Regularized Self-Intersection Local Times of Planar Brownian Motion

Abstract

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 $t_1 < \cdots < t_k$.