%instruction % Tex-Version: TeX, Version 3.1415 (C version 6.1) % %BODY \documentclass[12pt]{article} \usepackage{amssymb} \usepackage{amsfonts} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \def\stackunder#1#2{\mathrel{\mathop{#2}\limits_{#1}}} \def\dint{\mathop{\displaystyle \int}}% \def\QATOPD#1#2#3#4{{#3 \atopwithdelims#1#2 #4}}% \def\textrm#1{{\rm #1}} \def\text#1{\textrm#1} \begin{document} \title{The Renormalization of Self Intersection Local Times\\ I: The Chaos Expansion} \author{Margarida de Faria$^{*}$ \and Cust\'{o}dia Drumond$^{*}$ \and Ludwig Streit$% ^{*+}$ \\ %EndAName $^{*}$CCM, Universidade da Madeira, P 9000 Funchal\\ http://www.uma.pt/\\ $^{+}$BiBoS, Univ. Bielefeld, D 33615 Bielefeld} \date{February, 1998} \maketitle \section{Introduction} The intersections of Brownian motion paths have been investigated since the Forties \cite{levy}. One can consider intersections of sample paths with themselves or e.g. with other, independent Brownian motions \cite{wolp}, one can study simple \cite{dvor2} or n-fold intersections \cite{dvor3} and one can ask all of these questions for linear, planar, spatial or - in general - d-dimensional Brownian motion: evidently self-intersections become increasingly scarce as the dimension d increases. Intersection local times of Brownian motion were studied by many authors, see e.g. \cite{bass}, \cite{II}-\cite{gem}, \cite{imke}, \cite{legall}-\cite {yor2}. A more systematic review than we can give here of the subject will be found e.g. in the recent reference \cite{imke}. An informal but rather suggestive definition of self intersection local time of Brownian motion $B$ is in terms of an integral over Dirac's - or Donsker's - $\delta $-function \[ L\equiv \int d^2t\,\delta (B(t_2)-B(t_1)), \] intended to sum up the contributions from each pair of ''times'' $t_1,t_2$ for which the Brownian motion $B$ is at the same point. In Edwards' modeling of polymer molecules by Brownian motion paths, $L$ is used to model the ''excluded volume'' effect: different parts of the molecule should not be located at the same point in space. As another application, Symanzik introduced $L$ as a tool for constructive quantum field theory in \cite{sym}. A rigorous definition, such as e.g. through a sequence of Gaussians approximating the $\delta $-function or in terms of generalized Brownian functionals \cite{hiwa} \cite{16} \cite{watanabe}, will lead to increasingly singular objects and will necessitate various ''renormalizations'' as the dimension d increases. For $d>1$ the expectation will diverge in the limit and must be subtracted \cite{legall} \cite{varadhan}, clearly $L$ will then no more be positive. For $d>3,5,7,\ldots $ further subtractions have been proposed \cite{watanabe} that will make $L$ into a well defined generalized function of Brownian motion. For $d=3$ another renormalization has been constructed by Westwater to make the Gibbs factor $e^{-g\cdot L}$ of the polymer model well-defined \cite{west}, for yet another, recent approach see \cite{bass}. Yor, in \cite{yor2}, first suppresses the short time accumulation of self-intersections by the regularization \[ \delta (\overrightarrow{B}(t_2)-\overrightarrow{B}(t_1))\rightarrow \delta (% \overrightarrow{B}(t_2)-\overrightarrow{B}(t_1)+\overrightarrow{\varepsilon }% ) \] and shows, again for $d=3,$ that a multiplicative renormalization \[ r(\varepsilon )\left( L_{\!\,\varepsilon }-E(L_{\!\,\varepsilon })\right) \] gives rise to another, independent Brownian motion as the weak limit of regularized and subtracted approximations to $L.$ In the present note we study similar limits, for arbitrary $d\geq 3,$ using a Gaussian regularization of the $\delta $-function for which the chaos expansion of the corresponding regularized $L_{\!\,\varepsilon }$ is available \cite{hiwa}. For a suitably subtracted and renormalized local time, each term in this expansion converges in law to a Brownian motion$.$ We prepare and state these results in the following section 2, in section 3 we give their proofs. In a companion paper \cite{II} we extend these results to the corresponding series. \section{Definitions and Main Results} \subsection{White Noise Analysis and Local Times} We reproduce here some White Noise Analysis concepts as introduced in \cite {hiwa}, referring to \cite{8} for a systematic presentation. Brownian motions $B_i,i=1,...,d,$ have version in terms of white noise $% \omega _i$ via \[ B_i(t)=<\omega _i,1_{[0,t]}>=\int_0^t\omega _i(s)ds. \] Hence {we consider independent d-tuples of Gaussian white noise }$\mathbf{% \omega }=(\omega _1,...,\omega _d)$ and correspondingly, $d$-tuples of test functions $\mathbf{f}=(f_1,...,f_d)\in S(R,R^d),\ $and introduce the following notation: \[ \stackrel{\rightarrow }{n}=(n_1,\ldots ,n_d),\;\;n=\sum_1^dn_i,\;\;\stackrel{% \rightarrow }{n}!=\prod_1^dn_i! \] \[ <\mathbf{f,f>}=\sum_{i=1}^d\int dt\ f_i^2(t) \] \[ =\int d^nt\ F_{\stackrel{\rightarrow }{n}}(t_1,,...,t_n)\;\stackunder{i=1% }{\;\stackrel{d}{\otimes }}f_i^{\otimes n_i}(t_1,,...,t_n) \] and similarly for $<:\mathbf{\omega }^{\otimes \overrightarrow{n}}:,F_{% \overrightarrow{n}}>$ where for d-tuples of white noise the Wick product $% :\ldots :$ \cite{8} generalizes to \[ :\mathbf{\omega }^{\otimes \stackrel{\rightarrow }{n}}:=\stackunder{i=1}{% \stackrel{d}{\otimes }}:\omega _i^{\otimes n_i}:. \] The vector valued white noise $\mathbf{\omega }\ ${has the characteristic function \begin{equation} C(\mathbf{f})=E(e^{i<\mathbf{\omega },\mathbf{f}>})=\int_{S^{*}(R,R^d)}d\mu \left[ \mathbf{\omega }\right] e^{i<\mathbf{\omega },\mathbf{f}>}=e^{-\frac 12<\mathbf{f,f>}}, \end{equation} where }${<\mathbf{\omega },\mathbf{f}>\ =}\stackrel{d}{\stackunder{i=1}{\sum }}{<\omega }_i,{f}_i{>}$ and $f_i\in S(R,R).$ {\ The Hilbert space \ \[ (L^2)\equiv L^2(d\mu ).\ \] is canonically isomorphic to the }$d-$fold tensor product of {Fock spaces of symmetric square integrable functions: \begin{equation} (L^2)\simeq \stackunder{k=0}{(\stackrel{\infty }{\oplus }}% SyL^2(R^k,k!d^kt))^{\otimes d}\equiv \frak{F}, \label{4} \end{equation} for the general element of }${(L^2)}${\ this implies the chaos expansion \begin{equation} \varphi (\mathbf{\omega })=\stackrel{\infty }{\stackunder{\stackrel{% \rightarrow }{n}=0}{\sum }}<:\mathbf{\omega }^{\otimes \stackrel{\rightarrow }{n}}:,F_{\stackrel{\rightarrow }{n}}> \label{6} \end{equation} with kernel functions $F$ in }$\frak{F}${. } It is desirable to introduce regularizations for the intersection local time, with a view towards the construction of well-defined, ''renormalized'' intersection local times in higher dimensions where the latter do not exist without subtractions. A computationally simple regularization is, for $% \varepsilon >0,$ \[ L_{\!\,\varepsilon }\equiv \int_0^tdt_2\int_0^{t_2}dt_1\,\delta _{\!\,\varepsilon }\left( \mathbf{B}\left( t_2\right) -\mathbf{B}\left( t_1\right) \right) , \] with \begin{equation} \delta _{\!\,\varepsilon }\left( \mathbf{B}\left( t_2\right) -\mathbf{B}% \left( t_1\right) \right) \equiv \left( 2\pi \varepsilon \right) ^{-d/2}e^{-% \frac{\left( \mathbf{B}\left( t_2\right) -\mathbf{B}\left( t_1\right) \right) ^2}{2\varepsilon }}, \label{gauss} \end{equation} It has the following chaos expansion, which we quote here only for $d\geq 3:$ \begin{theorem} \cite{hiwa} For any $\varepsilon >0,$ $L_{\!\,\varepsilon }-E\left( L_{\!\,\varepsilon }\right) $ has kernel functions $F\in \frak{F}$ given by \begin{equation} \begin{array}{c} F_{\varepsilon ,\stackrel{\rightarrow }{n}}(s_1,\ldots ,s_n)=\left( -1\right) ^{\frac n2}\left( \kappa (\kappa +1)(2\pi )^{d/2}\,2^{\frac n2}\,% \stackrel{\rightarrow }{\frac n2}!\right) ^{-1}\cdot \\ \ \\ \Theta (u)\Theta (t-v)\cdot ((v-u+\varepsilon )^{-\kappa }+(t+\varepsilon )^{-\kappa }-(v+\varepsilon )^{-\kappa }-(t-u+\varepsilon )^{-\kappa }) \end{array} \label{f} \end{equation} if all n$_{i}$ are even, and zero otherwise, with $% v(s_1,...,s_n)\equiv max(s_1,...,s_n),$ $u(s_1,...,s_n)\equiv min(s_1,...,s_n),$ and $\kappa \equiv \left( n+d\right) /2-2.$ $\Theta $ is the Heaviside function. \end{theorem} Each kernel function is thus the sum of four terms. The first one is \begin{definition} \begin{eqnarray} M_t(d,\overrightarrow{n},\varepsilon ) &\equiv &\int_{[0,t]^n}d^ns(v-u+\varepsilon )^{-\varkappa }:\omega ^{\otimes \overrightarrow{n}}(s): \label{m} \\ &=&\int_{[0,t]^n}d^ns(v-u+\varepsilon )^{-\varkappa }\bigotimes_{i=1}^d:\omega _i(s_1^i)...\omega _i(s_{n_i}^i): \nonumber \\ &=&\sum_{i=1}^d\sum_{m=1}^{n_i}\int_0^tds_m^i\int_0^{s_m^i}d^{n-1}s(v-u+% \varepsilon )^{-\varkappa }\bigotimes_{i=1}^d:\omega _i(s_1^i)...\omega _i(s_{n_i}^i): \nonumber \\ &=&\sum_{i=1}^dn_i\int_0^tdB_i(\tau )\int_0^\tau d^{n-1}s(\tau -u+\varepsilon )^{-\varkappa }:\omega ^{\otimes \overrightarrow{n-\delta _i}% }(s): \nonumber \\ &=&\sum_{k=1}^d\int_0^tdB_k(v)m_k(v) \\ &\equiv &\sum_{k=1}^dM_{k,t} \end{eqnarray} The others give \[ N_t(d,\overrightarrow{n},\varepsilon )\equiv \int_{[0,t]^n}d^ns\left( (t+\varepsilon )^{-\kappa }-(v+\varepsilon )^{-\kappa }-(t-u+\varepsilon )^{-\kappa }\right) :\omega ^{\otimes \overrightarrow{n}}(s): \] \end{definition} All of the above processes are continuous. \begin{definition} We denote the $\stackrel{\rightarrow }{n}$ $^{th}$ order contribution to the regularized local time $L_{\!\,\varepsilon }$ by \[ K_t(d,\overrightarrow{n},\varepsilon )\equiv \left( -1\right) ^{\frac n2}\left( \kappa (\kappa +1)(2\pi )^{d/2}\,2^{\frac n2}\,\stackrel{% \rightarrow }{\frac n2}!\right) ^{-1}\left( M_t(d,\overrightarrow{n}% ,\varepsilon )+N_t(d,\overrightarrow{n},\varepsilon )\right) \] \end{definition} \begin{enumerate} \begin{remark} Our key observation is that, as $\varepsilon $ goes to zero, $M$ is more singular than $N,$ and that it is a Brownian martingale. \end{remark} \end{enumerate} \subsection{The main theorems} \begin{theorem} \label{th1}: For $d\geq 3$, the renormalized $M$ converge in distribution to independent Brownian motions $\beta _i$: \[ \left( rM_i,;i=1,...,d\right) \stackrel{\frak{L}}{\stackunder{\varepsilon \rightarrow +0}{\rightarrow }}\left( \frac{n_i}nk_n\beta _i;i=1,...,d\right) \] with \begin{equation} k_n^2=\overrightarrow{n}!n(n-1)\left\{ \begin{array}{r} 1\hfill \text{ if }d=3 \\ \frac 1{(n-1)n...(n+d-5)}\text{ if }d>3 \end{array} \right. \end{equation} if \[ r(\varepsilon )=\QATOPD\{ . {\left| ln\varepsilon \right| ^{-1/2}}{\varepsilon ^{(d-3)/2}} \] \end{theorem} \begin{theorem} \label{th2}:For $d\geq 3$, the renormalized $\stackrel{\rightarrow }{n}$ $% ^{th}$ order contributions to the regularized local time $L_{\!\,\varepsilon }$ converge in distribution to Brownian motions $\beta $% \[ rK(d,\overrightarrow{n},\varepsilon ;)\stackrel{\frak{L}}{\stackunder{% \varepsilon \rightarrow +0}{\rightarrow }}c_{n,d}\beta \] with \[ c_{\overrightarrow{n},d}^2=k_n^2\left( \kappa (\kappa +1)(2\pi )^{d/2}\,2^{\frac n2}\,\stackrel{\rightarrow }{\frac n2}!\right) ^{-2} \] \end{theorem} \section{Proofs} \begin{proposition} $M_{k,t}$ are orthogonal Brownian martingales \end{proposition} \noindent Proof: Orthogonality is obvious. For the martingale property see \cite{Hida} and \cite{dpv}, it is a consequence of the fact that the kernel functions of M$_t$ in (\ref{m}) do not depend on $t$ (except through the limit of integrations).$\blacksquare $ Their limiting behavior, as $\varepsilon \rightarrow +0,$ is studied in the following lemma (from now on we shall consider only the situations which require renormalisation, i.e., $d\geq 3.)$ \begin{lemma} As $\varepsilon \rightarrow +0,$% \[ \left\| M_{k,t}\right\| _2^2=\frac{n_k}nk_n^2\left( t+o(1)\right) \QATOPD\{ . {\left| ln\varepsilon \right| \text{ for }d=3}{\varepsilon ^{-(d-3)}% \text{for}d>3} \] \end{lemma} Proof: \[ \left\| M_{k,t}\right\| _2^2=\frac{n_k}n\overrightarrow{n}!\left\| (v-u+\varepsilon )^{-\varkappa }\right\| _{L^2([0,t]^n)}^2 \] For $d>3$% \begin{eqnarray*} \left\| (v-u+\varepsilon )^{-\varkappa }\right\| _{L^2([0,t]^n)}^2 &=&\int_0^td^ns(v-u+\varepsilon )^{-2\varkappa } \\ &=&n(n-1)\int_0^tdv\int_0^vdu\frac{(v-u)^{n-2}}{(v-u+\varepsilon )^{2\varkappa }} \\ &=&n(n-1)\varepsilon ^{3-d}\int_0^tdv\int_0^{v/\varepsilon }dx\frac{x^{n-2}}{% (x+1)^{n+d-4}} \\ &=&\varepsilon ^{3-d}t\left( \frac{n(n-1)}{(n-1)n...(n+d-5)}+o(1)\right) \end{eqnarray*} while for $d=3$% \[ \left\| (v-u+\varepsilon )^{-\varkappa }\right\| _{L^2([0,t]^n)}^2=n(n-1)t\left| \ln \varepsilon \right| \left( 1+o(1)\right) \] $\blacksquare $ To show convergence of these martingales by Theorem VIII.2.17 of \cite{J-C} we must show convergence of characteristics. Since the processes $M$ are continuous this reduces to showing convergence in probability of $% \left\langle rM_i,rM_k\right\rangle _t$ as $\varepsilon \rightarrow +0.$ This will be taken care of by proposition \ref{ms} which we prepare now: \[ \left\langle rM_i,rM_k\right\rangle _t=r^2\delta _{ik}\int_0^tdv\left( m_i(v)\right) ^2 \] and we need to estimate \[ \int_0^tdv\left( m_i(v)\right) ^2\equiv \sum_{\overrightarrow{k}}\langle :\omega ^{\otimes \overrightarrow{k}}:,G_{\overrightarrow{k}}^{(i)}\rangle \] Let us note that \[ E\left( \left\langle rM_i,rM_k\right\rangle _t\right) =E\left( r^2M_iM_k\right) =\delta _{ik}r^2\frac{n_k}n\overrightarrow{n}!\left\| (v-u+\varepsilon )^{-\varkappa }\right\| _{L^2([0,t]^n)}^2 \] \[ E\left( \left\langle rM_i,rM_k\right\rangle _t\right) =\frac{n_k}% nk_n^2\left( t+o(1)\right) \QATOPD\{ . {\left| ln\varepsilon \right| \text{ for }d=3}{\varepsilon ^{-(d-3)}\text{for }d>3} \] Next we intend to show that the rhs gives indeed that the rest of $% \left\langle rM_i,rM_k\right\rangle _t$ goes to zero. The kernel of the highest order is \begin{equation} G_{2\overrightarrow{n}-2\overrightarrow{\delta }_i}^{(i)}(s_1,s_1^{\prime };,\ldots ;s_{n-1},s_{n-1}^{\prime })=\int_0^td\tau \Theta \left( \tau -v\vee v^{\prime }\right) (\tau -u+\varepsilon )^{-\varkappa }(\tau -u^{\prime }+\varepsilon )^{-\varkappa } \label{G} \end{equation} $v^{\left( \prime \right) }$ and $u^{\left( \prime \right) }$ are the largest and the smallest of the $s_i^{\left( \prime \right) },$ and $\left( \overrightarrow{\delta }_i\right) _k\equiv \delta _{ik}.$ (For $n=2$: $u^{\left( \prime \right) }=v^{\left( \prime \right) }=s^{\left( \prime \right) }$). \begin{remark} The integral (\ref{G}) can be calculated in closed form (using e.g. \cite {G-R} nos. 2.15 and 2.263.4), but one gets a useful approximation by introducing the following auxiliary functions: \begin{eqnarray*} &&H_{2n-2}(s_1,...,s_{n-1};s_1^{\prime },...,s_{n-1}^{\prime };\varepsilon ;d) \\ &\equiv &(v\vee v^{\prime }-u\vee u^{\prime }+\varepsilon )^{-\left( n+d\right) /2+3}(v\vee v^{\prime }-u\wedge u^{\prime }+\varepsilon )^{-\left( n+d\right) /2+2} \end{eqnarray*} where $v\equiv max(s_i),u\equiv min(s_i)$ and $v^{\prime }\equiv max(s_i^{\prime }),u^{\prime }\equiv min(s_i^{\prime }).$ These functions majorize the kernel functions \end{remark} \begin{lemma} For $n>2,$ and $\kappa >1$ \begin{eqnarray} 0 &\leq &G_{2\overrightarrow{n}-2\overrightarrow{\delta }_i}^{(i)}(s_1,s_1^{% \prime };,\ldots ;s_{n-1},s_{n-1}^{\prime }) \label{h} \\ &\leq &\frac 1{\kappa -1}H_{2n-2}(s_1,...,s_{n-1};s_1^{\prime },...,s_{n-1}^{\prime };\varepsilon ;d) \nonumber \end{eqnarray} \end{lemma} Proof: \begin{eqnarray*} 0 &\leq &\int_0^td\tau \Theta \left( \tau -v\vee v^{\prime }\right) (\tau -u+\varepsilon )^{-\varkappa }(\tau -u^{\prime }+\varepsilon )^{-\varkappa } \\ &\leq &\int_{v\vee v^{\prime }}^td\tau (\tau -u\vee u^{\prime }+\varepsilon )^{-\varkappa }(v\vee v^{\prime }-u\wedge u^{\prime }+\varepsilon )^{-\varkappa } \\ &\leq &\frac 1{\kappa -1}(v\vee v^{\prime }-u\vee u^{\prime }+\varepsilon )^{-\varkappa +1}(v\vee v^{\prime }-u\wedge u^{\prime }+\varepsilon )^{-\varkappa } \end{eqnarray*} This estimate suffices to show \begin{lemma} For $2\overrightarrow{n}-\overrightarrow{e}_i-\overrightarrow{e}_k\neq 0$% \[ r^2G_{2\overrightarrow{n}-2\overrightarrow{\delta }_i}^{(i)}\text{ }% \rightarrow 0\text{ in }L^2(R^{2n-2}) \] as $\varepsilon $ goes to +0. \end{lemma} Proof: Consider first $n\geq 3$. By the above estimate it is sufficient to show that \[ \lim_{\varepsilon \rightarrow +0}r^4\left\| H_{2n-2}\right\| _{L^2([0,t]^{2n-2})}^2=0. \] \begin{eqnarray*} \left\| H_{2n-2}\right\| _{L^2(R^{2n-2})}^2 &=&\int d^{n-1}s\int d^{n-1}s^{\prime }(v\vee v^{\prime }-u\vee u^{\prime }+\varepsilon )^{2-2\kappa }(v\vee v^{\prime }-u\wedge u^{\prime }+\varepsilon )^{-2\kappa } \\ &=&c_n\int_0^tdv\int_0^vdv^{\prime }\int_0^vdu\int_0^{v^{\prime }}du^{\prime } \\ &&\frac{(v-u)^{n-3}(v^{\prime }-u^{\prime })^{n-3}}{(v\vee v^{\prime }-u\vee u^{\prime }+\varepsilon )^{n+d-6}(v\vee v^{\prime }-u\wedge u^{\prime }+\varepsilon )^{n+d-4}} \\ &\leq &c_n\varepsilon ^{8-2d}\int_0^{t/\varepsilon }dy\int_0^ydy^{\prime }\int_0^ydx\int_0^{y^{\prime }}dx^{\prime } \\ &&\frac 1{(y-x\vee x^{\prime }+1)^{d-3}(y-x\wedge x^{\prime }+1)^{d-1}} \end{eqnarray*} and then decompose the x-integration into the following two domains \[ x^{\prime }4 \end{array} \right. \] and one finds \[ I=\left\{ \begin{array}{l} O\left( 1\right) \text{ if }d=3 \\ O(\varepsilon ^{7-2d})\text{ if }d>3 \end{array} \right. \] Hence, in both cases, $r^4$ $I$ vanishes as $\varepsilon \rightarrow 0.$ The second case is \begin{equation} I=\varepsilon ^{8-2d}\int_0^{t/\varepsilon }dy\int_0^ydx\frac 1{(y-x+1)^{d-1}}\int_x^ydy^{\prime }\int_x^{y^{\prime }}dx^{\prime }\frac 1{(y-x^{\prime }+1)^{d-3}} \end{equation} Estimating the last two integrals we find \[ \int_x^ydy^{\prime }\int_x^{y^{\prime }}dx^{\prime }\frac 1{(y-x^{\prime }+1)^{d-3}}\leq \left\{ \begin{array}{l} \frac{(y-x)^2}2\text{ for }d=3 \\ const.\left( y-x\right) ^{5-d}\text{ for }d>3 \end{array} \right. \] Substitution of these estimates into the integrals over x and y gives for $% d>3$% \[ I\leq const.\varepsilon ^{8-2d}\int_0^{t/\varepsilon }dy\int_0^ydx\frac{% (y-x)^{n+2-d}}{(y-x+1)^{n+d-4}} \] where the integral over $x$ is bounded, so that again $I=O(\varepsilon% ^{7-2d}).$ For $d=3$ one finds similarly $I=O(\varepsilon ^0)\ $so in both cases, $r^4$ $I$ vanishes as $\varepsilon \rightarrow 0.$ For $n=2$ it is sufficient to use the estimate \begin{eqnarray*} 0 &\leq &G_2=\int_0^td\tau \Theta \left( \tau -u\vee u^{\prime }\right) (\tau -u+\varepsilon )^{1-d/2}(\tau -u^{\prime }+\varepsilon )^{1-d/2} \\ &\leq &\int_{u\vee u^{\prime }}^td\tau (\tau -u\vee u^{\prime }+\varepsilon )^{1-d/2}(u\vee u^{\prime }-u\wedge u^{\prime }+\varepsilon )^{1-d/2}=H_2(u,u^{\prime }) \end{eqnarray*} and to verify that \[ \lim_{\varepsilon \rightarrow +0}r^4\left\| H_2\right\| _{L^2([0,t]^2)}^2=0. \] $\blacksquare $ With this lemma we have established that the highest order term of the (renormalized) quadratic variation goes to zero in quadratic mean for any t% >0. The kernels $G_k$ of the other terms are obtained by integrating over pairs of $s,s^{\prime }$ such as e.g. in \[ Sym\int_0^tdsG_{2\left( \overrightarrow{n}-\overrightarrow{e}_i\right) }^{(i)}(s,s;s_2,s_2^{\prime };,\ldots ;s_{n-1},s_{n-1}^{\prime }), \] these new functions are in fact also bounded by an expression like (\ref{G}% ), and hence by (\ref{h}), so that for all $\varepsilon >0,$ \[ \left\| G_k\right\| \leq const.\left\| H_k\right\| \] With this goal we show \begin{lemma} Let $n\geq 2$ and $m>1$ , and let F$_{n,m}$ be a function symmetric in the variables $s$ and in the variables $s^{\prime },$ with \begin{eqnarray} 0 &\leq &F_{n,m}(s_1,\ldots ,s_{n-1},s_n,s_1^{\prime },\ldots ,s_{n-1}^{\prime },s_n^{\prime }) \\ &\leq &c\int_0^td\tau \Theta \left( \tau -\stackunder{i\leq n}{max}% (s,s^{\prime })\right) (\tau -\stackunder{i\leq n}{min}(s_i)+\varepsilon )^{-m}(\tau -\stackunder{i\leq n}{min}(s_i^{\prime })+\varepsilon )^{-m} \nonumber \end{eqnarray} Then $\exists c_m<\infty $ such that \begin{eqnarray*} 0&\leq& \int_0^tdsF_{n,m}(s_1,\ldots ,s,s_1^{\prime },\ldots ,s) \label{est} \\ &\leq& c_m\int_0^td\tau \Theta \left( \tau -\stackunder{i1/2.$ The second term \begin{eqnarray*} &&\dint\limits_{v\vee v^{\prime }}^td\tau \dint\limits_u^{u^{\prime }}ds\left( \tau -u+\varepsilon \right) ^{-m}\left( \tau -s+\varepsilon \right) ^{-m} \\ &\leq &\frac 1{m-1}\dint\limits_{v\vee v^{\prime }}^td\tau \left( \tau -u+\varepsilon \right) ^{-m}\left( \tau -u^{\prime }+\varepsilon \right) ^{-m+1} \\ &\leq &\frac 1{m-1}\dint\limits_0^td\tau \Theta \left( \tau -v\vee v^{\prime }\right) \left( \tau -u+\varepsilon \right) ^{-m+\frac 12}\left( \tau -u^{\prime }+\varepsilon \right) ^{-m+\frac 12}. \end{eqnarray*} using again $u4 \end{array} \right. \end{eqnarray*} which are suppressed by the renormalization \[ r^2(\varepsilon )=\QATOPD\{ . {\left| ln\varepsilon \right| ^{-1}\text{ for }% d=3}{\varepsilon ^{d-3}\text{ for }d>3}. \] A similar estimate holds for the third term of $N$, so that we have shown \begin{lemma} \[ ms-\lim_{\varepsilon \rightarrow +0}r(\varepsilon )N_t(d,\overrightarrow{n}% ,\varepsilon )=0. \] \end{lemma} In fact the convergence is uniform in any finite t-interval. Next we show \begin{lemma} \label{teit}The processes $\left\{ r(\varepsilon )N_{\cdot }(d,% \overrightarrow{n},\varepsilon ;):\varepsilon >0\right\} $, $\left\{ r(\varepsilon )M_{\cdot }(d,\overrightarrow{n},\varepsilon ;):\varepsilon >0\right\} $ and their linear combinations are tight. \end{lemma} \noindent Proof: A criterion for tightness of M (following \cite{ks}, p.64) is \begin{equation} \stackunder{\varepsilon >0}{\sup }E\left| rM_t-rM_s\right| ^{\!\,\alpha }\leq C_T\left( t-s\right) ^{1+\beta }, \label{eq23} \end{equation} $\forall T>0$ and $0\leq s0}{\sup }E\left| rM_t-rM_s\right| ^2\leq C_T\left( t-s\right) \label{24} \end{equation} by direct calculation: \begin{eqnarray*} E\left| M_t-M_s\right| ^2 &=&n\int_s^tdv\int_0^vd^{n-1}s\left( v-u\right) ^{-2\kappa } \\ &\leq &(t-s)n\int_0^td^{n-1}s\left( v-u+\varepsilon \right) ^{-2\kappa } \end{eqnarray*} This integral may be estimated as follows \begin{eqnarray*} \int_0^td^{n-1}s\left( v-u+\varepsilon \right) ^{-2\kappa } &=&c\int_0^tdu% \frac{(v-u)^{n-2}}{\left( v-u+\varepsilon \right) ^{n+d-4}} \\ &\leq &c\varepsilon ^{3-d}\int_0^{T/\varepsilon }dx\frac 1{\left( x+1\right) ^{d-2}} \\ &=&\left\{ \begin{array}{l} O(1)\cdot \ln T\text{ for }d=3 \\ O(\varepsilon ^{3-d})\text{ for }d>3 \end{array} \right. \end{eqnarray*} Renormalization of this estimate by the factor $r^2$ gives the desired estimate (\ref{24}), i.e. \begin{equation} \left\| rM_t-rM_s\right\| _2^2\leq c_T\left( t-s\right) \label{25} \end{equation} By the hypercontractivity of the Ornstein-Uhlenbeck semigroup (see e.g.\cite {8}, p.235), one has for n$^{th}$ order white noise monomials $\varphi \in (L^2)$ , and any $\alpha >2$% \[ \left\| \varphi \right\| _{\,\!\alpha }^{_\alpha }\leq c_{n,\alpha }\left\| \varphi \right\| _2^{_\alpha } \] For $\varphi =rM_t-rM_s$ ,and using the above estimate for the 2-norm, we get \[ E\left| rM_t-rM_s\right| ^{\,\!\alpha }\leq C_T\left( t-s\right) ^{\alpha /2} \] as required to ensure tightness. The estimates for $rN$ etc. are of the same kind.$\blacksquare $ \noindent \textbf{Proof of Theorem \ref{th2}}: We need to consider \[ rK=rM+rN \] knowing that, as $\varepsilon \rightarrow +0,$ the $rK$ are tight by Lemma \ref{teit}, the $rM$ converge in law, and the $rN_t$ go to zero in mean square. The latter two facts are sufficient, via the Cram\'{e}r-Wold device (see e.g. \cite{ks} p.61), for finite dimensional convergence of K; tightness then implies convergence in law. \textbf{Acknowledgements:} This work has had partial support from PAXIS XXI and FEDER. L. S. would like to express his appreciation for an inspiring discussion with L. Chen, for a very helpful remark of J. Potthoff, and for the splendid hospitality of the Grupo de F\'{i}sica Matem\'{a}tica da Universidade de Lisboa. \begin{thebibliography}{99} \bibitem{bass} Bass, R. F., Khoshnevisan, D.: \textit{Intersection Local Times and Tanaka Formulas.} Ann. Inst. H. 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