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January, 1988 Regularized Self-Intersection Local Times of Planar Brownian Motion
E. B. Dynkin
Ann. Probab. 16(1): 58-74 (January, 1988). DOI: 10.1214/aop/1176991885


Let $T^\varepsilon_k(\lambda; t_1,\ldots,t_k) = \rho(X_{t_1})q^\varepsilon(X_{t_2} - X_{t_1}) \cdots q^\varepsilon(X_{t_k} - X_{t_k - 1}),$ where $X_t$ is a Brownian motion in $\mathbb{R}^2, \lambda(dx) = \rho(x) dx$ and $q^\varepsilon$ converges to Dirac's delta function as $\varepsilon \downarrow 0$. The self-intersection local times of order $k$ are described by a generalized random field $T_k(\lambda; t_1,\ldots,t_k) = \lim_{\varepsilon\downarrow 0} T^\varepsilon_k(\lambda; t_1,\ldots,t_k), \quad\text{for} 0 < t_1 < \cdots < t_k.$ The field "blows up" as $t_i - t_j \rightarrow 0$ for some $i \neq j$. We show that with a proper choice of the coefficients $B^l_k(\varepsilon)$, a generalized random field $\mathscr{J}_k(\lambda; t_1,\ldots,t_k) = \lim_{\varepsilon\downarrow 0}\big\lbrack T^\varepsilon_k(\lambda; t_1,\ldots,t_k) + \sum^{k - 1}_{l = 1}\lbrack B^l_k(\varepsilon)T^\varepsilon_l\rbrack(\lambda; t_1,\ldots,t_k)\big\rbrack$ is well defined for all $0 \leq t_1 \leq \cdots \leq t_k$ and it coincides with $T_k(\lambda; t_1,\ldots,t_k)$ for $t_1 < \cdots < t_k$.


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E. B. Dynkin. "Regularized Self-Intersection Local Times of Planar Brownian Motion." Ann. Probab. 16 (1) 58 - 74, January, 1988.


Published: January, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0641.60085
MathSciNet: MR920255
Digital Object Identifier: 10.1214/aop/1176991885

Primary: 60G60
Secondary: 60J55 , 60J65

Keywords: Local times , moment functions of random fields , multiple points of the Brownian motion , regularization of generalized functions

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 1 • January, 1988
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