---
abstract: 'We study the chaos decomposition of self-intersection local times and their regularization, with a particular view towards Varadhan’s renormalization for the planar Edwards model.'
author:
- |
    **Jinky Bornales**\
    [Physics Department, MSU-IIT, Iligan City, The Philippines]{}\
    [jinky.bornales@g.msuiit.edu.ph]{}
- |
    **Maria João Oliveira**\
    [Universidade Aberta, P 1269-001 Lisbon, Portugal]{}\
    [CMAF, University of Lisbon, P 1649-003 Lisbon, Portugal]{}\
    [oliveira@cii.fc.ul.pt, mjoliveira@ciencias.ulisboa.pt (New)]{}
- |
    **Ludwig Streit**\
    [Forschungszentrum BiBoS, Bielefeld University, D 33501 Bielefeld, Germany]{}\
    [CCM, University of Madeira, P 9000-390 Funchal, Portugal]{}\
    [streit@physik.uni-bielefeld.de]{}
title: 'Chaos Decomposition and Gap Renormalization of Brownian Self-Intersection Local Times'
---

**Keywords:** Edwards model, self-intersection local time, Varadhan renormalization, white noise analysis

**Mathematics Subject Classifications (2010):** 28C20, 41A25, 60H40, 60J55, 60J65, 82D60

Introduction
============

The self-intersection local time of $d$-dimensional Brownian motion, informally, is given as $$L=\int_{0}^{T}dt_2\int_{0}^{t_2}dt_1\,\delta \left(\mathbf{B}(t_2)-\mathbf{B}(t_1)\right) .  \label{L}$$We shall see that, while “reasonably well defined” for $d=1$, these local times become more and more singular as the dimension $d$ increases. Intersections have thus been the object of extensive study by authors such as Dvoretzky, Erdös, Kakutani [@Dv], [@Dv1], [@Dv2], Varadhan [@v], Westwater [@Westwater], [@Westwater2], [@Westwater3], Le Gall [@LeGall], [@LeGall2], Rosen [@R1], [@R2], [@R3], Dynkin [@Dy1], [@Dy2], [@Dynkin], Watanabe [@Wat], Yor [@Yor1], [@Yor2], Imkeller et al. [@Imkeller], Albeverio et al. [@AOS], [@AlbHu]. For fractional Brownian motion there are papers e.g. by Rosen [@R4], Hu & Nualart [@Hu1], Grothaus et al. [@GOSS].

Apart from its intrinsic mathematical interest the self-intersection local time has played a role in constructive quantum field theory, and is a standard model in polymer physics for the self-repulsion (“excluded volume effect”) of chain polymers in solvents [@Schaefer].

Replacement of the Dirac delta function in (\[L\]) by a Gaussian $$\delta _{\varepsilon }(x):=\frac{1}{(2\pi \varepsilon )^{d/2}}e^{-\frac{|x|^{2}}{2\varepsilon }},\quad \varepsilon >0,$$ leads to regularized local times $$L_{\varepsilon }:=\int_{0}^{T}dt_2\ \int_{0}^{t_2}dt_1\,\delta _{\varepsilon
}(\mathbf{B}(t_2)-\mathbf{B}(t_1))   $$ and for $d=1$ one can show $L^{2}$ convergence w.r.t. white noise or Wiener measure space. But already for $d=2$ this fails since the expectation of $L_{\varepsilon }$ will diverge in the limit, asymptotically $$\mathbb{E}(L_{\varepsilon })\approx -\frac{T}{2\pi }\ln \varepsilon.$$ In this case it is sufficient to subtract the expectation, i.e. the centered regularized local time does have a well-defined $L^2$ limit: $$L_{\varepsilon ,c}:=L_{\varepsilon }-\mathbb{E}(L_{\varepsilon })\rightarrow
L_{c}.$$ Apart from the Gaussian regularization above, others have been considered to remove the singularity at $t_1=t_2$ in the integral (\[L\]). The “staircase regularization” avoids the line $t_1=t_2$ as in see e.g. Bolthausen [@Bolthausen] (Fig. 1).

![image](Fig1.jpg)

[: Domain of integration for the staircase-regularized local time.]{}

The widely used “gap regularization” does the same by omitting the strip $t_2-t_1<\Lambda$ in the integral. In the modelling of chain polymers the gap size $\Lambda$ will be a “microscopic” quantity, i.e. of the order of the inter-monomer distance, more precisely the “Kuhn” or “persistence” length. It plays an important role in renormalization group calculations [@Schaefer]: critical parameters are obtained from the postulate that macroscopic quantities do not depend on microscopic length scales.

Tools from White Noise Analysis [@hkps]
=======================================

Based on a $d$-tuple of independent Gaussian white noises $\bm{\omega}=(\omega _{1},...,\omega _{d})$ one defines a $d$-dimensional Brownian motion $\mathbf{B}$ through $$\mathbf{B}(t)\equiv \langle\bm{\omega},\I_{[0,t]}\rangle=\int_{0}^{t}ds\,
\bm{\omega }(s).$$

We shall use a multi-index notation $$\mathbf{n}=(n_{1},\ldots ,n_{d}),\;\;n=\sum_{i=1}^{d}n_{i},\;\;\mathbf{n}!=\prod_{i=1}^{d}n_{i}!$$and for $d$-tuples of Schwartz test functions $\mathbf{f}=(f_{1},\ldots,f_{d})\in 
S(\mathbb{R},\mathbb{R}^{d})$, $$\langle\mathbf{f},\mathbf{f}\rangle=\sum_{i=1}^{d}\int dt\, f_{i}^{2}(t)$$ $$\langle F_{\mathbf{n}},\mathbf{f}^{\otimes \mathbf{n}}\rangle=\int d^nt\, 
F_{\mathbf{n}}(t_1,\ldots,t_n) \underset{i=1}{\overset{d}{\otimes }}
f_i^{\otimes n_i}(t_1,\ldots,t_n)$$ and similarly for $\langle:\bm{\omega}^{\otimes \mathbf{n}}:,F_{\mathbf{n}}
\rangle$ where for $d$-tuples of white noise the Wick product $: \cdot :$ [@hkps] generalizes to $$:\bm{\omega }^{\otimes \mathbf{n}}:
=\underset{i=1}{\overset{d}{\otimes }}:\omega _{i}^{\otimes n_{i}}:$$ The vector valued white noise $\omega$ has the characteristic function $$C(\mathbf{f}):=\mathbb{E}(e^{i\langle\bm{\omega },\mathbf{f}\rangle})=\int_{S^{\ast }(\mathbb{R},\mathbb{R}^{d})}d\mu(\bm{\omega }) e^{i\langle\bm{\omega },\mathbf{f}\rangle}=e^{-\frac{1}{2}\langle\mathbf{f},\mathbf{f}\rangle},$$ where $\langle\bm{\omega},\mathbf{f}\rangle
=\sum_{i=1}^d\langle\omega_{i}, f_{i}\rangle$ and $f_{i}\in S(\R, \R)$.

Writing $$(L^{2}):= L^{2}(S^{\ast }(\R,\R^{d}), d\mu)$$ there is the Itô-Segal-Wiener isomorphism with the Fock space of symmetric square integrable functions: $$(L^{2})\simeq \left(\underset{k=0}{\overset{\infty }{\oplus }}\mathrm{Sym}\, 
L^{2}(\mathbb{R}^{k},k!d^{k}t)\right)^{\otimes d}.  $$ This implies the chaos expansion $$\varphi (\bm{\omega })=\sum_{\mathbf{n}\in\N_0^d}
\langle:\bm{\omega }^{\otimes \mathbf{n}}:,F_{\mathbf{n}}\rangle\textrm{ for  }
\varphi \in (L^{2})  $$ with kernel functions $F_{\mathbf{n}}$ in Fock space.

Generalized functionals are constructed via a Gel’fand triple $$(S)\subset (L^{2})\subset (S)^{\ast }.$$

The generalized functionals in ${(S)}^{\ast }$ are conveniently characterized by their action on exponentials. In particular we use the $$:\exp (\langle\bm{\omega},\mathbf{f}\rangle):\,= C(\mathbf{f})
\exp(\langle\bm{\omega},\mathbf{f}\rangle)\in (S)  $$ to make the

The transformation defined for all test functions $\mathbf{f}\in S(\R,\R^d)$ via the bilinear dual product on ${(S)}^{*}\times (S)$ by $$(S\Phi)(\mathbf{f})=
\langle\!\langle\Phi,:\exp (\langle\cdot,\mathbf{f}\rangle):\rangle\!\rangle$$ is called the S-transform of $\Phi\in {(S)}^{*}$.

The multilinear expansion of $S(\Phi)$ $$(S\Phi)(\mathbf{f})=\sum_{\mathbf{n}\in\N_0^d}\langle
\bm{\varphi}_{\mathbf{n}},\mathbf{f}^{\otimes \mathbf{n}}\rangle  $$ extends the chaos expansion to $\Phi \in (S)^{\ast }$, with distribution valued kernels $\varphi_{\mathbf{n}}$, such that $$\langle\!\langle\Phi,F\rangle\!\rangle=
\sum_{\mathbf{n}\in\N_0^d}\mathbf{n}!\langle\bm{\varphi}_{\mathbf{n}},
F_{\mathbf{n}}\rangle.$$

We shall indicate the projection onto chaos of order $n\geq k$ by a superscript $(k)$: $$\langle\!\langle\Phi ^{(k)},F\rangle\!\rangle=
\sum_{\mathbf{n}: n\geq k}\mathbf{n}!\langle\bm{\varphi }_{\mathbf{n}},
F_{\mathbf{n}}\rangle.$$

[@fhsw] $$\delta ^{(2N)}(\mathbf{B}(t_{2})-\mathbf{B}(t_{1}))\in (S)^{\ast }$$ with even kernel functions $$\bm{\psi }_{2\mathbf{n}}(u_{1},\ldots
,u_{2n};t_{1},t_{2})=\frac{1}{\mathbf{n}!}(2\pi )^{-d/2}
\left(\frac{1}{|t_{2}-t_{1}|}\right)^{\frac{d}{2}+n}
\left(-\frac{1}{2}\right)^{n}
\prod_{k=1}^{2n}\I_{[t_{1},t_{2}]}(u_{k}).  $$ All the kernel functions with odd indices vanish.

Setting $$\begin{aligned}
v &:=&\max (u_{1},\ldots ,u_{2n}) \\
u &:=&\min (u_{1},\ldots ,u_{2n})\end{aligned}$$one computes [@fhsw] the kernel functions of the truncated local time $L^{(2N)}$ for $2N>d-2$ by integration over $0<t_{1}<t_{2}<T$: $$\begin{aligned}
\varphi _{2\mathbf{n}}(u_{1},\ldots ,u_{2n})&=&\frac{(2\pi
)^{-d/2}}{\mathbf{n}!}\left(-\frac{1}{2}\right)
^{n}\int_{0}^{T}dt_{2}\int_{0}^{t_{2}}dt_{1}
\,(t_{2}-t_{1})^{-n-d/2}\I_{[t_{1},t_{2}]}^{\otimes 2n}(u_{1},\ldots ,u_{2n})\\ 
&=&(-1)^{n}\left((n+\frac{d}{2}-1)(n+\frac{d}{2}-2)(2\pi
)^{d/2}\,2^{n}\,\mathbf{n}!\right)^{-1}\cdot \Theta(u)\Theta (T-v)\cdot \\ 
&&\cdot (T^{-n-\frac{d}{2}+2}-v^{-n-\frac{d}{2}+2}-(T-u)^{-n-\frac{d}{2}+2}+(v-u)^{-n-\frac{d}{2}+2})$$ except for $2n=d=2$ where $$\varphi _2(u_{1},u_{2})=-\frac{1}{4\pi}\left(\ln v+\ln (T-u)-\ln
(v-u)-\ln T\right)\cdot\Theta(u)\Theta(T-v).$$ The Heaviside function $\Theta$ here is the indicator function of the positive half line.

Regularizations
===============

Replacement of the Dirac delta function by a Gaussian $$\delta _{\varepsilon }(x)=\frac{1}{(2\pi\varepsilon )^{d/2}}e^{-\frac{|x|^{2}}{2\varepsilon}},\quad \varepsilon >0,$$leads to regularized local times $$L_{\varepsilon }=\int_{0}^{T}dt_{2}\int_{0}^{t_{2}}dt_{1}\,\delta
_{\varepsilon }(\mathbf{B}(t_{2})-\mathbf{B}(t_{1}))   \label{3Eq2}$$ with kernel functions [@fhsw] $$\begin{aligned}
&&\varphi _{\varepsilon,2\mathbf{n}}(u_{1},\ldots ,u_{2n})\\
&=&\frac{(2\pi )^{-d/2}}{\mathbf{n}!}\left(-\frac{1}{2}\right)
^{n}\int_{0}^{T}dt_{2}\int_{0}^{t_{2}}dt_{1}\,(\varepsilon +\left|
t_{2}-t_{1}\right|)^{-n-d/2}\I_{[t_{1},t_{2}]}^{\otimes 2n}(u_{1},\ldots
,u_{2n}) \\ 
&=& (-1)^{n}\left((n+\frac{d}{2}-1)(n+\frac{d}{2}-2)(2\pi
)^{d/2}\,2^{n}\,\mathbf{n}!\right)^{-1}\cdot \Theta
(u)\Theta (T-v)\cdot \\ 
&&\cdot ((T+\varepsilon)^{-n-\frac{d}{2}+2}-(v+\varepsilon )^{-n-\frac{d}{2}
+2}-(T-u+\varepsilon)^{-n-\frac{d}{2}+2}+(v-u+\varepsilon )^{-n-\frac{d}{2}+2}),\end{aligned}$$ $$\begin{aligned}
&&\varphi _{\varepsilon,2}(u_{1},u_{2})\\
&=&-\frac{1}{4\pi}\left( \ln
(v+\varepsilon)+\ln (T-u+\varepsilon)-\ln (v-u+\varepsilon )-\ln
(T+\varepsilon )\right)\cdot\Theta(u)\Theta(T-v).\end{aligned}$$

Gap Regularization of Kernel Functions
--------------------------------------

In renormalization group studies of self-repelling Brownian motion another regularization is often used, see e.g. [@Schaefer] and the references there; it suppresses intersection in small time intervals $t_{2}-t_{1}$ between intersections by setting informally $$L(\Lambda ):=\underset{t_{2}-t_{1}>\Lambda }{\underset{0<t_{1}<t_{2}<T}
{\int d^{2}t}}\delta (\mathbf{B}(t_{2})-\mathbf{B}(t_{1})).$$

The expectation of $L(\Lambda)$ is equal to $$\mathbb{E}\left( L(\Lambda )\right) =\int_{\Lambda
}^{T}dt_{2}\int_{0}^{t_{2}-\Lambda }dt_1\,\bm{\psi }_{0}(t_{1},t_{2})=(2\pi
)^{-d/2}\int_{\Lambda }^{T}dt_{2}\int_{0}^{t_{2}-\Lambda
}dt_{1}\,(t_{2}-t_{1})^{-d/2}.$$ We note that for in particular $d=2$ $$\mathbb{E}\left( L(\Lambda )\right) =-\frac{T}{2\pi }\ln \Lambda +O(1).
\label{eqRef}$$ Recall [@fhsw] that for $\Lambda=0$ the kernel functions $\varphi_{2\mathbf{n}}$ of the truncated local time $L^{(2N)}$ are obtained by integrating the kernel functions $\psi_{2\mathbf{n}}(u_{1},\ldots,
u_{2n};t_{1},t_{2})$ over the rectangle $0<t_{1}<u$ and $v<t_{2}<T$, shaded grey in see Fig. 2. For $\Lambda >0$ the integration is further restricted to the light domain with $t_{2}-t_{1}>\Lambda$. This restriction is non-trivial when $v-u<\Lambda$. $$L^{(2N)}-L^{(2N)}(\Lambda )$$ thus has kernel functions $\rho_{2 \mathbf{n}}(u,v)$ with support on $$\left\{ 0<u<v<T\right\} \cap \left\{ v-u<\Lambda \right\}.$$ Fig. 2 is pertinent to the case where $\Lambda <v$ and $u<T-\Lambda$. In this case the kernel functions $\rho _{2 \mathbf{n}}(u,v)$ are obtained by integrating the $\psi _{2\mathbf{n}}(u_{1},\ldots, u_{2n};t_{1},t_{2})$ with respect to the $t_{i}$ over $v-\Lambda <t_{1}<u$ and $v\ <t_{2}<t_{1}+\Lambda$. Excepting the kernel function $\rho _{2}(u,v)$ for $d=2$, one finds, $$\begin{aligned}
\rho _{2\mathbf{n}}(u,v) &=&\int_{v-\Lambda}^{u}dt_{1}\int_{v}^{t_{1}+\Lambda }dt_{2}\,
\psi _{2\mathbf{n}}(u_{1},\ldots, u_{2n};t_{1},t_{2})\label{range} \\
&=&\frac{(2\pi)^{-d/2}}{\mathbf{n}!}\left(-\frac{1}{2}\right)^{n}
\frac{1}{d/2+n-1}\cdot\nonumber \\
&&\cdot\left(\frac{1}{d/2+n-2}\left((v-u)^{-d/2-n+2}-\Lambda^{-d/2-n+2}\right)+
(v-u-\Lambda)\Lambda^{-d/2-n+1}\right)\nonumber\end{aligned}$$ and for $2n=d=2$ $$\rho_{2}(u,v)=\frac{1}{4\pi}\left(\ln (v-u)-\ln \Lambda +\frac{\Lambda
-v+u}{\Lambda }\right).  \label{d}$$

![image](Fig2.jpg)

[: Domain of integration for kernels of the local time, light grey for the regularized local time, dark grey for the subtraction $\rho$ as in .]{}

Using $\tau =t_{2}-t_{1}$ we obtain the following estimate $$\begin{aligned}
\left| \rho _{2\mathbf{n}}(u,v)\right|  &=&\left|
\int_{v-\Lambda}^{u}dt_{1}\int_{v}^{t_{1}+\Lambda }dt_{2}\,
\psi _{2\mathbf{n}}(u_{1},\ldots, u_{2n};t_{1},t_{2})\right|  \label{range2} \\
&=&\frac{1}{2^n\mathbf{n}!}(2\pi)^{-d/2}
\int_{v-\Lambda}^{u}dt_{1}\int_{v-t_{1}}^{\Lambda }d\tau\, 
\left(\frac{1}{\tau }\right)^{\frac{d}{2}+n} \nonumber\\
&\leq &\frac{1}{2^n\mathbf{n}!}(2\pi)^{-d/2}
\frac{1}{d/2+n-1}\int_{v-\Lambda}^{u}dt_{1}\,\left( v-t_{1}\right) ^{-d/2-n+1} \nonumber\\
&\leq &\frac{1}{2^n\mathbf{n}!}(2\pi)^{-d/2}
\frac{1}{d/2+n-1}\frac{1}{d/2+n-2}\left(v-u\right)^{-d/2-n+2}\nonumber\end{aligned}$$ while for $d=2n=2$ one readily finds from (\[d\]) $$\left| \rho_2(u,v)\right|  
\leq \frac{1}{4\pi}\left| \ln (v-u)\right| \label{ln}$$ when $v-u<\Lambda \ll 1$.

For very small or very large $u, v$, i.e. $0<u<v<\Lambda$ or $T-\Lambda
<u<v<T$, respectively, the range of integrations in (\[range\]) and (\[range2\]) for $\psi _{2\mathbf{n}}(u_{1},\ldots, u_{2n};t_{1},t_{2})$ will be $0<t_{1}<u$ and $v<t_{2}<t_{1}+\Lambda$, or $v<t_{2}<T$ and $t_{2}-\Lambda <t_{1}<u$ respectively, see Fig. 3.

![image](Fig3.jpg)

[: Modified integration domains for $\rho$ when $u, v$ are close to zero or to $T$ respectively.]{}

Computations as in (\[range\]) are again straightforward, we note here only that the estimate of (\[range2\]) is true also in these two cases.

For $N>0$ and $0<\Lambda\ll T$ the chaos expansion of the gap-regularized local time $L^{(2N)}(\Lambda)$ has the kernel functions $$\varphi_{\Lambda,2\mathbf{n}}(u_{1},\ldots ,u_{2n})=
\varphi _{2\mathbf{n}}(u_{1},\ldots ,u_{2n})-
\Theta(\Lambda -(v-u)) \rho_{2\mathbf{n}}(u_{1},\ldots ,u_{2n}),
\label{jinky}$$ and zero otherwise, while $$L^{(2N)}-L^{(2N)}(\Lambda )$$ has the kernel functions $\rho_{2\mathbf{n}}$ for $0<u<v<T$, $n\geq N$.

The Heaviside function $\Theta$ exhibits the support property of the $\rho_{2\mathbf{n}}$, i.e. in the gap regularization the kernel functions of the local time are only modified when all arguments $u_{k}$ are close to each other.

With these results one can in particular estimate the rate of convergence for the centered self-intersection local time in $d=2$. Apart from the term $n=1$ the sum $$\left\|L^{(2)}-L^{(2)}(\Lambda)\right\|^2_{(L^2)}
=\sum_{\mathbf{n}: n\geq 1}(2\mathbf{n})!\left\|\rho_{2\mathbf{n}}\right\|^2_{L^2([0,T]^{2n})}$$ can be estimated as follows: $$\begin{aligned}
\sum_{\mathbf{n}: n>1}\left( 2\mathbf{n}\right)!
\left\|\rho _{2\mathbf{n}}\right\|_{L^{2}(\left[ 0,T\right] ^{2n})}^{2}
&\leq&(2\pi )^{-d}\sum_{\mathbf{n}: n> 1}
\frac{\left( 2\mathbf{n}\right) !}{(\mathbf{n}!) ^{2}}
\left(\frac{1}{2}\right)^{2n}\left( \frac{1}{d/2+n-1}\frac{1}{d/2+n-2}\right) ^{2} \\
&&\underset{v-u<\Lambda}{\int_{0}^{T}}d^{2n}u_{k}\,
\left(\frac{1}{v-u}\right)^{d+2n-4}. \end{aligned}$$ We can integrate out the $2n-2$ variables $u_{k}$ with $u<u_{k}<v$ that lie between the smallest and the largest and obtain in this way $$\begin{aligned}
\sum_{\mathbf{n}: n>1}\left( 2\mathbf{n}\right)!
\left\|\rho _{2\mathbf{n}}\right\|_{L^{2}(\left[ 0,T\right] ^{2n})}^{2}
&\leq &(2\pi)^{-2}\sum_{\mathbf{n}: n>1}
\frac{\left( 2\mathbf{n}\right) !}{( \mathbf{n}!)^{2}}
\left(\frac{1}{2}\right)^{2n}\left( \frac{1}{n}\frac{1}{n-1}\right) ^{2} \\
&&\cdot 2n(2n-1)\underset{v-u<\Lambda }{\int_{0}^{T}dv\int_{0}^{v}du}\\
&\leq&\frac{\Lambda T}{2\pi^2}\sum_{\mathbf{n}: n>1}\frac{
\left( 2\mathbf{n}\right) !}{2^{2n}\left(\mathbf{n}!\right)^{2}}
\frac{1}{n}\frac{2n-1}{(n-1)^2}.\end{aligned}$$ The series is convergent (Stirling’s formula). From it is straightforward to estimate the remaining term with $n=1$: $$\left\|\rho_2\right\|^2_{L^2([0,T]^2)}\leq
\left(\frac{1}{4\pi}\right)^2\int_0^Tdt\int_0^\Lambda
d\tau\,\ln^2\tau=0(T\Lambda \ln ^{2}\Lambda).$$ So we have shown

\[Prop\] For $d=2$ $$\left\| L^{(2)}-L^{(2)}(\Lambda )\right\|_{\left( L^{2}\right)
}^{2}=0(T\Lambda \ln ^{2}\Lambda)\text{ as }\Lambda \searrow 0.$$

A similar improvement of the rate of convergence has been found in the Gaussian regularization in [@Bock].

Varadhan Renormalization
========================

The model proposed by Edwards [@Edwards] for self-repelling Brownian motion suppresses self-crossings, modifying the Brownian path (or white noise) measure by a density function, informally $$\varrho =Z^{-1}\ \exp \left( -gL\right)$$ with $g>0$ $$Z=\mathbb{E}\left( \exp \left( -gL\right) \right) .$$

There is no problem for $d=1$ since $L$ is a positive random variable and hence $\exp \left( -gL\right) <1$. For $d=2$ however we should replace $L$ by the centered $L_{c}$ and this then is no more positive, so that $\exp
\left( -gL_{c}\right)$ is unbounded. The point of Varadhan’s theorem is to show that this happens only on small sets so that

(Varadhan [@v]) For $d=2$ $$\varrho =Z^{-1}\ \exp \left( -gL_{c}\right)$$ with $$Z=\mathbb{E}\left( \exp \left( -gL_{c}\right) \right)$$ is integrable.

Varadhan defines the centered local time as the limit of Gaussian approximations as in (\[3Eq2\]) and uses the Chebyshev inequality to show that $\exp \left( -gL_{c}\right)$ is integrable for $0<g<\frac{\pi }{T}$.

A similar slightly stronger result can be obtained using Varadhan’s technique with the approximation $$L^{(2)}(\Lambda )\rightarrow L^{(2)}=L_{c}.$$

Fix $0<\Lambda <1$. By (\[eqRef\]) there exists a positive constant $k$ such that $$L^{(2)}(\Lambda )\geq -\mathbb{E}(L(\Lambda ))\geq -k-\frac{T}{2\pi }
|\ln (\Lambda )|.$$ For any constant $N\geq k+\frac{T}{2\pi }|\ln (\Lambda)|$ one has $$\begin{aligned}
\mathbb{P}(L_{c}\leq -N) &=&\mathbb{P}(L_{c}-L^{(2)}(\Lambda )\leq
-N-L^{(2)}(\Lambda )) \\
&\leq &\mathbb{P}\left( |L^{(2)}(\Lambda )-L_{c}|\geq N-k-\frac{T}{2\pi }|\ln (\Lambda )|\right) .\end{aligned}$$ An application of Chebyshev’s inequality, using Theorem \[Prop\] then yields $$\mathbb{P}(L_{c}\leq -N)\leq \frac{\mathbb{E}(|L^{(2)}(\Lambda )-L_{c}|^{2})
}{\left( N-k-\frac{T}{2\pi }|\ln (\Lambda )|\right) ^{2}}\leq K\frac{\Lambda 
\ln ^{2}\Lambda}{\left( N-k-\frac{T}{2\pi }|\ln (\Lambda )|\right) ^{2}}.$$ In particular, for $$\Lambda =\exp \left( -\alpha (N-k)\right),\quad 0<\alpha < \frac{2\pi }{T}$$ one obtains $$\mathbb{P}(L_{c}\leq -N)\leq 
\frac{K\alpha^2}{\left(1-\frac{T}{2\pi}\alpha\right)^{2}}
\exp \left( -\alpha(N-k)\right).$$ Hence, $\exp \left( -gL_{c}\right) $ is integrable for $g<\frac{2\pi }{T}$.

For the Gaussian regularization an analogous result can be found in [@Bock].

Concluding Remarks
==================

[@FDS], [@fhsw], [@Hu1], [@Imkeller], [@Jenane], [@Sandra], [@NV], in the present context of regularizations and the rate of convergence see in particular [@Bock]. Yet another regularization of the self-intersection local time is suggested by (\[jinky\]), namely $$L_{\Lambda}^{(2N)}:=\sum_{\mathbf{n}: n\geq N} \langle
:\bm{\omega}^{\otimes 2\mathbf{n}}:,\varphi^{(\Lambda)}_{2\mathbf{n}}\rangle$$ with kernel functions $$\varphi^{(\Lambda)}_{2\mathbf{n}}(u_{1},\ldots ,u_{2n}):=
\Theta(v-u-\Lambda) \varphi _{2\mathbf{n}}(u_{1},\ldots ,u_{2n})$$ where simply the range $v-u<\Lambda$ is cut out. Details of this regularization will be discussed elsewhere.

Acknowledgments {#acknowledgments .unnumbered}
---------------

This work was financed by Portuguese national funds through FCT - Funda[ç]{}ão para a Ciência e Tecnologia, within the project PTDC/MAT-STA/1284/2012.

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