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A CLT for the third integrated moment of Brownian local time increments
Jay Rosen
Mathematics , 2009,
Abstract: 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 $t$ $${\int (L^{x+h}_t- L^x_t)^3 dx-12h\int (L^{x+h}_t - L^x_t)L^x_t dx-24h^{2}t\over h^2} \stackrel{\mathcal{L}}{\Longrightarrow}\sqrt{192}(\int (L^x_t)^3dx)^{1/2}\eta$$ as $h\to 0$, where $\eta$ 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
Lectures on Isomorphism Theorems
Jay Rosen
Mathematics , 2014,
Abstract: 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.
A random walk proof of the Erdos-Taylor conjecture
Jay Rosen
Mathematics , 2005,
Abstract: For the simple random walk in Z^2 we study those points which are visited an unusually large number of times, and provide a new proof of the Erdos-Taylor conjecture describing the number of visits to the most visited point.
A stochastic calculus proof of the CLT for the L^{2} modulus of continuity of local time
Jay Rosen
Mathematics , 2009,
Abstract: We give a stochastic calculus proof of the Central Limit Theorem \[ {\int (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}} \stackrel{\mathcal{L}}{\Longrightarrow}c(\int (L^{x}_{t})^{2} dx)^{1/2} \eta\] as $h\to 0$ for Brownian local time $L^{x}_{t}$. Here $\eta$ is an independent normal random variable with mean zero and variance one.
Intersection local times for interlacements
Jay Rosen
Mathematics , 2013,
Abstract: We define renormalized intersection local times for random interlacements of L\'evy processes in R^{d} and prove an isomorphism theorem relating renormalized intersection local times with associated Wick polynomials.
Asymptotic expansions for functions of the increments of certain Gaussian processes
Michael Marcus,Jay Rosen
Mathematics , 2007,
Abstract: Let $G=\{G(x),x\ge 0\}$ be a mean zero Gaussian process with stationary increments and set $\sigma^2(|x-y|)= E(G(x)-G(y))^2$. Let $f$ be a function with $Ef^{2}(\eta)<\ff$, where $\eta=N(0,1)$. When $\sigma^2$ is regularly varying at zero and \[ \lim_{h\to 0}{h^2\over \sigma^2(h)}= 0\qquad {and}\qquad \lim_{h\to 0}{\sigma^2(h)\over h}= 0 \quad {but} \quad ({d^{2}\over ds^2}\sigma^2(s))^{j_0} \] 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)}{\sigma (h)}) dx \label{abst}\nn &&\qquad = \sum_{j=0}^{j_0} (h/\sigma(h))^{j} {E(H_{j}(\eta) f(\eta))\over\sqrt {j!}} :(G')^{j}:(I_{[a,b]}) +o({h\over\sigma (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 \[ E(G'(g)G'(\wt g)) = \int \int \rho (x-y)g(x)\wt g(y) dx dy\label{3.7bqs}, \] where $ \rho(s)={1/2}{d^{2}\over ds^2}\sigma^2(s)$.
Continuous Differentiability of Renormalized Intersection Local Times in R^{1}
Jay S. Rosen
Mathematics , 2009, DOI: 10.1214/09-AIHP338
Abstract: We study $\gamma_{k}(x_2,...,x_k;t)$, the k-fold renormalized self-intersection local time for Brownian motion in $R^1$. Our main result says that $\gamma_{k}(x_2,...,x_k;t)$ is continuously differentiable in the spatial variables, with probability 1.
Large deviations and renormalization for Riesz potentials of stable intersection measures
Xia Chen,Jay Rosen
Mathematics , 2009,
Abstract: We study the object formally defined as \gamma\big([0,t]^{2}\big)=\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-\sigma} dr ds-E\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-\sigma} dr ds, where $X_{t}$ is the symmetric stable processes of index $0<\beta\le 2$ in $R^{d}$. When $\beta\le\sigma<\displaystyle\min \Big\{{3\over 2}\beta, d\Big\}$, 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 $\gamma$. This is applied to obtain results about stable processes in random potentials.
An almost sure invariance principle for renormalized intersection local times
Richard F. Bass,Jay Rosen
Mathematics , 2004,
Abstract: Let \beta_k(n) be the number of self-intersections of order k, appropriately renormalized, for a mean zero random walk X_n in Z^2 with 2+\delta moments. On a suitable probability space we can construct X_n and a planar Brownian motion W_t such that for each k\geq 2, |\beta_k(n)-\gamma_k(n)|=O(n^{-a}), a.s. for some a>0 where \gamma_k(n) is the renormalized self-intersection local time of order k at time 1 for the Brownian motion W_{nt}/\sqrt n.
A CLT for the $L^{2}$ moduli of continuity of local times of Levy processes
Michael B. Marcus,Jay Rosen
Mathematics , 2009,
Abstract: Let $X=\{X_{t},t\in R_{+}\}$ be a symmetric L\'evy process with local time $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$. When the L\'evy exponent $\psi(\la)$ is regularly varying at infinity with index $1<\beta\leq 2$ and satisfies some additional regularity conditions && \sqrt{h\psi^{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_{\beta,1})^{1/2} \eta (\int (L_{1}^{x})^{2} dx)^{1/2} \nn, as $h\rar 0$, where $\eta$ is a normal random variable with mean zero and variance one that is independent of $L^{x}_{t}$, and $c_{\beta,1}$ is a known constant.
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