ON THE SELF-INTERSECTION LOCAL TIME OF BROWNIAN MOTION-VIA CHAOS EXPANSION

We discuss the weak compactness problem related to the selfintersection local time of Brownian motion. We also propose a regular renormalization for self-intersection local time of higher dimensional Brownian motion.


Introduction
Let B t = (B 1 t , . . ., B d t ), 0 ≤ t < ∞ be the d-dimensional standard Brownian motion starting at 0 and let (Ω, F, W ) be the associated canonical probability space with the natural filtration (F t ) t≥0 .Let T > 0 be fixed throughout this paper.The following formal expression is called the self-intersection local time of the Brownian motion: (1.1) where δ is the Dirac delta function on R d and (1.1) is interpreted as the limit T 0 t 0 P ε (B t − B s ) ds dt as ε → 0, where P ε is the heat kernel.There are many references on it.In recent years there have been some results on the smoothness and renormalization of (1.1).When d = 2, Nualart and Vives [NV92] (see also [NV94]) proved that Y. Hu is in D α,2 for all α < 1/2, where D α,2 is the so-called Meyer-Watanabe distribution space.It was proved later by Imkeller, Pérez-Abreu and Vives [IPV93] that (1.2) is in D α,2 for all α < 1.In [AHZ95], Albeverio, Hu and Zhou proved that (1.2) is not in D α,2 when α ≥ 1, giving a complete description of the smoothness of (1.2) for d = 2. Motivated by a result of Yor [Yo85], Imkeller, Pérez-Abreu and Vives [IPV93] also proved that when d = 3, In this paper we shall prove that when As a by-product, we also prove the result of Imkeller, Pérez-Abreu and Vives in a simpler manner (we believe).Since when d ≥ 4, (1.4) is singular (the element of D α,2 is a distribution when α < 0), we propose a new renormalization scheme (see (5.2) below) to replace (1.4).
In section 2, we give a quick derivation of the chaos expansion of (1.1) and its approximation that we are going to use throughout this paper.In section 3, we provide some necessary estimates.In section 4, we prove the weak compactness result.In section 5, we propose a new (more regular) renormalization formula.

Chaos expansion
We will use the heat kernel to approximate the Dirac delta function δ on R d , where d is an integer ≥ 3. (We will concern with the case d ≥ 3.But the approach of this section works also for d = 1, 2.) Denote Remark 2.1.We will make use of the following simple facts without further mention: For f sufficiently regular We are going to deduce an explicit chaos expansion for the approximation of the self-intersection local time of Brownian motion.Denote (2.1) Given u ≥ s ≥ 0 and f ∈ C ∞ , consider the process Applying the Itô's formula to the above process, we obtain where we used the Einstein's convention on summation 2), we have Applying (2.2) to the integrand ∇ i P t−u1+ε in (2.3), noting ∆p t = 0 and repeating this we obtain Lemma 2.2. where
Theorem 2.3.Using the notations above, we have where J α (f ε α (T )) is defined as (see also [HM88]) Proof: It is obvious that the chaos of odd terms are zero.Let 0 Since it admits a chaos expansion according to the Wiener-Itô chaos expansion theorem.Since (2.10) is orthogonal to the remaining term obtained from (2.4), we see that (2.10) is the 2n-th chaos expansion of Remark 2.4.Letting ε → 0, we get a formal expansion of the self-intersection of local time (1.1).This formula was obtained in [FHSW94], [HWYY94] using the so-called S-transform.It was also obtained in [AHZ95] with another simpler technique when d = 2 and used to prove a non differentiability theorem.The obtention above seems to be the simplest.Let us also point out that the explicit chaos expansion of E ε (T ) is already known in [NV92], [NV94] and [IPV93] in terms of Hermite polynomials.The method used here appeared in [Hu94] to obtain the Isobe-Sato formula.

Some estimates
Let A n and B n , n = 1, 2, . . .be two sequences of real numbers.We denote A n ≈ B n iff there are two positive constants p > 0 and q > 0 (independent of n and T ) such that pA n ≤ B n ≤ qA n .
The following result should be found in literature and is stated explicitly in ([AHZ95]) when d = 2.However, we still give a simple proof.

Y. Hu
This shows that when d ≥ 4, proving (3.3).Similarly, we can prove that when d = 3, we have lim as ε → 0, where the last identity follows from (3.2) and (3.5).This gives the estimate for the L 2 norm of g ε n (T ).Now we have to estimate the L 2 norm of G ε n (T ).It is easy to see that for some positive constant 0 < µ < ∞.We should dominate the two terms arising from (3.9).When d ≥ 4, we have for n ≥ 1 By (3.3), we see that when d ≥ 4 lim ε→0 Similarly, we can prove by (3.6) that when d = 3, This gives the estimate arising from the first member of the RHS of (3.9).The same argument (using (3.4) and (3.7)) implies that the same conclusion holds for the second member of the RHS of (3.9).Thus we have lim ε→0 Thus we obtain Theorem 3.3.We have 1) when d = 3, (3.10) lim

Renormalization I
We denote F 2 := Ω |F (ω)| 2 P (dω).We need the following Definition 4.1.Let F = ∞ n=0 F n be the chaos expansion of F , where F n is the n-th chaos of F .We say that It is easy to see that D θ,2 is a Hilbert space.We will not discuss this space here.However, we refer to [Wa84] for more details.

Renormalization II
In [Yo85], Yor showed that (1.3) (when d = 3) converges in distribution to a Brownian motion which is independent of the original Brownian motions.In this section we will exclusively discuss the case d ≥ 4. It is unknown what is the limit of (1.4) (when d ≥ 4).In fact from Theorem 4.3, we see that (1.4) is unbounded in any space D θ,2 for θ ≥ 0. Therefore (1.4) is not regular (i.e. in the sense that it is not in the Meyer-Watanabe test functional space).For any θ ∈ R, let us introduce (5.1) t − B s ) ds dt − E T 0 t 0 δ(B t − B s ) ds dt *The author also holds a position in the Young Scientist Laboratory of Mathemtical Physics, Wuhan Institute of Mathematical Sciences, Chinese Academy of Sciences, Wuhan 430071, China.
d are integer or half integer.Using the explicit expression for P t