Continuous differentiability of renormalized intersection local times in <span class="mathjax-formula">$R^1$</span>

www.imstat.org/aihp 2010, Vol. 46, No. 4, 1025–1041

DOI:10.1214/09-AIHP338

© Association des Publications de l’Institut Henri Poincaré, 2010

Continuous differentiability of renormalized intersection local times in R ¹

Jay S. Rosen

Department of Mathematics, College of Staten Island, CUNY, Staten Island, NY 10314. E-mail:[email protected] Received 15 July 2009; revised 19 August 2009; accepted 1 September 2009

Abstract. We studyγ_k(x₂, . . . , x_k;t), thek-fold renormalized self-intersection local time for Brownian motion inR¹. Our main result says thatγ_k(x₂, . . . , x_k;t)is continuously differentiable in the spatial variables, with probability 1.

Résumé. Nous étudions γ_k(x₂, . . . , x_k;t), le temps local renormalisé d’auto-intersection d’ordrek du mouvement brownien dansR¹. Notre résultat principal montre queγ_k(x₂, . . . , x_k;t)est presque sûrement continûment différentiable dans les variables spatiales.

MSC:60J55; 60J65

Keywords:Continuous differentiability; Intersection local time; Brownian motion

1. Introduction

The object of this paper is to establish the almost sure continuous differentiability of renormalized intersection local time for the multiple intersections of Brownian motion inR¹.

Intersection local times were originally envisioned as a means of “measuring” the amount of self-intersections of Brownian motionWt∈R^m. Formally, thek-fold intersection local time is

α_k(x₂, x₃, . . . , x_k;t )=

· · ·

{0≤t₁≤···≤t_k≤t}

k

j=2

δ(W_tj−W_tj−1−x_j)dt1· · ·dtk,

whereδ(x)denotes theδ-function. Intuitively,α_k(0,0, . . . ,0;t )measures the “amount” ofk-fold intersections.

More precisely, we can set α_k,ε(x2, x3, . . . , x_k;t )

=

· · ·

{0≤t₁≤···≤t_k≤t}

k

j=2

f_ε(W_tj−W_tj−1−x_j)dt1· · ·dtk, (1.1) wheref_εis an approximateδ-function, and try to take theε→0 limit.

In two dimensions, limε→0α_k,ε(x₂, x₃, . . . , x_k;t ) will not exist unless all x_i =0! This gave rise to the prob- lem of renormalization: to subtract fromαk,ε(x2, x3, . . . , xk;t )terms involving lower order intersection local times,

1Supported in part by grants from PSC-CUNY and the NSF.

α_j,ε(x₂, x₃, . . . , x_k;t )forj < k, in such a way that a finite and continuous,ε→0 limit results. This was originally done for double intersections of Brownian motion by Varadhan [14], and gave rise to a large literature, see Bass and Khoshnevisan [1], Dynkin [2], Le Gall [3–5] and Rosen [7,8,10–12].

In this paper we are concerned with one dimensional Brownian motion. In this case, as we show below, the limit αk(x2, x3, . . . , xk;t )=lim

ε→0αk,ε(x2, x3, . . . , xk;t ) (1.2)

exists a.s. Although, as we will see,α_k(x₂, x₃, . . . , x_k;t )is almost surely continuous, it is notC¹in the spatial variable.

It is here that renormalization enters in the one dimensional case.

Let g(x)=

_∞

0

e^{−t /2}p_t(x)dt=e^−|x|, (1.3)

wherep_t(x)is the Brownian density function. We define the renormalized k-fold intersection local time forx = (x₂, . . . , x_k)∈R^k−1by

γk(x;t )=

A⊆{2,...,k}

(−1)^|A|

j∈A

g(xj)

αk−|A|(xA^c;t ), (1.4)

where for anyB= {i₁<· · ·< i_|B|} ⊆ {2, . . . , k}

x_B=(x_i1, x_i2, . . . , x_i|B|). (1.5)

Here, we use the conventionα₁(t)=t. Simple combinatorics then show that α_k(x;t )=

A⊆{2,...,k}

j∈A

g(x_j)

γ_k−|A|(x_A^c;t ). (1.6)

The renormalization (1.4) used here is similar to that used in [11] and [1] for two dimensional Brownian motion, but in that caseg(x)=_∞

0 e^{−t /2}p_t(x)dt is infinite whenx =0, compare (1.3). One key result of those papers is that γ_k(x;t )has a continuous extension from(R²− {0})^k−1×R₊to(R²)^k−1×R₊.

Here is our main result.

Theorem 1. For Brownian motion inR¹ α_k(x;t )=lim

ε→0α_k,ε(x;t ) (1.7)

exists and is jointly continuous a.s.Furthermore,γ_k(x;t )is differentiable inx and∇xγ_k(x;t )is jointly continuous with probability1.

Fork=2, this was established in [9] by very different techniques. It seems impossible to use those techniques for k >3.

In [13] we use Theorem1to give a simple proof of the CLT for theL²modulus of continuity of local time.

Note that g(x) is continuously differentiable for x =0. Equation (1.6) then exhibits precisely the non- differentiability ofαk(x;t ). This justifies our choice of renormalization (1.4). Simple combinatorics show that we obtain a similar result if we add anyC¹function tog(x). We have choseng(x)as a potential density to simplify our proofs.

Our paper is organized along the lines of [11]. That paper was concerned with the continuity ofγ_k(x₂, x₃, . . . , x_k;t ) in two dimensions, for Brownian motion and stable processes. Our challenge here is to study differentiability, and for ease of exposition we consider only Brownian motion. After laying the groundwork in Section2, we establish the exis- tence and almost sure continuity ofα_k(x₂, x₃, . . . , x_k;ζ )in Section3, whereζ is an independent mean-2 exponential random variable. In Section4 we show that the renormalizedk-fold intersection local time γ_k(x₂, x₃, . . . , x_k;ζ )is almost surely differentiable with a continuous derivative, again at an independent exponential time. In Section5we use martingale techniques to obtain a.s. joint continuty.

2. Intersection local times: moments

LetW_t denote Brownian motion inR¹with transition densitesp_t(x). In this section we introduce approximate inter- section local times forW_tand present formulas for the expectations of their moments.

Letf denote a smooth positive function supported on[−1,1], such that

f (x)dx=1, and for anyε >0 let f_ε(y)=1

εf y

ε

andf_ε,x(y)=f_ε(y−x). We define the approximate intersection local time of orderkas α_k,ε(x₂, x₃, . . . , x_k;t )

=

· · ·

{0≤t₁≤···≤t_k≤t}

k

j=2

f_ε,xj(W_tj−W_tj−1)dt1· · ·dtk. (2.1)

We often abbreviate this asα_k,ε(x;t )wherex=(x₂, x₃, . . . , x_k)∈R^k−1. Letg(x)=_∞

0 e^{−t /2}p_t(x)dt=e^−|x|denote the Green’s function forWt, and letζ denote a mean-2 exponential random variable independent ofWt. The following follows appear in [11], Section 2, and are reproduced here for the convenience of the reader. The first formula follows easily from the Markov property forWt.

Lemma 1.

E _n

i=1

α_ki,εi xⁱ;ζ

=

v∈V

ⁿ

i=1 ki

j=2

f_ε

i,x_jⁱ y_jⁱ −y_jⁱ₋₁^k

p=1

g(w_v(p)−w_v(p−1))dw1· · ·dwk, (2.2)

wherexⁱ=(x₂ⁱ, x₃ⁱ, . . . , x_kⁱ

i),k=n

i=1ki,(w1, . . . , wk)=(y¹, . . . , yⁿ)∈(R¹)^kandVis the set of permutationsvof {1,2, . . . , k}such that wheneverw_v(p)=y_jⁱ, w_v(p)˜ =yⁱ_˜

jwe havep >p˜⇐⇒j >j˜. A change of variables leads to the following more useful formula.

Lemma 2.

E _n

i=1

α_ki,εi xⁱ;ζ

=

s∈S

n i=1

k_i

j=2

f y_j^ik

p=1

g

z_s(p)+

c(p)

j=2

ε_s(p)y_j^s(p)+x_j^s(p)

−

z_s(p−1)+

c(p−1) j=2

ε_s(p−1)y_j^s(p−1)+x_j^s(p−1)

dy_jⁱdz1· · ·dzn, (2.3)

wherexⁱ=(xⁱ₂, x₃ⁱ, . . . , x_kⁱ

i),k=_n

i=1k_i,Sis the set of mappingss:{1,2, . . . , k} → {1, . . . , n}such that|s⁻¹(i)| = ki,∀1≤i≤n,andc(p)= |{u≤p|s(u)=s(p)}|.

3. Intersection local times: existence and continuity at exponential times

We first consider the intersection local timeα_kat an independent mean-2 exponential timeζ.

A functionZ_ε(x)indexed byε∈(0,1]andx in a topological spaceS will be said to converge locally uniformly inx asε→0 if for any compactK∈S,Z_ε(x)converges uniformly inx∈Kasε→0.

Theorem 2. Almost surely,αk,ε(x;ζ )converges locally uniformly inxasε→0.Hence αk(x;ζ ):=lim

ε→0αk,ε(x;ζ ) (3.1)

is continuous.

Furthermore,the occupation density formula holds:

Φ(x2, . . . , x_k)α_k(x2, . . . , x_k;ζ )dx2· · ·dxk

=

· · ·

{0≤t₁≤···≤t_k≤ζ}Φ(W_t2−W_t1, . . . , W_tk−W_tk−1)dt1· · ·dtk (3.2) for all bounded Borel measurable functionsΦonR^k−1.

Remark 1. The occupation density formula(3.2)shows thatαk(x;ζ )is independent of the particularf used to define αk,ε(x;ζ ).

Proof of Theorem2. We will show that forneven andγ >0 we can findδ >0 such that E α_k,ε(x;ζ )−α_k,ε x;ζn

≤c_n,γ(ε, x)− ε, x^δn/2 (3.3) for all 0< ε, ε≤γ /2 and allx, x∈R^k−1. The multidimensional version of Kolmogorov’s lemma, [6], Chapter 1, Theorem 2.1, then gives us that for anyδ< δand anyM <∞we have

α_k,ε(x;ζ )−α_k,ε x;ζ≤c_n,γ(ω)(ε, x)− ε, x^δ/2 (3.4)

for all rational 0< ε, ε≤γ /2 and all rationalx, x∈R^k−1,|x|,|x| ≤M. Sinceαk,ε(x;ζ )is clearly continuous as long asε >0, this will establish the statements concerning (3.1).

To establish (3.3) we first handle the variation inε. Ifhis a function ofε, let

ε,εh=h(ε)−h ε . From Lemma 2 we have

E _n

i=1

α_k,εi xⁱ;ζ

−α_k,ε

i xⁱ;ζn

= n

i=1 ε_i,ε_iE

_n

i=1

α_k,εi xⁱ;ζ

= n

i=1 ε_i,ε_i

s∈S

n i=1

k

j=2

f y_j^ink

p=1

g

z_s(p)+

c(p)

j=2

ε_s(p)y_j^s(p)+x_j^s(p)

−

zs(p−1)+

c(p−1) j=2

εs(p−1)y_j^s(p−1)+x_j^s(p−1)

dy_jⁱdz1· · ·dzn, (3.5)

where we eventually set all(ε_i, ε_i)=(ε, ε). We expand this as a sum of many terms using

ε,ε(uv)=( _ε,εu)v(ε)+u ε

( _ε,εv) (3.6)

so that each term contains for each 1≤i≤na single difference of the form _εi,ε

ig. Since each g factor in (3.5) involves at most two i’s, whenever our procedure gives two differences involving the sameg factor we write one of the differences as two terms. The upshot is that after setting all(ε_i, ε_i)=(ε, ε), the expectationE({α_k,ε(x;ζ )− α_k,ε(x;ζ )}ⁿ)can be written as a sum of many terms of the form appearing in (2.3) except that at leastn/2 of theg factors have been replaced by factors of the form

ε,ε,jg

z_s(p)+

c(p)

j=2

˜

ε_s(p)y_j^s(p)+x^s(p)_j

−

z_s(p−1)+

c(p−1) j=2

˜

ε_s(p−1)y_j^s(p−1)+x_j^s(p−1)

, (3.7)

whereε˜ can be variouslyε, εand the notation ε,ε,j denotes a difference between twogfactors of the above form in which one of theε’s, multiplyingy^s(p)_j ory_j^s(p−1)has been replaced byε.

Our result then follows using

g(x)−g(y)=e^−|x|−e^−|y|≤ |x−y| g(x)+g(y)

(3.8) for the variation inε, and the variation inxis handled similarly. This completes the proof of (3.1).

To prove the occupation density formula (3.2) we note that

Φ(x2, . . . , xk)αk,ε(x2, . . . , xk;ζ )dx2· · ·dxk

=

· · ·

{0≤t₁≤···≤t_k≤ζ}Φ∗F_ε(W_t2−W_t1, . . . , W_tk−W_tk−1)dt1· · ·dtk, (3.9) whereF_ε(x₂, . . . , x_k)=_k

j=2f_ε(x_j). Hence, by what we have established above, we can take theε→0 limit in (3.9) to yield (3.2) wheneverΦ is a bounded continuous function. The monotone convergence theorem then allows us to obtain (3.2) for all bounded Borel measurableΦ. This completes the proof of Theorem 2.

4. Renormalized intersection local times: continuous differentiability at exponential times

We have defined the renormalizedk-fold intersection local time forx=(x2, . . . , xk)∈R^k−1by γ_k(x;t )=

A⊆{2,...,k}

(−1)^|A|

j∈A

g(x_j)

α_k−|A|(x_A^c;t ), (4.1)

where for anyB= {i₁<· · ·< i_|B|} ⊆ {2, . . . , k}

x_B=(x_i1, x_i2, . . . , x_i|B|). (4.2)

Here, we use the conventionα1(t)=t. Define the approximate renormalizedk-fold intersection local time forx= (x₂, . . . , x_k)∈R^k−1by

γ_k,ε(x;t )=

A⊆{2,...,k}

(−1)^|A|

j∈A

g_ε(x_j)

α_k−|A|,ε(x_A^c;t ), (4.3)

whereg_ε(x)=f_ε∗g(x). Clearlyγ_k,ε(x;ζ )∈C¹for fixedε >0 so that for anyy_l, z_l γ_k,ε(x₂, . . . , x_l−1, z_l, x_l+1, . . . , x_k;ζ )

−γ_k,ε(x₂, . . . , x_l−1, y_l, x_l+1, . . . , x_k;ζ )= _zl

y_l

∂

∂x_lγ_k,ε(x;ζ )dxl. (4.4)

By Theorem2and the continuity ofgit follows that almost surelyγ_k,ε(x;ζ )→γ_k(x;ζ )asε→0, locally uni- formly inx. It follows from the next theorem that _∂x^∂

lγ_k,ε(x;ζ )→ψ_k(x;ζ )asε→0, locally uniformly inx, for some continuousψ_k(x). Hence it follows from (4.4) that for anyy_l, z_l

γ_k(x₂, . . . , x_l−1, z_l, x_l+1, . . . , x_k;ζ )

−γ_k(x2, . . . , x_l−1, y_l, x_l+1, . . . , x_k;ζ )= z_l

yl

ψ_k(x;ζ )dxl. (4.5)

This implies thatγ_k(x;ζ )is differentiable with respect tox_l, and _∂x^∂

lγ_k(x;ζ )=ψ_k(x;ζ ), hence is continuous inx. Theorem 3. Almost surely,for each2≤l≤k

∂

∂x_lγ_k,ε(x;ζ ) (4.6)

converges locally uniformly asε→0.Hence the limit is continuous inx.

Proof. As in the proof of Theorem2it suffices to show that forneven andγ >0 we can findδ >0 such that E

∂

∂x_lγk,ε(x;ζ )− ∂

∂x_lγ_k,ε(x;ζ ) n

≤cn,γ(ε, x)− ε, x^δn/2 (4.7)

for all 0< ε, ε≤γ /2 and allx, x∈R^k−1. Note that

∂

∂x_lα_k,ε(x₂, x₃, . . . , x_k;t )

= −

· · ·

{0≤t1≤···≤tk≤t} l−1

j=2

f_ε(W_tj−W_tj−1−x_j)

×(f_ε)(W_tl−W_tl−1−x_l) k

j=l+1

f_ε(W_tj−W_tj−1−x_j)dt1· · ·dtk. (4.8)

Note also that sinceg(x)is differentiable for allx=0, withg(x)= −g(x)forx >0 andg(x)=g(x)forx <0, it follows that for any compactly supportedf ∈C¹(R¹)

f(x)g(x)dx

=lim

ε→0

₋ε

−∞f(x)g(x)dx+ _ε

−ε

f(x)g(x)dx+ _∞

ε

f(x)g(x)dx

=lim

ε→0

f (−ε)g(−ε)− ₋ε

−∞f (x)g(x)dx

+lim

ε→0

ε

−ε

f(x)g(x)dx +lim

ε→0

−f (ε)g(ε)+ _∞

ε

f (x)g(x)dx

= − ₀

−∞f (x)g(x)dx+ _∞

0

f (x)g(x)dx

= − _∞

−∞f (x)g(x)dx, (4.9)

where for definiteness we setg(0)=0. Hence as in (2.3), and using the product rule for differentiation E

_n

i=1

∂

∂x_lⁱα_ki,εi xⁱ;ζ

=(−1)ⁿ

s∈S

n i=1

_k i

j=2,j=l

fε_i y_jⁱ

(fε_l) yⁱ_l

× k

p=1

g

z_s(p)+

c(p)

j=2

y_j^s(p)+x_j^s(p)

−

z_s(p−1)+

c(p−1) j=2

y_j^s(p−1)+x_j^s(p−1)

dy_jⁱdz1· · ·dzn

=

s∈S,a∈A

(−1)^a¯2 n

i=1 ki

j=2

f_εi(yⁱ_j)

× nk

p=1

g^{(a1(p)+a2(p))}

zs(p)+

c(p)

j=2

y_j^s(p)+x_j^s(p)

−

z_s(p−1)+

c(p−1) j=2

y_j^s(p−1)+x_j^s(p−1)

dy_jⁱdz1. . .dzn, (4.10)

whereg⁽⁰⁾=g, g⁽¹⁾=g, g⁽²⁾=g, andAis the set of mapsa=(a₁, a₂):[1, . . . , kn] → {0,1} × {0,1}such that:

• _nk

p=1a₁(p)+a₂(p)=n,

• for each 1≤i≤n nk

p=1

a1(p)1_{s(p)=i}+a2(p)1_{s(p−1)=i}=1,

• ifa₁(p)=1, thenc(p)≥l,

• ifa₂(p)=1, thenc(p−1)≥l.

In other words, ifs(p)=ithena1(p)=1 if and only if, after using the product rule for differentiation, thepthg factor is the onlygfactor to which ^∂

∂yⁱ_l has been applied. Similarly, ifs(p−1)=ithena₂(p)=1 if and only if, after using the product rule for differentiation, thepthgfactor is the onlygfactor to which ^∂

∂yⁱ_l has been applied. In (4.10),

¯

a2=nk

p=1a2(p).

Then by scaling E

_n

i=1

∂

∂x_lⁱαk_i,ε_i xⁱ;ζ

=

s∈S,a∈A

(−1)^a¯2 n

i=1 k_i

j=2

f y_jⁱ

× k

p=1

g^{(a1(p)+a2(p))}

z_s(p)+

c(p)

j=2

ε_s(p)y_j^s(p)+x_j^s(p)

−

zs(p−1)+

c(p−1) j=2

εs(p−1)y_j^s(p−1)+x_j^s(p−1)

dy_jⁱdz1· · ·dzn, (4.11)

where we eventually set allε_i=ε.

We note in particular that ifpis a “bad integer,” i.e.s(p)=s(p−1), theg^(m)factor in the above product has the form

g^(m) ε_s(p)y_c(p)^s(p)+x^s(p)_c(p)

(4.12) and in this casem=0 or 1.

Let us now analyze the changes which occur in (4.11) when we replace the factor _∂x^∂r

l αk_r,ε_r(x^r;ζ ) by

∂

∂x_l^r{(

j∈Bg_εr(x_j^r))α_kr−|B|,εr(x_B^rc;ζ )}. Keeping in mind (4.12) we see that nows runs over thoses∈S such that s(p)=r, c(p)∈B⇒s(p−1)=r, i.e. suchp’s are bad, and in the integrand on the right hand side of (4.11), aside from the factor

j∈Bg^(·)(ε_ry_j^r+x_j^r), all other occurences ofε_ry_i^r+x_i^r, i∈B are deleted.

Ifh(x)is any function of the variablex we use the notation Dxh=h(x)−h(0)

for the difference between the value ofhatxand it’s value atx=0. Ifs∈Swe setBs= {p|s(p)=s(p−1)}. The upshot is that we have

E _n

i=1

∂

∂x_lⁱγk_i,ε xⁱ;ζ

=

s∈S,a∈A

(−1)^a¯2 n

i=1 k_i

j=2

f y_jⁱ

p∈Bs

g^{(a1(p)+a2(p))} ε_s(p)y_c(p)^s(p)+x_c(p)^s(p)

×

p∈B_s

D_ε

s(p)y_j^s(p)+x^s(p)_c(p)

p∈B_s^c

g^{(a1(p)+a2(p))}

z_s(p)+

c(p)

j=2

ε_s(p)y_j^s(p)+x_j^s(p)

−

z_s(p−1)+

c(p−1) j=2

ε_s(p−1)y_j^s(p−1)+x_j^s(p−1)

dy_jⁱdz1· · ·dzn, (4.13) where we eventually set all(ε_i, ε_i)=(ε, ε).

To establish (4.7) we first handle the variation inε. By (4.13) E

_n

i=1

∂

∂x_lⁱγ_ki,ε xⁱ;ζ

− ∂

∂x_lⁱγ_ki,ε xⁱ;ζ

= n

i=1 εi,ε_i

s∈S,a∈A

(−1)^{a¯2 n}

i=1 ki

j=2

f y_jⁱ

×

p∈B_s

g^{(a1(p)+a2(p))} ε_s(p)y_c(p)^s(p)+x^s(p)_c(p)

p∈B_s

D_ε

s(p)y_j^s(p)+x_c(p)^s(p)

×

p∈B_s^c

g^{(a1(p)+a2(p))}

z_s(p)+

c(p)

j=2

ε_s(p)y_j^s(p)+x_j^s(p)

−

z_s(p−1)+

c(p−1) j=2

ε_s(p−1)y_j^s(p−1)+x_j^s(p−1)

dy_jⁱdz1· · ·dzn, (4.14)

where we eventually set all(ε_i, ε_i)=(ε, ε). We will show that this is bounded in absolute value byc_n|ε−ε|^δn/2 for all|ε|,|ε| ≤γ. (At this stage, and for ease in generalizing to the variation in x, we allow ε, ε to be zero or