On the self-intersection local time of Brownian motion-via chaos expansion

Publicacions Matem`atiques, Vol 40 (1996), 337–350. ON THE SELF-INTERSECTION LOCAL TIME OF BROWNIAN MOTION-VIA CHAOS EXPANSION Yaozhong Hu* Abstract 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. 1. Introduction Let Bt=(B1 t,... ,Bd 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 (Ft)t≥0. Let T>0be fixed throughout this paper. The following formal expression is called the self-intersection local time of the Brownian motion: (1.1) T 0t 0 δ(Bt−Bs)ds dt, where δis the Dirac delta function on Rdand (1.1) is interpreted as the limit T 0t 0Pε(Bt−Bs)ds dt as ε→0, where Pεis 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 (1.2) T 0t 0 δ(Bt−Bs)ds dt −ET 0t 0 δ(Bt−Bs)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. 338 Y. Hu is in Dα,2for all α<1/2, where Dα,2is the so-called Meyer-Watanabe distribution space. It was proved later by Imkeller, P´erez-Abreu and Vives [IPV93] that (1.2) is in Dα,2for all α<1. In [AHZ95], Albeverio, Hu and Zhou proved that (1.2) is not in Dα,2when α≥1, giving a complete description of the smoothness of (1.2) for d= 2. Motivated by a result of Yor [Yo85], Imkeller, P´erez-Abreu and Vives [IPV93] also proved that when d=3, (1.3) 1 log(1/ε)T 0t 0 Pε(Bt−Bs)ds dt −ET 0t 0 Pε(Bt−Bs)ds dt is weakly compact in Dα,2for all α<1/2 and when d≥4, (1.4) εd−4 2T 0t 0 Pε(Bt−Bs)ds dt −ET 0t 0 Pε(Bt−Bs)ds dt is weakly compact in Dα,2for all α<4−d 2. In this paper we shall prove that when d= 3 (1.3) is unbounded in Dα,2when α≥1/2 and when d≥4, (1.4) is unbounded in Dα,2when α≥4−d 2. As a by-product, we also prove the result of Imkeller, P´erez-Abreu and Vives in a simpler manner (we believe). Since when d≥4, (1.4) is singular (the element of Dα,2is 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. 2. Chaos expansion We will use the heat kernel to approximate the Dirac delta function δ on Rd, where dis an integer ≥3. (We will concern with the case d≥3. But the approach of this section works also for d=1,2.) Denote Pt(x)=(2πt)−d/2e−|x|2 2t;Pt(x, y)=Pt(x−y),t>0,x,y∈Rd, Self-intersection local time, chaos expansion 339 ∇if:= ∂ ∂xi f, Pεf(x)=Rd Pε(x, y)f(y)dy. Remark 2.1. We will make use of the following simple facts without further mention: For fsufficiently regular ∂ ∂tPt(x, y)=1 2 d  i=1 ∇iPt(x, y); ∂ ∂xi Ptf=Pt∂ ∂xi f,i=1,... ,d. Rd Ps(x, z)Pt(z,y)dz =Pt+s(x, y),P 0f(x) := lim t→0Ptf(x)=f(x). We are going to deduce an explicit chaos expansion for the approximation of the self-intersection local time of Brownian motion. Denote (2.1) Eε(T)=T 0t 0 Pε(Bt−Bs)ds. Given u≥s≥0 and f∈C∞, consider the process Xv:= Pu−vf(Bv−Bs),s≤v≤u. Applying the Itˆo’s formula to the above process, we obtain (2.2) f(Bu−Bs)=Pu−sf(0) + u s ∇iPu−u1f(Bu1−Bs)dBi u1, where we used the Einstein’s convention on summation for i=1,... ,d, i.e. aibi:= d i=1 aibi. Taking f=Pεin (2.2), we have (2.3) Pε(Bt−Bs)=Pt−s+ε(0) + t s ∇iPt−u1+ε(Bu1−Bs)dBi u1. Applying (2.2) to the integrand ∇iPt−u1+εin (2.3), noting ∆pt= 0 and repeating this we obtain Lemma 2.2. Pε(Bt−Bs)=Pt−s+ε(0) + t s ∇iPt−u1+ε(0) dBi u1+··· +s≤u1<···<un≤t ∇α1···∇ αnPt−s+ε(0) dBα1 u1···dBαn un +s≤u1<···<un+1≤t ∇α1···∇ αn+1 Pt−u1+ε(Bu1−Bs)(2.4) dBα1 u1···dBαn+1 un+1 =Pt−s+ε(0) + I1+···+In+In+1, 340 Y. Hu where Ij:=s≤u1<···<uj≤t ∇α1···∇ αjPt−s+ε(0) dBα1 u1···dBαj un,j=1,2,...,n; and In+1 :=s≤u1<···<un+1≤t ∇α1···∇ αn+1 Pt−u1+ε(Bu1−Bs)dBα1 u1···dBαn+1 un+1 . For a fixed 0 ≤j≤n,Ijis orthogonal to Ik,k=0,1,... ,j −1, j+1,... ,n+ 1, i.e. EIjIk=0. SoInis the n-th Itˆo-Wiener chaos of Pε(Bt−Bs). Now we compute ∇α1···∇ αnPt−s+ε(0) in (2.4). First we define for j=1,... ,d and α=(α1,... ,α 2n), we set 2nj:= 2nj(α):= #{αi;αi=j}and n=n1+···+nd.nj,j =1,... ,d are integer or half integer. Using the explicit expression for Pt Pt−s+ε(x)=[2π(t−s+ε)]−d/2e−|x|2 2(t−s+ε) =[2π(t−s+ε)]−d/2 ∞  n=0 (−1)n|x|2n 2nn!(t−s+ε)n, we have for nj,j=1,... ,d which are integer (2.5) ∇α1···∇ α2nPt−s+ε(0) = (−1)n(2n1)! ···(2nd)! (2π)d/22nn1!···nd!(t−s+ε)n+d/2. Other derivatives vanish at 0. We also say that ∅is also a multi-index with length |α|= 0. We set when |α|=0, C∅=   1 8π2when d=4; (2π)−d/2when d=4 and f∅=     2ε−1T−log T+ε εwhen d=4; 2(d−2)−12(d−4)−1ε−d/2+2 −(T+ε)−d/2+2−ε−d/2+1T when d=4. And we introduce for |α|≥1, (2.6) Cα=(−1)n(2n1)! ···(2nd)! (2π)d/22nn1!···nd!(n+d/2−1)−1(n+d/2−2)−1 and (2.7) fε α(T)(u1,... ,u 2n)=[−(u2n+ε)−n−d/2+2 +(T+ε)−n−d/2+2 +(u2n−u1+ε)−n−d/2+2 −(T−u1+ε)−n−d/2+2]. We will use |α|=2nto denote the sum over all α=(α1,... ,α 2n) such that #{αi;αi=j}is even. Self-intersection local time, chaos expansion 341 Theorem 2.3. Using the notations above, we have (2.8) Eε(T)= ∞  n=0  |α|=2n CαJα(fε α(T)), where Jα(fε α(T)) is defined as (see also [HM88]) (2.9) Jα(fε α(T)):=0≤u1<···<u2n≤T fε α(T)(u1,... ,u 2n)dBα1 u1···dBα2n u2n. Proof: It is obvious that the chaos of odd terms are zero. Let 0 ≤ u1<···<u 2n≤T. Then it is easy to show that (2.10) T 0t 0s≤u1<···<un≤t ∇α1···∇ αnPt−s+ε(0) dBα1 u1···dBαn un =CαJα(fε α(T)). Since T 0t 0Pε(Bt−Bs)ds dt is in L2, it admits a chaos expansion according to the Wiener-Itˆo 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 T 0t 0Pε(Bt−Bs)ds dt. 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 Eε(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. 3. Some estimates Let Anand Bn,n=1,2,... be two sequences of real numbers. We denote An≈Bniff there are two positive constants p>0 and q>0 (independent of nand T) such that pAn≤Bn≤qAn. The following result should be found in literature and is stated explicitly in ([AHZ95]) when d= 2. However, we still give a simple proof. 342 Y. Hu Lemma 3.1. Let Cαbe given by (2.6). Then (3.1)  |α|=2n C2 α≈(2n)!nd 2−5. Proof: By the Stirling formula n!=(2π)1/2nn+1/2e−n(1 + O(1/n)), i.e. n!≈nn+1/2e−nwe see that (2n1)! ···(2nd)! 22nn1!2···nd!2≈(n1+1) −1/2···(nd+1) −1/2. (We allow n1,... ,n dto be 0.) On the other hand it is easy to have  n1+···+nd=n (n1+1) −1/2···(nd+1) −1/2 ≈u1,... ,ud−1≥0 (u1+1) −1/2···(nd−1+1) −1/2 (n−u1−···−ud−1+1) −1/2du1···dud−1 ≈nd/2−1. Thus  |α|=2n C2 α≈ n1+···+nd=n (2n)! (2n1)! ···(2nd)! (2n1)! ···(2nd)! 2nn1!2···nd!22 n−4 ≈(2n)!  n1+···+nd=n (n1+1) −1/2···(nd+1) −1/2≈(2n)!nd/2−5 proving the lemma. We need the elementary computation Lemma 3.2. Let d≥3and n≥1. Then when d≥4, (3.2) lim K→∞ K−10≤u<v≤K (v−u+1) −2n−d+4(v−u)2n−2du dv =∞ 0 (x+1) −2n−d+4x2n−2dx; (3.3) lim K→∞ K−10≤u<v≤K (v+1) −2n−d+4(v−u)2n−2du dv =0; Self-intersection local time, chaos expansion 343 (3.4) lim K→∞ K−10≤u<v≤K (K−u+1) −2n−d+4(v−u)2n−2du dv =0; When d=3, we have (3.5) lim K→∞(Klog K)−10≤u<v≤K (v−u+1)−2n+1(v−u)2n−2du dv =1; (3.6) lim K→∞(Klog K)−10≤u<v≤K (v+1) −2n+1(v−u)2n−2du dv =0; (3.7) lim K→∞(Klog K)−10≤u<v≤K (K−u+1)−2n+1(v−u)2n−2du dv =0. Proof: Making the transformation u=yand v−u=x, we have 0≤u<v<K (v−u+1) −2n−d+4(v−u)2n−2du dv =K 0 dx K x dy(x+1) −2n−d+4x2n−2 =K 0 (K−x)(x+1) −2n−d+4x2n−2dx =KK 0 (x+1) −2n−d+4x2n−2dx −K 0 (x+1) −2n−d+4x2n−1dx =: AK+BK. Let K→∞. When d≥4, we have lim K→∞ K−1AK=∞ 0 (x+1) −2n−d+4x2n−2dx and limK→∞ K−1BK= 0. This gives (3.2). When d= 3, one can see that limK→∞(Klog K)−1AK= 1 and limK→∞(Klog K)−1BK=0. This proves (3.5). Now 0≤u<v≤K (v+1) −2n−d+4(v−u)2n−2du dv ≤2 2n−2+1K 0 (v+1)−d+3 dv ≤3 2n−2+1(K+1)−d+4 log(K+1). 344 Y. Hu This shows that when d≥4, lim K→∞ K−10≤u<v≤K (v+1) −2n−d+4(v−u)2n−2du dv =0, proving (3.3). Similarly, we can prove that when d=3,wehave lim K→∞(Klog K)−10≤u<v≤K (v+1) −2n+1(v−u)2n−2du dv =0, proving (3.6). Now we estimate 0≤u1<···<u2n≤T|fε α(T)(u1,... ,u 2n)|2du1···du2n. Denote for n≥1 gε n(T)(u1,... ,u 2n):=(u2n−u1+ε)−n−d/2+2 and Gε n(T)(u1,... ,u 2n) =−(u2n+ε)−n−d/2+2 +(T+ε)−n−d/2+2 −(T−u1+ε)−n−d/2+2 so that fε α(T)=gε n(T)+Gε n(T). First we have (3.8) 0≤u1<···<u2n≤T |gε α(T)(u1,... ,u 2n)|2du1···du2n =0≤u1<···<u2n≤T (u2n−u1+ε)−2n−d+4 du1···du2n =1 (2n−2)!0≤u1<u2n≤T (u2n−u1)−2n−d+4(u2n−u1+ε)2n−2du1du2n =ε4−d (2n−2)! 0≤u<v≤T/ε (v−u+1) −2n−d+4(v−u)−2n−2du dv =T (2n−2)! ∞ 0(x+1) −2n−d+4x2n−2dx ·O(ε3−d)d≥4 T (2n−2)! ·Olog 1 ε,d=3 as ε→0, where the last identity follows from (3.2) and (3.5). This gives the estimate for the L2norm of gε n(T). Now we have to estimate the L2 norm of Gε n(T). It is easy to see that (3.9) |Gn(u1,... ,u 2n)|≤µ[(u2n+ε)−n−d/2+2 +(T−u1+ε)−n−d/2+2] Self-intersection local time, chaos expansion 345 for some positive constant 0 <µ<∞. We should dominate the two terms arising from (3.9). When d≥4, we have for n≥1 0≤u1<···<u2n≤T (u2n+ε)−2n−d+4 du1···du2n =0≤u1<u2n≤T (u2n+ε)−2n−d+4(u2n−u1)2n−2du1du2n =ε4−d (2n−2)! 0≤u<v≤T/ε (v+1) −2n−d+4(v−u)−2n−2du dv. By (3.3), we see that when d≥4 lim ε→0εd−30≤u1<···<u2n≤T (u2n+ε)−2n−d+4 du1···du2n=0. Similarly, we can prove by (3.6) that when d=3, 1 log(1/ε)0≤u1<···<u2n≤T (u2n+ε)−2n+1 du1···du2n=0. 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→0εd−30≤u1<···<u2n≤T |Gε n(T)|2du1···du2n= 0 when d≥4 and 1 log(1/ε)0≤u1<···<u2n≤T |Gε n(T)|2du1···du2n= 0 when d=3. Thus we obtain Theorem 3.3. We have 1) when d=3, (3.10) lim ε→0 1 log(1/ε)0≤u1<···<u2n≤T |fε α(T)(u1,... ,u 2n)|2du1···du2n =T (2n−2)!. 2) When d≥4, (3.11) lim ε→0εd−30≤u1<···<u2n≤T |fε α(T)(u1,... ,u 2n)|2du1···du2n =T (2n−2)! ∞ 0 (x+1) −2n−d+4x2n−2dx.