_{1}α)^{2},(G_{2}+c_{2}α)^{2})$ is infinitely divisible for all $|α|\leq α_{0}$ but not for any $|\al|>\al_{0}$.

math.PR↗