Renormalized self-intersection local times and Wick power chaos processes
著者
書誌事項
Renormalized self-intersection local times and Wick power chaos processes
（Memoirs of the American Mathematical Society, no. 675）
American Mathematical Society, 1999
大学図書館所蔵 全19件
注記
"November 1999, volume 142, number 675 (first of 4 numbers)" -- T.p
Includes bibliographical references (p. 124-125)
内容説明・目次
内容説明
Sufficient conditions are obtained for the continuity of renormalized self-intersection local times for the multiple intersections of a large class of strongly symmetric Levy processes in $R^m$, $m=1,2$. In $R^2$ these include Brownian motion and stable processes of index greater than 3/2, as well as many processes in their domains of attraction. In $R^1$ these include stable processes of index $3/4<\beta\le 1$ and many processes in their domains of attraction.Let $(\Omega, \mathcal F(t),X(t), P^{x})$ be one of these radially symmetric Levy processes with 1-potential density $u^1(x,y)$. Let $\mathcal G^{2n}$ denote the class of positive finite measures $\mu$ on $R^m$ for which $\int\!\!\int (u^1(x,y))^{2n}\,d\mu(x)\,d\mu(y)<\infty$. For $\mu\in\mathcal G^{2n}$, let $\alpha_{n,\epsilon}(\mu,\lambda) \overset\text{def}\to=\int\!\!\int_{\{0\leq t_1\leq \cdots \leq t_n\leq \lambda\}} f_{\epsilon}(X(t_1)-x)\prod_{j=2}^n f_{\epsilon}(X(t_j)- X(t_{j-1}))\,dt_1\cdots\,dt_n\,d\mu(x)$ where $f_{\epsilon}$ is an approximate $\delta-$function at zero and $\lambda$ is an random exponential time, with mean one, independent of $X$, with probability measure $P_\lambda$.The renormalized self-intersection local time of $X$ with respect to the measure $\mu$ is defined as $\gamma_{n}(\mu)=\lim_{\epsilon\to 0}\,\sum_{k=0}^{n-1}(-1)^{k} {n-1 \choose k}(u^1_{\epsilon}(0))^{k} \alpha_{n-k,\epsilon}(\mu,\lambda)$ where $u^1_{\epsilon}(x)\overset\text{def}\to= \int f_{\epsilon}(x-y)u^1(y)\,dy$, with $u^1(x)\overset\text{def} \to= u^1(x+z,z)$ for all $z\in R^m$. Conditions are obtained under which this limit exists in $L^2(\Omega\times R^+,P^y_\lambda)$ for all $y\in R^m$, where $P^y_\lambda\overset\text{def}\to= P^y\times P_\lambda$. Let $\{\mu_x,x\in R^m\}$ denote the set of translates of the measure $\mu$.The main result in this paper is a sufficient condition for the continuity of $\{\gamma_{n}(\mu_x),\,x\in R^m\}$ namely that this process is continuous $P^y_\lambda$ almost surely for all $y\in R^m$, if the corresponding 2$n$-th Wick power chaos process, $\{:G^{2n}\mu_x:,\,x\in R^m\}$ is continuous almost surely. This chaos process is obtained in the following way.A Gaussian process $G_{x,\delta}$ is defined which has covariance $u^1_\delta(x,y)$, where $\lim_{\delta\to 0}u_\delta^1(x,y)=u^1(x,y)$. Then $:G^{2n}\mu_x:\overset\text{def}\to= \lim_{\delta\to 0}\int:G_{y,\delta}^{2n}:\,d\mu_x(y)$ where the limit is taken in $L^2$. ($:G_{y,\delta}^{2n}:$ is the 2$n$-th Wick power of $G_{y,\delta}$, that is, a normalized Hermite polynomial of degree 2$n$ in $G_{y,\delta}$). This process has a natural metric $d(x,y)\overset\text{def}\to= \frac1 {(2n)!}\(E(:G^{2n}\mu_x:-:G^{2n}\mu_y:)^2\)^{1/2} =\(\int\!\! \int \(u^1(u,v)\)^{2n} \left(d(\mu_x(u)-\mu_y(u)) \right) \left(d(\mu_x(v)-\mu_y(v)) \right)\)^{1/2}$. A well known metric entropy condition with respect to $d$ gives a sufficient condition for the continuity of $\{:G^{2n}\mu_x:,\,x\in R^m\}$ and hence for $\{\gamma_{n}(\mu_x),\,x\in R^m\}$.
目次
Introduction Wick products Wick power chaos processes Isomorphism theorems Equivalence of two versions of renormalized self-intersection local times Continuity Stable mixtures Examples A large deviation result Appendix A. Necessary conditions Appendix B. The case $n=3$ Bibliography.
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