Fluctuations of Brownian motion with drift
Abstract
Consider 3 dimensional Brownian motion started on the unit sphere {|x| = 1} with initial density ρ. Let ρt be the first hitting density on the sphere {|x| = t + 1}, t.
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Publicacions Matem`atiques, Vol 43 (1999), 85–125. FLUCTUATIONS OF BROWNIAN MOTION WITH DRIFT Joseph G. Conlon∗and Peder Olsen∗∗ Abstract Consider 3 dimensional Brownian motion started on the unit sphere {|x|=1}with initial density ρ. Let ρtbe the first hitting density on the sphere {|x|=t+1},t>0. Then the linear operators Ttdefined by Ttρ=ρtform a semigroup with an infinitesimal generator which is approximately the square root of the Laplacian. This paper studies the analogous situation for Brownian motion with a drift b, where bis small in a suitable scale invariant norm. Chapter 1. Introduction In two previous papers [CR], [CO] we studied the Dirichlet problem for an elliptic equation on a domain in R3. Let BRbe the ball of radius R in R3centered at the origin, 0 <R<∞. Consider the problem (1.1) (−∆−b(x)·∇)u(x)=f(x),x∈BR, u(x)=0,x∈∂BR. A function g:R3→Cis said to be in the Morrey space Mq p, 1≤p≤q<∞,if|g|pis locally integrable and there is a constant C such that (1.2) Q|g|pdx ≤Cp|Q|1−p/q, for all cubes Q⊂R3. Here |Q|denotes the volume of Q. The norm of g,gq,p is defined as the minimum Cfor which (1.2) holds. In [CR]we proved the following: ∗Research partially supported by the U.S. National Science Foundation under grants DMS 9403399 and DMS 9404197. ∗∗ Research supported by NFR - Norges Forskingsr˚ad (Norwegian Research Council) grant 100222/40.
86 J. G. Conlon, P. Olsen Theorem 1.1. Suppose 1<r<p≤q,1<p≤3, and |b|∈M3 p, f∈Mq r, for some qwith q>3/2. Then there exists >0depending only on r,p,qsuch that if b3,p <then the Dirichlet problem (1.1) is solvable. Further, there is a constant Cdepending only on r,p,qsuch that the solution uof (1.1) satisfies the inequality, u∞≤CR2−3/qfq,r. The condition b3,p small for some p,1<p≤3, includes the two important cases when |b(x)|=/|x|and |b|∈L3(R3), b3=, 1. Theorem 1.1 is a perturbative result. Writing the solution of (1.1) as a perturbation series in bone can show that the series converges if b3,p 1. Observe also that Theorem 1.1 is a scale invariant theorem. The result for general Rcan be obtained by a scaling argument from the result for a particular value of R. Since all the results in this paper have the same scaling property we shall take R=1/4 from here on. It is well known [SV] that the solution of the Dirichlet problem (1.1) has a representation as an expectation value with respect to Brownian motion with drift b. Let Xb(t), t≥0, denote the drift process started at time 0. If Xb(0) ∈B1/4let τbe the minimum time tsuch that Xb(t)∈∂B1/4. Then the solution uof (1.1) is given by u(x)=Exτ 0 f(Xb(t)) dt,x∈B1/4, where Exdenotes that the expectation value is taken conditioned on Xb(0) = x. Theorem 1.1 tells us therefore something about the behavior of the diffusion process Xb(t) when b3,p 1. It gives us similar estimates on the expected time the diffusion spends in a subset of B1/4 before exiting ∂B1/4to those one has for standard Brownian motion. In [CO] we proved a nonperturbative version of Theorem 1.1, allowing bto be only locally in a Morrey space M3 pwith small norm. A key ingredient in the proof of this theorem was the fact that the fluctuations of Xb(t) did not increase as tincreases, provided b3,p 1. To be specific, suppose 0 <ρ≤1/2, and the process Xb(t) starts on the sphere ∂B(1−ρ)/4with initial density function f, and fρ,bis the first hitting density on the sphere ∂B1/4. It is evident, by conservation of probability, that the average value of fis the same as the average value of fρ,b,Avf =Avfρ,b.In[CO] we proved the following:
Fluctuations of BM 87 Theorem 1.2. Suppose 0<ρ≤1/2and 1<p≤3. Consider f and fρ,bto be functions on the unit sphere S, with f∈L2(S). Then for δ>0there exists , depending only on p,ρ,δsuch that, if b3,p < and f−Avf2≤δ|Avf|, one has fρ,b∈L2(S)and fρ,b−Avfρ,b2≤ δ|Avfρ,b|. It is easy to see that Theorem 1.2 holds uniformly in ρas ρ→0 for the case of Brownian motion, b≡0. Since bwith b3,p 1 is perturbative to Brownian motion it is natural to expect a similar uniformity when b3,p is small. In this paper we prove a uniform version of Theorem 1.2. We cannot however use the L2norm to measure the oscillation of fand fρ,b. We must use a finer norm which weights high Fourier modes more than low Fourier modes. This subtlety is closely related to the extra complication in the proof of Theorem 1.2 over Theorem 1.1. To prove Theorem 1.1 one shows that a certain integral operator is bounded on Morrey spaces. To prove Theorem 1.2 one needs to know that this same integral operator is bounded on a weighted Morrey space, where the weight of a point x∈B1/4decreases as xgets close to ∂B1/4. To define our new norm on functions fwith domain S, let ∆Sbe the Laplace operator on the unit sphere. For k=1,2,..., let Ekbe the L2projection operator onto the space spanned by the eigenfunctions of −∆Swith eigenvalues λ2satisfying 2k−1<λ≤2k. Let E0be the projection onto the constant function. For f:S→Cand ν>0we define f2,ν by f2,ν = sup k≥0 2νkEkf2. We then have the following: Theorem 1.3. Suppose 0<ρ≤1/2,1<p≤3, and ν>0is sufficiently small, depending only on p. Then for δ>0there exists >0 depending only on p,δsuch that if b3,p <and f−Avf2,ν ≤δ|Avf|, one has fρ,b−Avfρ,b2,ν ≤δ|Avfρ,b|. We consider the relationship between the proof of Theorem 1.1 and the proof of Theorem 1.2. Let GDbe the Dirichlet Green’s function for −∆onB1/4, whence GDis given explicitly by the formula, GD(x, y)= 1 4π|x−y|−1 16π|y| 1 |x−¯y|, where ¯yis the reflection of yin ∂B1/4. Let Tbe the integral operator on functions with domain B1/4given by (1.3) Tf(x)=B1/4 b(x)·∇xGD(x, y)f(y)dy, x ∈B1/4.
88 J. G. Conlon, P. Olsen It was shown in [CR] that Theorem 1.1 is a consequence of the fact that Tis a bounded operator on the Morrey space Mq rwith norm, T, satisfying T≤Cb3,p for some constant Cdepending only on p,q,r. For ga function with domain ∂B1/4let v(y)=Pg(y), y∈B1/4,be the solution of the Dirichlet problem, ∆v(y)=0,y∈B1/4, v(y)=g(y),y∈∂B1/4. The function vis given explicitly by the Poisson formula, (1.4) Pg(y)= 1 π∂B1/4 1/16 −|y|2 |y−x|3g(x)dx, y ∈B1/4. We can formally define an integral operator Qon the functions gby (1.5) Qg(x)=B1/4 dy GD(x, y)(I−T)−1b·∇Pg(y),x∈B1/4, where Tis given by (1.3). This operator induces an operator on functions with domain Sas follows: Let f:S→Cand denote also by fthe function with domain ∂B1/4naturally induced by f. Consider now the function Qρfwith domain ∂B(1−ρ)/4defined by (1.6) Qρf(x)=Qf(x),x∈∂B(1−ρ)/4. We can think of Qρfas a function with domain S, whence Qρis an operator on functions with domain S.In[CO] we proved that Qρis a bounded operator on L2(S), 0 <ρ≤1/2, with norm Qρsatisfying Qρ≤Cb3,p for some constant Cdepending only on ρ,p, provided b3,p <for some depending only on p. Theorem 1.2 is a consequence of this fact. Observe that the proof of Theorem 1.2 must be more difficult than the proof of Theorem 1.1 since in the definition of Qρone assumes that the inverse (I−T)−1exists. To prove Theorem 1.3 we need to know not only that Qρis bounded on L2(S) for 0 <ρ≤1/2 but also to have a bound which is uniform as ρ→0. Let ,denote the scalar product on L2(S). In section 2 we prove the following:
Fluctuations of BM 89 Theorem 1.4. Let 0<ρ≤1/2,1<p≤3. Then there exists C, >0depending only on p, such that |f,Qρg| ≤ Cb3,pf2g2, provided b3,p ≤,f,g∈L2(S). In order to prove Theorem 1.3 we need to know more detailed properties of Qρthan those given in Theorem 1.4. In particular we must know that if Qρacts on a slowly varying function gthen the slowly varying component of Qρghas norm bounded linearly in ρas ρ→0. We also need to know that if gis highly oscillatory then the slowly varying component of Qρghas small norm. These properties of Qρare summarised in the following: Theorem 1.5. Let 0<ρ≤1/2,1<p≤3. Then there exists C,, η>0depending only on psuch that b3,p <implies that |Ekf,QρEkg| ≤ Cb3,pEkf2Ekg2min[ρ2k,1] min[2η(k−k),1] 0≤k, k<∞. We prove Theorem 1.5 in section 3. Theorem 1.3 is a simple consequence of Theorems 1.4 and 1.5. It is proved in section 4. Chapter 2. Configuration Space Localization To prove Theorem 1.4 we shall modify the proof in [CO] so that the estimates are uniform in ρas ρ→0. Define an operator Aon functions gwith domain S1/4={|x|=1/4} by (2.1) Ag(y)=|b(y)|1/2Pg(y),|y|<1/4. Proposition 2.1. Ais a bounded linear operator from L2(S1/4)to L2(B1/4). There is a constant Cdepending only on p>1such that A≤Cb1/2 3,p .
90 J. G. Conlon, P. Olsen We shall prove Proposition 2.1 by following the general lines of the proof of Theorem 1.2 of [CR]. If a function uis defined on the sphere S1/4 then Au is defined on the ball B1/4. Let Q0be a cube with side of length 1 and for n=1,2,... let Qnbe the dyadic subcubes of Q0with side of length 2−n. We define uQnas follows: uQn=0 ifQn∩S1/4is empty, uQn=|Qn|−2/3Qn∩S1/4|u(x)|dx, otherwise. For ξ∈R3let Q0(ξ) be the unit cube centered at ξwith corresponding dyadic subcubes Qn(ξ). We then have the following: Lemma 2.1. There exists a universal constant Csuch that (2.2) B1/4|Au(y)|2dy ≤CB1/4 dξ ∞ n=0 Qn(ξ)⊂Q0(ξ) u2 Qn(ξ)Qn(ξ)|b(y)|dy. Proof: We have from the definition (2.1) that Au(y)=|b(y)|1/21 16πS1/4 1−16|y|2 |y−x|3u(x)dx. We estimate Au(y) in the annulus Rm=y:1 4(1 −2−m)≤|y|<1 4(1 −2−(m+1)),m=0,1,2,... . Thus |Au(y)|≤C|b(y)|1/22−m m n=0 23nS1/4∩{|x−y|<2−n−1}|u(x)|dx, for some universal constant C. Choosing α>0 and applying the Schwarz inequality to the RHS of the previous expression we have |Au(y)|2 ≤Cα|b(y)|2−2m(1−α) m n=0 22(3−α)nS1/4∩{|x−y|<2−n−1}|u(x)|dx2 ,
Fluctuations of BM 91 for some constant Cαdepending only on α.Thus B1/4|Au(y)|2dy =∞ m=0 Rm|Au(y)|2dy ≤Cα 0≤n≤m<∞ 2−2m(1−α)22(3−α)nRm dy|b(y)| S1/4∩{|x−y|<2−n−1}|u(x)|dx2 . Let Un=∪∞ m=nRm,n≥0. Then it is clear from the previous expression that if α<1 there is a constant Cαdepending only on αsuch that B1/4|Au(y)|2dy ≤Cα ∞ n=0 an, where an=Un dy|b(y)|22nS1/4∩{|x−y|<2−n−1}|u(x)|dx2 . It is clear now that there exists a universal constant Csuch that an≤CB1/4 dξ Qn(ξ)⊂Q0(ξ) u2 Qn(ξ)Qn(ξ)|b(y)|dy, n ≥0. The result follows then from the last two inequalities. Next we shall show that for fixed ξthe RHS of (2.2) is bounded by the L2norm of u. Lemma 2.2. Let Q0be a cube in R3with side of length 1and dyadic subcubes Qnwith side of length 2−n,n=1,2,... . Then there is a constant Cdepending only on p>1such that (2.3) ∞ n=0 Qn⊂Q0 u2 QnQn|b(y)|dy ≤Cb3,p u2 2. Evidently Proposition 2.1 follows from Lemma 2.1 and Lemma 2.2. Observe that for fixed none has Qn⊂Q0 u2 QnQn|b(y)|dy ≤ Qn⊂Q0 u2 Qnb3,p 2−2n ≤Cb3,p Qn⊂Q0Qn∩S1/4|u(x)|2dx ≤Cb3,p u2 2,
92 J. G. Conlon, P. Olsen for some universal constant C. In order to do the summation with respect to nin (2.3) we need to resort to a Calderon-Zygmund decomposition. First we have Lemma 2.3. Let Qbe a cube in R3with side of length 2−nQ, where nQis a nonnegative integer. Suppose for some ε>0one has |Q|εuQ≤ |Q|εuQfor all dyadic subcubes Qof Q. Then if εis sufficiently small there exists a constant Cdepending only on p>1such that Q⊂Q u2 QQ|b(y)|dy ≤Cb3,p |Q|2/3u2 Q. Proof: We have for fixed n≥nQ, Qn⊂Q u2 QnQn|b(y)|dy ≤26(n−nQ)εu2 Q {Qn:Qn⊂Q,Q n∩S1/4=∅}Qn|b(y)|dy ≤26(n−nQ)εu2 QQ∩{y:1>|y|>1−2−n√3}|b(y)|dy ≤26(n−nQ)εu2 Qmeas Q∩{y:1>|y|>1−2−n√3}1−1/p Q|b(y)|pdy1/p ≤C26(n−nQ)εu2 Q2−2nQ2−n1−1/pb3,p |Q|1/p−1/3 =Cb3,p |Q|2/3u2 Q2(n−nQ)(6ε+1/p−1), where Cis a universal constant. If we now choose εto satisfy ε<(1 −1/p)/6, then we have Q⊂Q u2 QQ|b(y)|dy ≤Cb3,p |Q|2/3u2 Q ∞ n=nQ 2(n−nQ)(6ε+1/p−1) ≤Cpb3,p |Q|2/3u2 Q, where the constant Cpdepends only on p>1.
Fluctuations of BM 93 Let Q0be a unit cube in R3. We make a Calderon-Zygmund decomposition of Q0based on the criterion in Lemma 2.3. In particular we define a sequence of families Fjof dyadic subcubes of Q0,j=0,1,2,... as follows: F0={Q0}. Let G1⊂Q0be defined as G1={x∈Q0:|Q|εuQ≤|Q0|εuQ0 for all dyadic subcubes Qof Q0with x∈Q}. Then there is a unique finite family F1of disjoint dyadic subcubes of Q0 such that Q∈F1 Q=Q0\G1. Proceeding by induction as in section 2 of [CR], we can construct sets Gjand families Fj,j≥1, with the properties (a) ∞ j=1 Gj=Q0. (b) Q∈FkQ=Q0\k j=1 Gj. (c) For any Q∈F klet ¯ Q∈F k−1be the unique dyadic subcube containing Q. Then |Q|εuQ>|¯ Q|εu¯ Q. It is clear now from Lemma 2.3 that there is a constant Cdepending only on p>1 such that ∞ n=0 Qn⊂Q0 u2 QnQn|b(y)|dy ≤Cb3,p ∞ j=0 Q∈Fj|Q|2/3u2 Q. The proof of Lemma 2.2 will be complete if we can show Lemma 2.4. There exists a constant Cdepending only on p>1such that (2.4) ∞ j=0 Q∈Fj|Q|2/3u2 Q≤Cu2 2.
100 J. G. Conlon, P. Olsen Proof: We have ∞ n=nQ Snh(x) 2 ≤2∞ k=nQ Skh(x) k n=nQ Snh(x). It follows from Holder’s inequality that Snh(x)≤22n d(Qn)Qn∩B1/4 (1/4−|y|)q|h(y)|qdy1/qQn|b(y)|q/2dy1/q , where 1/q +1/q= 1. Since 1/q < 1−1/2pimplies q/2<pwe conclude that Snh(x)≤b1/2 3,p |Qn|1/6hQnd(Qn),x∈Qn, whence (2.8) (1/4−|x|)Snh(x)≤b1/2 3,p |Qn|1/6hQn,x∈Qn. Now if we use (2.7) we have from the previous inequality (1/4−|x|) k n=nQ Snh(x)≤b1/2 3,p k n=nQ|Qn|1/6hQn, ≤b1/2 3,p |Q|1/6hQ k n=nQ 23ε(n−nQ) ≤Cεb1/2 3,p |Q|1/6hQ23ε(k−nQ) for some constant Cεdepending only on ε>0. Hence (2.9) Q∩B1/4 (1/4−|x|)2 ∞ n=nQ|b(x)|1/2Snh(x) 2 dx ≤2Cεb1/2 3,p |Q|1/6hQ ∞ k=nQ 23ε(k−nQ) Q∩B1/4 (1/4−|x|)|b(x)|Skh(x)dx. For man integer let Embe the set Em=x∈R3:2 m−1<|b(x)|≤2m.
Fluctuations of BM 101 For m,kintegers and k≥nQlet am,k be given by am,k = Qk⊂Q|Em∩Qk|Qk (1/4−|y|)|b(y)|1/2|h(y)|dy. Then we have (2.10) ∞ k=nQ 23ε(k−nQ)Q∩B1/4 (1/4−|x|)|b(x)|Skh(x)dx ≤∞ m=−∞ ∞ k=nQ 23ε(k−nQ)2m+2kam,k. There are two estimates on am,k which we use. The first follows from (2.7). Thus from (2.8) we have am,k ≤ Qk⊂Q|Em∩Qk|b1/2 3,p 2−5k/2hQk, and then (2.7) implies that am,k ≤|Em∩Q|b1/2 3,p 2−5k/22(1/2+3ε)(k−nQ)hQ. The second estimate is obtained by using the fact that |Em∩Qk|≤|Qk|. Thus am,k ≤2−3kQ (1/4−|y|)|b(y)|1/2|h(y)|dy. If we again apply (2.8) we have that am,k ≤2−3kb1/2 3,p |Q|5/6hQ. If follows now that for any α,0<α<1, we have ∞ m=−∞ ∞ k=nQ 23ε(k−nQ)2m+2kam,k ≤∞ m=−∞ ∞ k=nQ 23ε(k−nQ)2m+2k2−3kb1/2 3,p |Q|5/6hQα |Em∩Q|b1/2 3,p 2−5k/22(1/2+3ε)(k−nQ)hQ1−α.
102 J. G. Conlon, P. Olsen For any α>0 one can find sufficiently small ε>0 such that the sum with respect to kabove converges. Thus there is a constant Cα>0 depending only on α,εand (2.11) ∞ m=−∞ ∞ k=nQ 23ε(k−nQ)2m+2kam,k ≤Cαb1/2 3,p |Q|α+1/6hQ ∞ m=−∞ 2m|Em∩Q|1−α. Let m0be an arbitrary integer. Evidently one has m0 m=−∞ 2m|Em∩Q|1−α≤|Q|1−α m0 m=−∞ 2m=2|Q|1−α2m0. Using the fact that 2mp |Em∩Q|≤2pbp 3,p|Q|1−p/3, it follows that if α>0 satisfies (1 −α)p>1 then ∞ m=m0+1 2m|Em∩Q|1−α≤Cαbp(1−α) 3,p |Q|(1−p/3)(1−α)2m0(1+αp−p), for some finite constant Cα. Hence, setting λ=2 m0, it follows that ∞ m=−∞ 2m|Em∩Q|1−α ≤2|Q|1−αλ+Cαbp(1−α) 3,p |Q|(1−p/3)(1−α)λ(1+αp−p). The RHS of the last inequality is minimised when λ∼b3,p|Q|−1/3. We conclude therefore that ∞ m=−∞ 2m|Em∩Q|1−α≤Cαb3,p |Q|2/3−α, for some finite constant Cα. Putting this last inequality together with (2.9), (2.10), and (2.11) we conclude that Q∩B1/4 (1/4−|x|)2 ∞ n=nQ|b(x)|1/2Snh(x) 2 dx ≤C2b1/2 3,p |Q|1/6hQb1/2 3,p |Q|α+1/6hQb3,p |Q|2/3−α =C2b2 3,p|Q|h2 Q, for some constant Cdepending only on pand q.
Fluctuations of BM 103 Proposition 2.2 follows now from (2.6) and Lemma 2.5 just in the same way as Theorem 1.2 of [CR] follows from Lemma 2.4 of [CR]. Next we consider the third stage in section 4 of the proof that the operator Qρ is bounded on L2. Let Kρbe an operator on functions h:B1/4→C defined by (2.12) Kρh(x)=B1/4 dy GD(x, y)|b(y)|1/2h(y),|x|=1 4(1 −ρ), where 0 <ρ<1/2. Proposition 2.4. For 0<ρ<1/2,Kρis a bounded operator from L2 weight(B1/4)to L2(∂B(1−ρ)/4)and the norm of Kρsatisfies an inequality Kρ≤Cb1/2 3,p , where the constant Cdepends only on p>1. Proof: We write the operator Kρas a sum Kρ=∞ n=0 Kρ,n, where Kρ,0h(x)=B1/4∩{|x−y|<1 5ρ}dy GD(x, y)|b(y)|1/2h(y), Kρ,nh(x)=B1/4∩{1 5ρ2n−1≤|x−y|<1 5ρ2n}dy GD(x, y)|b(y)|1/2h(y), n=1,2,... . Evidently |Kρ,0h(x)|≤B1/4∩{|x−y|<1 5ρ}|b(y)|1/2|h(y)| |x−y|dy ≤{|x−y|<1 5ρ}|b(y)| |x−y|dy1/2 B1/4∩{|x−y|<1 5ρ}|h(y)|2 |x−y|dy1/2 , by the Schwarz inequality. Since b∈M3 pthere is a universal constant C such that {|x−y|<1 5ρ}|b(y)| |x−y|dy ≤Cb3,p ρ.
104 J. G. Conlon, P. Olsen Hence |x|=1 4(1−ρ)|Kρ,0h(x)|2dx ≤Cb3,pρ{|x|=1 4(1−ρ)} dx{|x−y|<1 5ρ}|h(y)|2 |x−y|dy ≤C1b3,p ρ2{1/4−|y|>ρ/20}∩B1/4|h(y)|2dy ≤C2b3,p B1/4 (1/4−|y|)2|h(y)|2dy =C2b3,p h2 2,weight where C1and C2are universal constants. Hence we have shown that Kρ,0is a bounded operator and that Kρ,0≤C1/2 2b1/2 3,p . Next we consider Kρ,n for n≥1. We use the fact that there is a universal constant Csuch that if |x|=(1−ρ)/4, then GD(x, y)≤Cρ(1/4−|y|)/(ρ2n)3,y∈B1/4∩1 5ρ2n−1≤|x−y|<1 5ρ2n. Applying the Schwarz inequality as we did before we have |Kρ,nh(x)| ≤C1ρ (ρ2n)3b1/2 3,p ρ2nB1/4∩{1 5ρ2n−1≤|x−y|<1 5ρ2n} (1/4−|y|)2|h(y)|2dy1/2 , for some universal constant C1. Hence there is a universal constant C2 such that |x|=1 4(1−ρ)|Kρ,nh(x)|2dx ≤C2b3,p 2−2nh2 2,weight. It follows that Kρ,n is a bounded operator and Kρ,n≤C1/2 2b1/2 3,p 2−n,n≥1. Now the boundedness of Kρfollows from the Minkowski inequality Kρ≤ ∞ n=0 Kρ,n≤C1/2 2b1/2 3,p ∞ n=0 2−n=2C1/2 2b1/2 3,p .
Fluctuations of BM 105 Proof of Theorem 1.4: Suppose gis in L2(∂B1/4). Then from Proposition 2.1 and Harnack the function h(y)=|b(y)|1/2n(y)·∇Pg(y)isin L2 weight (B1/4) and h2,weight ≤Cb1/2 3,p g2, where Cis a constant depending only on p>1. By Proposition 2.2 the function udefined by u=(I−Tsym)−1his in L2 weight(B1/4) and u2,weight ≤C1h2,weight, for some constant C1depending only on p>1 provided b3,p is sufficiently small. Finally we have Qρg=Kρu. Hence Proposition 2.4 and the previous two inequalities tells us that Qρis a bounded operator from L2(∂B1/4)toL2(∂B(1−ρ)/4) provided b3,p is sufficiently small and Qρ≤Cb3,p for some constant Cdepending only on p>1. Chapter 3. Fourier Space Localisation Our first goal is to prove a version of Theorem 1.5 which takes account of the location of gin Fourier space. Theorem 3.1. Suppose f,gare in L2(S). Then there exists ε>0 and a constant C>0depending only on p>1such that b3,p <ε implies (3.1) |f,QρEkg| ≤ Cb3,p f2Ekg2ρ2k,0≤k<∞. We can prove Theorem 3.1 by slightly modifying the proof of Theorem 1.4. The main point to observe is that the quantity 2kin the estimate (3.1) replaces the weighting factor in the L2norm of Proposition 2.2. We shall therefore need to apply Proposition 2.3 here instead of Proposition 2.2. First we define an operator which is analogous to the operator Aof (2.1). Thus for 0 <ρ<1/2 let Aρf(y) be defined on functions fwith domain ∂B(1−ρ)/4by (3.2) Aρf(y)=|b(y)|1/24 ρ|x|=(1−ρ)/4 f(x)GD(x, y)dx, |y|<1/4. It is evident that lim ρ→0Aρf(y)=Af(y),|y|<1/4. The following proposition is therefore a generalization of Proposition 2.1.
106 J. G. Conlon, P. Olsen Proposition 3.1. For 0<ρ<1/2,Aρis a bounded linear operator from L2(∂B(1−ρ)/4)to L2(B1/4). There is a constant Cdepending only on p>1such that Aρ≤Cb1/2 3,p . Proof: Let Uρbe the spherical shell Uρ=y:1 4(1 −3ρ/2) <|y|<1 4 and χρbe the characteristic function of the set Uρ. Let us first consider the operator Kρdefined by Kρf(y)=χρ(y)Aρf(y),y∈B1/4. We show that Kρis bounded by arguing as in Proposition 2.4. Thus we write Kρ=∞ n=0 Kρ,n, where Kρ,0f(y)=χρ(y)|b(y)|1/24 ρ{|x|=1 4(1−ρ),|x−y|<ρ}f(x)GD(x, y)dx, Kρ,nf(y)=χρ(y)|b(y)|1/24 ρ{|x|=1 4(1−ρ),ρ2n−1≤|x−y|<ρ2n} f(x)GD(x, y)dx, n≥1. We have now from the Schwarz inequality that |Kρ,0f(y)|2≤Cχ ρ(y)|b(y)|ρ−1{|x|=1 4(1−ρ),|x−y|<ρ}|f(x)|2 |x−y|dx, for some universal constant C. Hence Kρ,0f(y)2 2≤Cρ −1|x|=1 4(1−ρ) dx|f(x)|2|x−y|<ρ |b(y)| |x−y|dy ≤C1b3,p f2 2 for some constant C1. We conclude that Kρ,0≤C1/2 1b1/2 3,p .To estimate Kρ,n for n≥1 we use the bound GD(x, y)≤Cρ/(ρ2n)2,|x|=1 4(1 −ρ),ρ2n−1≤|x−y|<ρ2n,
Fluctuations of BM 107 where Cis a universal constant. Hence we have |Kρ,nf(y)|2 ≤C1χρ(y)|b(y)|1 (ρ2n)2{|x|=1 4(1−ρ),ρ2n−1≤|x−y|<ρ2n}|f(x)|2dx, for some universal constant C1. Hence Kρ,nf2 2≤C1 (ρ2n)2|x|=1 4(1−ρ) dx|f(x)|2Uρ∩{ρ2n−1≤|x−y|<ρ2n}|b(y)|dy. We have now Uρ∩{ρ2n−1≤|x−y|<ρ2n}|b(y)|dy ≤meas Uρ∩{ρ2n−1≤|x−y|<ρ2n}1−1/p|x−y|<ρ2n|b(y)|pdy1/p ≤C(ρ322n)1−1/p b3,p (ρ2n)3/p−1 =Cb3,p ρ22n(1+1/p), for some universal constant C. From the last two inequalities we conclude that there is a universal constant C1such that Kρ,n≤C1/2 1b1/2 3,p 2−n(1−1/p)/2,n≥1. Hence (3.3) Kρ≤C1/2 1b1/2 3,p ∞ n=0 2−n(1−1/p)/2≤C1/2 2b1/2 3,p for some constant C2depending only on p>1. Next we consider the operator A0defined by A0f(y)=[1−χρ(y)] Aρf(y),y∈B1/4. It is easy to see that there is a universal constant Csuch that GD(x, y)≤Cρ(1/4−|y|)/|x−y|3,y∈B1/4\Uρ,|x|=1 4(1 −ρ).
108 J. G. Conlon, P. Olsen Furthermore for y∈B1/4\Uρone has 1/4−|y|≤C11 4(1 −ρ)−|y|for a suitable constant C1. Hence the operator A0has a kernel which is bounded by a constant times the kernel of A. Applying Proposition 2.1 we conclude that A0is bounded and A0≤C1/2 2b1/2 3,p . Since Aρ= A0+Kρthe result follows from this last inequality and (3.3). Proposition 3.1 enables us to pull out the factor ρin the inequality (3.1). Next we address the problem of how to pull out the factor 2kin (3.1). The factor occurs due to the effect of the gradient in the expression ∇(PEkg). Suppose y=(y1,y 2,y 3)∈R3. We shall want to show that (3.4) |y|∂ ∂yi PEkg(y)≃2kPh i(y),|y|<1/4,1≤i≤3, where hiis an L2function on ∂B1/4with norm comparable to g. Furthermore we shall need to show that hiis approximately concentrated in Fourier space on the range of Ek. To do this we introduce polar coordinates (r, θ, ϕ) on the ball, 0 ≤r<1/4, 0 <θ<π,0<ϕ<2π.We may also assume that i= 3 in (3.4) and the representation ∂ ∂y3 = cos θ∂ ∂r −sin θ1 r ∂ ∂θ. Let Y!,m(θ,ϕ) be the spherical harmonics on the unit sphere. Thus :is a nonnegative integer, mis an integer satisfying −:≤m≤:and −∆SY!,m =:(:+1)Y!,m, 1 i ∂ ∂ϕ Y!,m =mY !,m. Now the Poisson kernel applied to the boundary data Y!,m yields PY!,m(r, θ, ϕ)=(4r)!Y!,m(θ,ϕ). Thus |y|∂ ∂y3 PY !,m(r, θ, ϕ)=:(4r)!cos θY !,m(θ,ϕ)−(4r)!sin θ∂ ∂θ Y!,m(θ,ϕ). Let P!,m(z), :=0,1,2,...,0≤m≤:be the associated Legendre functions. Then one has [M, p. 495], for m≥0, Y!,±m(θ,ϕ)=σ±m,! (2:+ 1)(:−m)! 4π(:+m)! 1/2 P!,m(cos θ)e±imϕ,
Fluctuations of BM 109 where |σ±m,!|= 1. If we use the relations (2:+1)zP !,m(z)=(:+m)P!−1,m(z)+(:+1−m)P!+1,m(z), (z2−1) d dz P!,m(z)=(:−m+1)P!+1,m(z)−(:+m+1)zP!,m(z), to be found in [H, p. 289,290], we may conclude that cos θY!,m(θ,ϕ)=(:+m)(:−m) (2:+ 1)(2:−1)1/2 Y!−1,m(θ,ϕ) +(:+1+m)(:+1−m) (2:+ 3)(2:+1) 1/2 Y!+1,m(θ,ϕ), sin θ∂ ∂θY!,m(θ,ϕ)=−(:+m+1)(:+m)(:−m) (2:+ 1)(2:−1)1/2 Y!−1,m(θ,ϕ) +(:−m)(:+1+m)(:+1−m) (2:+ 3)(2:−1) 1/2 Y!+1,m(θ,ϕ). Hence we have shown that |y|∂ ∂y3 PY!,m(r, θ, ϕ)=:ar PY!−1,m(r, θ, ϕ)+br−1PY!+1,m(r, θ, ϕ), where a,bare constants which are bounded by 16 in absolute value. It is easy now to state a rigorous version of (3.4). Lemma 3.1. Let gbe square integrable on the sphere ∂B1/4, such that Ekg=gfor some k≥1. Then for any i,1≤i≤3, there exist functions h+,h−on ∂B1/4such that |y|∂ ∂yi Pg(y)=2 k(4|y|)Ph−(y)+2 k(1/4|y|)Ph+(y),|y|<1/4, and the functions h+,h−satisfy (Ek+Ek+1)h+=h+,(Ek+Ek−1)h−=h−, h+2≤Cg2,h−2≤Cg2, where Cis a universal constant.
116 J. G. Conlon, P. Olsen Observe that it is sufficient to prove (3.9) under the condition 2−k<ρ/2. In that case we write (3.10) Mρf(y)= ∞ n=0 gn(y),y∈U 0,k, where g0(y)= {|x|=1 4(1−ρ),|x−y|<ρ}dxf(x)GD(x, y)|b(y)|1/2(1/4−|y|)−1, gn(y)= {|x|=1 4(1−ρ),2n−1ρ<|x−y|<2nρ}dxf(x)GD(x, y)|b(y)|1/2(1/4−|y|)−1 . Since 2−k<ρ/2wehave (3.11) |gn(y)|≤ Cρ (ρ2n)3|b(y)|1/2{|x|=1 4(1−ρ),|x−y|<2nρ}|f(x)|dx ≤C1ρ (ρ2n)2|b(y)|1/2{|x|=1 4(1−ρ),|x−y|<2nρ}|f(x)|2dx1/2 , n=0,1,2,... , for some universal constants C,C1by the Schwarz inequality. This last inequality implies U0,k |gn(y)|2dy ≤C2 1ρ2 (ρ2n)4{|x|=1 4(1−ρ)}dx|f(x)|2U0,k∩{|x−y|<2nρ}|b(y)|dy. Now if we estimate U0,k∩{|x−y|<2nρ}|b(y)|dy ≤meas U0,k ∩{|x−y|<2nρ}1−1/pb3,p (2nρ)3/p−1 ≤Cb3,p (2nρ)1+1/p 2−k(1−1/p),
Fluctuations of BM 117 we can conclude that (3.12) U0,k |gn(y)|2dy1/2 ≤C1b1/2 3,p f2(2−k/ρ)(1−1/p)/22−n(3−1/p)/2, for some universal constant C1. Hence from (3.10) and the Minkowski inequality it follows that (3.9) holds with δ=(1−1/p)/2. We can assume now that 2−k>ρ,k>k . Let fk(x) be the function fk=2 −4k(−∆S+2 2k)2Ekf, where ∆Sis the Laplacian on the unit sphere. Evidently there is a universal constant Csuch that fk2≤CEkf2. Furthermore Ekf can be written in terms of fkby Ekf(x)=|x|=1 4(1−ρ) Hk(x, x)fk(x)dx, where Hk(x, x) is the kernel of the operator 24k(−∆S+2 2k)−2.Itis well known [CH] that there are constants C,c>0 such that 0≤Hk(x, x)≤C22kexp[−c|x−x|/2−k]. Again we write MρEkf(y)= ∞ n=0 gn(y),y∈U 0,k, where g0(y)={|x|=1 4(1−ρ),|x|=1 4(1−ρ),|x−y|<2−k} dx dxHk(x, x)fk(x)GD(x, y)|b(y)|1/2(1/4−|y|)−1, gn(y)={|x|=1 4(1−ρ),|x|=1 4(1−ρ),2n−1−k<|x−y|<2n−k} dx dxHk(x, x)fk(x)GD(x, y)|b(y)|1/2(1/4−|y|)−1,
118 J. G. Conlon, P. Olsen if n≥1. Observe now that |x|=1 4(1−ρ) dxHk(x, x)GD(x, y)(1/4−|y|)−1 ≤C22k|x|=1 4(1−ρ) dx GD(x, y)(1/4−|y|)−1≤C122k, for some universal constant C1. Hence (3.13) |g0(y)|≤C122k{|x|=1 4(1−ρ),|x−y|<2−k}dx|fk(x)||b(y)|1/2. Nowif2 n−1−k<|x−y|<2n−k,n≥1, then one easily sees that |x|=1 4(1−ρ) dxHk(x, x)GD(x, y)(1/4−|y|)−1≤C22k−3n, for some universal constant C2. Hence if n≥1 we have the inequality (3.14) |gn(y)| ≤C222k−3n{|x|=1 4(1−ρ),|x−y|<2n−k}dx|fk(x)||b(y)|1/2. We can now bound the integrals of |gn(y)|2over U0,k exactly as in (3.12) by using the inequalities (3.13), (3.14). There is therefore a universal constant C1such that U0,k |gn(y)|2dy1/2 ≤C1b1/2 3,p fk22(k−k)(1−1/p)/22−n(3−1/p)/2,n≥0. Since fk2≤CEkf2the result follows as before.
Fluctuations of BM 119 Proof of Theorem 3.2: Let h(y)=|b(y)|1/2n(y)·∇PEkg(y), y∈B1/4. Then from the Harnack principle and Lemma 3.2 it is easy to see that h is in the space L2 k,δ,weight(B1/4) for every δ>0 and h2,k,δ,weight ≤Cδ,pb1/2 3,p Ekg2, where the constant Cδ,p depends only on δ,p>1. Now by Proposition 3.2 one sees that if b3,p is sufficiently small then the function ξ(y)=(1/4−|y|)(I−Tsym)−1h(y)isinL2 k,δ(B1/4) and ξ2,k,δ ≤Cδ,p b1/2 3,p Ekg2, for some suitable constant Cδ,p. Observe next that Ekf,QρEkg=B1/4 ξ(y)MρEkf(y)dy ≤ k r=0 Ur,k |ξ(y)|2dy1/2Ur,k |MρEkf(y)|2dy1/2 . If we use now Proposition 3.4 we have that Ekf,QρEkg≤ k r=0 2−rδξ2,k,δ Ur,k |MρEkf(y)|2dy1/2 ≤ k−k r=0 2−rδξ2,k,δ C1/2b1/2 3,p Ekf22−(k−k−r)δ + k r=k−k+1 2−rδξ2,k,δ C1/2b1/2 3,p Ekf2 ≤C1b1/2 3,p ξ2,k,δ Ekf22−(k−k)δ ≤C2b3,p Ekg2Ekf22−(k−k)δ for constants C1,C2depending only on p>1 provided we choose 0<δ<δ appropriately.
120 J. G. Conlon, P. Olsen The proof of Theorem 1.5 will be complete if we can prove: Theorem 3.3. Suppose f,gare in L2(S). Then there exists ε>0 and constants η,C>0depending only on p>1such that if b3,p <ε then Ekf,QρEkg≤Cb3,p Ekf2Ekg2ρ2k2η(k−k),0≤k, k<∞. The main work to be done to prove this last proposition is to show that Proposition 3.4 also holds for the operator Aρdefined by (3.2). Thus we have the following: Proposition 3.5. Suppose k,kare nonnegative integers and 0<ρ<1/2. Then there exists δ>0and a constant Cdepending only on p>1such that for f∈L2(∂B(1−ρ)/4), U0,k AρEkf(y) 2dy ≤Cb3,p Ekf2 22−2(k−k)δ. Proof: We proceed in the same way as in Proposition 3.4. By Proposition 3.1 we can assume that k≥k. Next we show that one may also assume 2−k>ρ. This follows from the inequality (3.15) U0,k Aρf(y) 2dy ≤Cb3,p f2 2(2−k/ρ)2δ. Observe that it is sufficient to prove (3.15) under the condition 2−k< ρ/2. In that case we write Aρf(y)= ∞ n=0 gn(y),y∈U 0,k, where g0(y)=4 ρ{|x|=1 4(1−ρ),|x−y|<ρ}dxf(x)GD(x, y)|b(y)|1/2, gn(y)=4 ρ{|x|=1 4(1−ρ),2n−1ρ<|x−y|<2nρ}dxf(x)GD(x, y)|b(y)|1/2,n≥1. Since 2−k<ρ/2wehave |gn(y)|≤ Cρ (ρ2n)3|b(y)|1/2{|x|=1 4(1−ρ),|x−y|<2nρ}|f(x)|dx.
Fluctuations of BM 121 Since this last inequality is exactly the same as (3.11) we can conclude that the theorem holds in the case of 2−k≤ρ. To deal with the case of 2−k>ρwe proceed again as in Proposition 3.4. The functions gnare now defined by g0(y)=4 ρ{|x|=1 4(1−ρ),|x|=1 4(1−ρ),|x−y|<2−k} dx dxHk(x, x)fk(x)GD(x, y)|b(y)|1/2, gn(y)=4 ρ{|x|=1 4(1−ρ),|x|=1 4(1−ρ),2n−1−k<|x−y|<2n−k} dx dxHk(x, x)fk(x)GD(x, y)|b(y)|1/2, if n≥1. Evidently one has 4 ρ|x|=1 4(1−ρ) dxHk(x, x)GD(x, y) ≤4C22k ρ|x|=1 4(1−ρ) dxGD(x, y)≤C122k, for some universal constant C1. Similarly 4 ρ|x|=1 4(1−ρ) dxHk(x, x)GD(x, y)≤C222k−3n for some universal constant C2if 2n−1−k<|x−y|<2n−k,n≥1. Now, using these last two estimates the proof of the theorem is identical to the proof of Proposition 3.4. Proof of Theorem 3.3: Let h(y)=|b(y)|1/2n(y)·∇PEkg(y), y∈B1/4. From Lemmas 3.1, 3.2 it follows that his in the space L2 k,δ(B1/4) for every δ>0 and h2,k,δ ≤Cδ,pb1/2 3,p 2kEkg2, where the constant Cδ,p depends only on δ,p>1. Now by Proposition 3.3 one sees that if b3,p is sufficiently small then the function ξ(y)=(I−Tsym)−1h(y)isinL2 k,δ(B1/4) and (3.16) ξ2,k,δ ≤Cδ,pb1/2 3,p 2kEkg2,
122 J. G. Conlon, P. Olsen for some suitable constant Cδ,p. Observe next that Ekf,QρEkg= ρB1/4 ξ(y)AρEkf(y)dy . The rest of the proof follows now from (3.16) and Proposition 3.5 in exactly the same way as Theorem 3.2 follows from Proposition 3.4. Chapter 4. Proof of Theorem 1.3 We first define the density fρ,bin terms of the density f. To do this we consider the Dirichlet problem (∆+b(y)·∇)v(y)=0,y∈B1/4, v(y)=g(y),y∈∂B1/4. Formally v(y) is given by the formula v(y)=Pg(y)+Qg(y),y∈B1/4, where Pis the Poisson integral (1.4) and Qg is defined by (1.5). Now vcan be represented in terms of the diffusion process Xb(t) by the expression v(y)=Ey[g(Xb(τ))],y∈B1/4, where τis the first hitting time on ∂B1/4for the process started at Xb(0) = y. It is clear then that if we regard the density fas a function on ∂B(1−ρ)/4and the density fρ,bas a function on ∂B1/4, then ∂B(1−ρ)/4 f(y)v(y)dy /normalisation =∂B1/4 fρ,b(y)g(y)dy /normalisation, where the normalisations are chosen so that the measures are probability measures. It follows therefore, on going back to regarding fand fρ,bas functions on the unit sphere Sthat fρ,b,g=f,Pρg+Qρg,g∈L2(S), where Pρg(y)=Pg(y), y∈∂B(1−ρ)/4. Hence fρ,b=P∗ ρf+Q∗ ρf, where P∗ ρand Q∗ ρare the formal adjoints of Pρ,Qρrespectively.
Fluctuations of BM 123 We can analyse the operator Pρprecisely since we know its eigenfunctions. In fact if Yl,m(θ, φ), 0 ≤θ<π,0≤φ<2π, is a spherical harmonic and we take g=Yl,m then Pg(y)=(4|y|)lYl,m(θ,φ), y∈B1/4. Hence (4.1) PρYl,m =(1−ρ)lYl,m. It follows in particular that Pρis selfadjoint, whence Pρ=P∗ ρ. We also have that Pρ1=PρY0,0= 1. Hence for any f∈L2(S) we have that (4.2) Pρf−Avf2≤(1 −ρ)f−Avf2. Theorem 1.2 follows from (4.2) and the fact that Qρ≤Cb3,p where Cdepends only on p,ρ. In fact fρ,b−Avfρ,b2=Pρf+Q∗ ρf−Avf2 ≤Pρf−Avf2+Q∗ ρf2 ≤(1 −ρ)f−Avf2+Cb3,pf2. Suppose now f−Avf2≤δ|Avf|. Then f2≤(1+δ)|Avf|. Hence the last inequality yields fρ,b−Avfρ,b2≤(1 −ρ)δ|Avf|+Cb3,p(1+δ)|Avf| =[1−ρ+Cb3,p(1+δ−1)]δ|Avfρ,b|, since Avf =Avfρ,b. Theorem 1.2 follows from the last inequality by choosing b3,p sufficiently small. Theorem 1.3 follows by a similar argument from Theorem 1.5. Since Yl,m is an eigenfunction of −∆Swith eigenvalue l(l+1),l =0,1,2,..., it follows from (4.1) that there is a universal constant c>0 with (4.3) EkPρf2≤{1−cmin[ρ2k,1]}Ekf2,k≥1. The inequality (4.3) plays the same role in the proof of Theorem 1.3 as (4.2) plays in the proof of Theorem 1.2. By the Minkowski inequality we have Ekfρ,b2=EkPρf+EkQ∗ ρf2 ≤EkPρf2+EkQ∗ ρf2.
124 J. G. Conlon, P. Olsen Theorem 1.5 yields an appropriate estimate on EkQ∗ ρf2, which when combined with (4.3) proves Theorem 1.3. To see this let us assume f−Avf2,ν ≤δ|Avf|. Then (4.4) Ekf2≤δ|Avf|/2νk,k=1,2,... . Now, for some gsatisfying Ekg2=1, (4.5) EkQ∗ ρf2=f,QρEkg≤ ∞ k=0 |Ekf,QρEkg|. Hence from Theorem 1.5 and (4.4) we have, EkQ∗ ρf2≤Cb3,p|Avf|min[ρ2k,1]2−ηk + k k=1 Cb3,p δ|Avf|2−νkmin[ρ2k,1]2η(k−k) +∞ k=k+1 Cb3,pδ|Avf|2−νkmin[ρ2k,1]. Note that the first term on the right in the last inequality comes from the k= 0 term on the right in (4.5). Hence if η>νwe have that EkQ∗ ρf2≤Cb3,p|Avf|min[ρ2k,1] 2−ηk +δ2(η−ν)(k+1)2−ηk/[2(η−ν)−1] + δ2−ν(k+1)/[1 −2−ν]. We conclude that EkQ∗ ρf2≤C(δ)b3,pδ|Avf|min[ρ2k,1]2−νk,k≥1, where the constant C(δ) depends only on δ. Combining this last inequality with (4.3) we have that Ekfρ,b2≤{1+[C(δ)b3,p −c] min[ρ2k,1]}δ|Avf|2−νk,k≥1. The theorem follows now by choosing b3,p sufficiently small so that C(δ)b3,p <c. Acknowledgement. The authors would like to thank Paul Federbush and Mark Pinsky for helpful conversations on spherical harmonics.
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