FLUCTUATIONS OF BROWNIAN MOTION WITH DRIFT

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 > 0. Then the linear operators Tt defined by Tt ρ = ρt form 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 b is small in a suitable scale invariant norm. ∗Research partially supported by the U.S. National Science Foundation under grants DMS 9403399 and DMS 9404197. ∗∗Research supported by NFR Norges Forskingsr̊ad (Norwegian Research Council) grant 100222/40.


Chapter 1. Introduction
In two previous papers [CR], [CO] we studied the Dirichlet problem for an elliptic equation on a domain in R 3 .Let B R be the ball of radius R in R 3 centered at the origin, 0 < R < ∞.Consider the problem (1.1) A function g : R 3 → C is said to be in the Morrey space M q p , 1 ≤ p ≤ q < ∞, if |g| p is locally integrable and there is a constant C such that (1.2) for all cubes Q ⊂ R 3 .Here |Q| denotes the volume of Q.The norm of g, g q,p is defined as the minimum C for which (1.2) holds.In [CR] we proved the following: Theorem 1.1.Suppose 1 < r < p ≤ q, 1 < p ≤ 3, and |b| ∈ M 3 p , f ∈ M q r , for some q with q > 3/2.Then there exists > 0 depending only on r, p, q such that if b 3,p < then the Dirichlet problem (1.1) is solvable.Further, there is a constant C depending only on r, p, q such that the solution u of (1.1) satisfies the inequality, u ∞ ≤ CR 2−3/q f q,r .
The condition b 3,p small for some p, 1 < p ≤ 3, includes the two important cases when |b(x)| = /|x| and |b| ∈ L 3 (R 3 ), b 3 = , 1. Theorem 1.1 is a perturbative result.Writing the solution of (1.1) as a perturbation series in b one can show that the series converges if b 3,p 1. Observe also that Theorem 1.1 is a scale invariant theorem.The result for general R can 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 X b (t), t ≥ 0, denote the drift process started at time 0. If X b (0) ∈ B 1/4 let τ be the minimum time t such that X b (t) ∈ ∂B 1/4 .Then the solution u of (1.1) is given by where E x denotes that the expectation value is taken conditioned on X b (0) = x.Theorem 1.1 tells us therefore something about the behavior of the diffusion process X b (t) when b 3,p 1.It gives us similar estimates on the expected time the diffusion spends in a subset of B 1/4 before exiting ∂B 1/4 to those one has for standard Brownian motion.
In [CO] we proved a nonperturbative version of Theorem 1.1, allowing b to be only locally in a Morrey space M 3 p with small norm.A key ingredient in the proof of this theorem was the fact that the fluctuations of X b (t) did not increase as t increases, provided b 3,p 1.To be specific, suppose 0 < ρ ≤ 1/2, and the process X b (t) starts on the sphere ∂B (1−ρ)/4 with initial density function f , and f ρ,b is the first hitting density on the sphere ∂B 1/4 .It is evident, by conservation of probability, that the average value of f is the same as the average value of f ρ,b , Avf = Avf ρ,b .In [CO] we proved the following: Theorem 1.2.Suppose 0 < ρ ≤ 1/2 and 1 < p ≤ 3. Consider f and f ρ,b to be functions on the unit sphere S, with f ∈ L 2 (S).Then for δ > 0 there exists , depending only on p, ρ, δ such that, if b 3,p < and It is easy to see that Theorem 1.2 holds uniformly in ρ as ρ → 0 for the case of Brownian motion, b ≡ 0. Since b with b 3,p 1 is perturbative to Brownian motion it is natural to expect a similar uniformity when b 3,p is small.In this paper we prove a uniform version of Theorem 1.2.We cannot however use the L 2 norm to measure the oscillation of f and f ρ,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 ∈ B 1/4 decreases as x gets close to ∂B 1/4 .
To define our new norm on functions f with domain S, let ∆ S be the Laplace operator on the unit sphere.For k = 1, 2, . . ., let E k be the L 2 projection operator onto the space spanned by the eigenfunctions of −∆ S with eigenvalues λ 2 satisfying 2 k−1 < λ ≤ 2 k .Let E 0 be the projection onto the constant function.For f : S → C and ν > 0 we define f 2,ν by We then have the following: Theorem 1.3.Suppose 0 < ρ ≤ 1/2, 1 < p ≤ 3, and ν > 0 is sufficiently small, depending only on p. Then for δ > 0 there exists > 0 depending only on p, δ such that if b 3,p < and We consider the relationship between the proof of Theorem 1.1 and the proof of Theorem 1.2.Let G D be the Dirichlet Green's function for −∆ on B 1/4 , whence G D is given explicitly by the formula, where ȳ is the reflection of y in ∂B 1/4 .Let T be the integral operator on functions with domain B 1/4 given by It was shown in [CR] that Theorem 1.1 is a consequence of the fact that T is a bounded operator on the Morrey space M q r with norm, T , satisfying T ≤ C b 3,p for some constant C depending only on p, q, r.
For g a function with domain ∂B 1/4 let v(y) = P g(y), y ∈ B 1/4 , be the solution of the Dirichlet problem, The function v is given explicitly by the Poisson formula, We can formally define an integral operator Q on the functions g by where T is given by (1.3).This operator induces an operator on functions with domain S as follows: Let f : S → C and denote also by f the function with domain ∂B 1/4 naturally induced by f .Consider now the function We can think of Q ρ f as a function with domain S, whence Q ρ is an operator on functions with domain S. In [CO] we proved that Q ρ is a bounded operator on L 2 (S), 0 < ρ ≤ 1/2, with norm Q ρ satisfying Q ρ ≤ C b 3,p for some constant C depending only on ρ, p, provided b 3,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 Q ρ one assumes that the inverse (I − T ) −1 exists.To prove Theorem 1.3 we need to know not only that Q ρ is bounded on L 2 (S) for 0 < ρ ≤ 1/2 but also to have a bound which is uniform as ρ → 0. Let , denote the scalar product on L 2 (S).In section 2 we prove the following: In order to prove Theorem 1.3 we need to know more detailed properties of Q ρ than those given in Theorem 1.4.In particular we must know that if Q ρ acts on a slowly varying function g then the slowly varying component of Q ρ g has norm bounded linearly in ρ as ρ → 0. We also need to know that if g is highly oscillatory then the slowly varying component of Q ρ g has small norm.These properties of Q ρ are summarised in the following: 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 ρ as ρ → 0.
Define an operator A on functions g with domain S 1/4 = {|x| = 1/4} by (2.1) We shall prove Proposition 2.1 by following the general lines of the proof of Theorem 1.2 of [CR].If a function u is defined on the sphere S 1/4 then Au is defined on the ball B 1/4 .Let Q 0 be a cube with side of length 1 and for n = 1, 2, . . .let Q n be the dyadic subcubes of Q 0 with side of length 2 −n .We define u Qn as follows: For ξ ∈ R 3 let Q 0 (ξ) be the unit cube centered at ξ with corresponding dyadic subcubes Q n (ξ).We then have the following: Lemma 2.1.There exists a universal constant C such that (2.2) |b(y)| dy.
Proof: We have from the definition (2.1) that We estimate Au(y) in the annulus for some universal constant C. Choosing α > 0 and applying the Schwarz inequality to the RHS of the previous expression we have for some constant C α depending only on α.Thus Then it is clear from the previous expression that if α < 1 there is a constant C α depending only on α such that where It is clear now that there exists a universal constant C such that The result follows then from the last two inequalities.
Next we shall show that for fixed ξ the RHS of (2.2) is bounded by the L 2 norm of u.Lemma 2.2.Let Q 0 be a cube in R 3 with side of length 1 and dyadic subcubes Q n with side of length 2 −n , n = 1, 2, . . . .Then there is a constant C depending only on p > 1 such that Evidently Proposition 2.1 follows from Lemma 2.1 and Lemma 2.2.Observe that for fixed n one has for some universal constant C. In order to do the summation with respect to n in (2.3) we need to resort to a Calderon-Zygmund decomposition.First we have Lemma 2.3.Let Q be a cube in R 3 with side of length 2 −n Q , where n Q is a nonnegative integer.Suppose for some ε > 0 one has where C is a universal constant.
If we now choose ε to satisfy ε < (1 − 1/p)/6, then we have where the constant C p depends only on p > 1.
Let Q 0 be a unit cube in R 3 .We make a Calderon-Zygmund decomposition of Q 0 based on the criterion in Lemma 2.3.In particular we define a sequence of families F j of dyadic subcubes of Q 0 , j = 0, 1, 2, . . .as follows: Then there is a unique finite family F 1 of disjoint dyadic subcubes of Q 0 such that Proceeding by induction as in section 2 of [CR], we can construct sets G j and families F j , j ≥ 1, with the properties (a) It is clear now from Lemma 2.3 that there is a constant C depending only on p > 1 such that The proof of Lemma 2.2 will be complete if we can show Lemma 2.4.There exists a constant C depending only on p > 1 such that Proof: We shall assume without lost of generality that u ∞ < ∞.Hence there exists an integer t ≥ 1 such that F t is empty, whence We wish to estimate the ratio for some universal constant c > 0. We conclude therefore that Next we write for any positive sequence a m .We choose a m to be given by Then, in view of the last two inequalities we have for some constant C depending only on ε > 0. We conclude then that Now if we sum this last inequality with respect to j and use the fact that the sets G j are disjoint we obtain the inequality (2.4).
We return now to the operator Q ρ of (1.6).The proof in section 4 of [CO] that Q ρ is a bounded operator on L 2 had three ingredients.The first stage was to show that the function b • ∇P g with domain B 1/4 is in an appropriate Morrey space, This Morrey space, M 3/2 r (B 1/4 ), is defined to be the set of functions h : We see now how this follows from Proposition 2.1.By the Harnack inequality there is a constant C such that Hence for any cube Q one has for some constant C depending only on p > 1.
The second stage was to prove that, for T the integral operator given by (1.3), the function ( B 1/4 ).Thus the perturbation due to the integral operator T is small in the sense that it preserves the space in which b • ∇P g lies.We shall do something analagous here.For y ∈ R 3 let us define a vector n(y) by Evidently one has |n(y)| ≤ 1 for all y ∈ R 3 .Let T sym be the integral operator on functions with domain B 1/4 which has kernel Then we may formally write From Proposition 2.1 and Harnack the function |b| 1/2 n • ∇P g is in this space.
Proposition 2.2.The operator T sym is bounded on the space L 2 weight (B 1/4 ) and there exists a constant C > 0 depending only on p > 1 such that Remark 2.1.It is interesting to compute the information that the results of Olsen [O] give about the operator T sym .If we apply Theorem 2 of [O] then , and u, q can be taken slightly larger than 2, 1 respectively.Hence T g f ∈ M t s where We can do better than this by combining various theorems of [O].
Proposition 2.3.The operator T sym is bounded on the space L 2 (B 1/4 ) and there exists a constant C > 0 depending only on p > 1 such that Proof: We have , where 1/q + 1/q = 1.We choose q in the range 2 < q < 2p whence q < 2. Consider now the function g 1 (x) defined by Now |b| q /2 is in the Morrey space M 6/q 2p/q .Hence by Theorem 9 of [O] g 1 is in M t s where 1/t = q /6 − 1/3, s = t(2p/q )/(6/q ) = tp/3, 3,p for some constant C depending only on p, q .Next let g(x) be given by . By Holder's inequality for Morrey spaces, Lemma 11 of [O], we have that g is in M 3 p since q/6 + q/q t = q/6 + (q/q )[q /6 − 1/3] = 1/3, q/2p + q/q s = q/2p + (q/q )(3/p)[q /6 − 1/3] = 1/p.Furthermore, Now |h| q is in the space L 2/q (B 1/4 ).Hence by Corollary 3 of [O] T g |h| q is also in the space L 2/q (B 1/4 ) provided 2/q < p. Furthermore for some constant C 1 depending only on q, p.Now from (2.5) and the previous inequality we conclude that The result follows since if p > 1 the two inequalities 2 < q < 2p, 2/q < p can be simultaneously satisfied.In fact q can be chosen in the range Remark 2.2.Theorem 9 of [O] was originally proved in [A].A different proof was given in [CF].
The proof of Proposition 2.3 gives us an important insight into how the proof of Proposition 2.2 should go.We shall follow the lines of the proof of Theorem 1.2 of [CR] and Proposition 2.1 of [CO], but the Calderon-Zygmund decomposition will be based on taking averages of |h(y)| q .Let Q 0 be a unit cube and for n = 1, 2, . . .Q n be the dyadic subcubes of Q 0 with side of length 2 −n .Let d(Q n ) be given by For n = 0, 1, 2, . . .define an operator S n on functions h with domain B 1/4 by Let T Q0 be the operator given by Then Jensen's inequality implies there is a universal constant C such that if Q 0 (ξ) is the unit cube centered at ξ, (2.6) Hence it is sufficient to prove Proposition 2.2 with the operator T sym replaced by T Q0 .The key lemma analagous to Lemma 2.4 of [CR] is Then there are constants C, ε > 0 depending only on p and q such that It follows from Holder's inequality that , where 1/q + 1/q = 1.Since 1/q < 1 − 1/2p implies q /2 < p we conclude that Now if we use (2.7) we have from the previous inequality for some constant C ε depending only on ε > 0. Hence (2.9) For m an integer let E m be the set For m, k integers and k ≥ n Q let a m,k be given by Then we have (2.10) There are two estimates on a m,k which we use.The first follows from (2.7).Thus from (2.8) we have and then (2.7) implies that The second estimate is obtained by using the fact that If we again apply (2.8) we have that If follows now that for any α, 0 < α < 1, we have For any α > 0 one can find sufficiently small ε > 0 such that the sum with respect to k above converges.Thus there is a constant C α > 0 depending only on α, ε and Let m 0 be an arbitrary integer.Evidently one has Using the fact that The RHS of the last inequality is minimised when λ ∼ b 3,p |Q | −1/3 .We conclude therefore that for some finite constant C α .Putting this last inequality together with (2.9), (2.10), and (2.11) we conclude that for some constant C depending only on p and q.
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 Q ρ is bounded on L 2 .Let K ρ be an operator on functions h : B 1/4 → C defined by (2.12) ) and the norm of K ρ satisfies an inequality 3,p , where the constant C depends only on p > 1.
Proof: We write the operator K ρ as a sum where where C 1 and C 2 are universal constants.Hence we have shown that K ρ,0 is a bounded operator and that Next we consider K ρ,n for n ≥ 1.We use the fact that there is a universal constant C such that if |x| = (1 − ρ)/4, then Applying the Schwarz inequality as we did before we have , for some universal constant C 1 .Hence there is a universal constant C 2 such that It follows that K ρ,n is a bounded operator and Now the boundedness of K ρ follows from the Minkowski inequality Proof of Theorem 1.4: Suppose g is in L 2 (∂B 1/4 ).Then from Proposition 2.1 and Harnack the function h where C is a constant depending only on p > 1.By Proposition 2.2 the function u defined by u = (I − T sym ) −1 h is in L 2 weight (B 1/4 ) and for some constant C 1 depending only on p > 1 provided b 3,p is sufficiently small.Finally we have Q ρ g = K ρ u.Hence Proposition 2.4 and the previous two inequalities tells us that Q ρ is a bounded operator from L 2 (∂B 1/4 ) to L 2 (∂B (1−ρ)/4 ) provided b 3,p is sufficiently small and Q ρ ≤ C b 3,p for some constant C depending 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 g in Fourier space.
Theorem 3.1.Suppose f , g are in L 2 (S).Then there exists ε > 0 and a constant C > 0 depending only on p > 1 such that b 3,p < ε implies We can prove Theorem 3.1 by slightly modifying the proof of Theorem 1.4.The main point to observe is that the quantity 2 k in the estimate (3.1) replaces the weighting factor in the L 2 norm 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 A of (2.1).Thus for 0 < ρ < 1/2 let A ρ f (y) be defined on functions f with domain The following proposition is therefore a generalization of Proposition 2.1.
Proposition 3.1.For 0 < ρ < 1/2, A ρ is a bounded linear operator from L 2 (∂B (1−ρ)/4 ) to L 2 (B 1/4 ).There is a constant C depending only Proof: Let U ρ be the spherical shell and χ ρ be the characteristic function of the set U ρ .Let us first consider the operator K ρ defined by We show that K ρ is bounded by arguing as in Proposition 2.4.Thus we write where We have now from the Schwarz inequality that for some universal constant C. Hence where C is a universal constant.Hence we have for some universal constant C 1 .Hence We have now for some universal constant C. From the last two inequalities we conclude that there is a universal constant C 1 such that for some constant C 2 depending only on p > 1.
Next we consider the operator A 0 defined by It is easy to see that there is a universal constant C such that Hence the operator A 0 has a kernel which is bounded by a constant times the kernel of A. Applying Proposition 2.1 we conclude that A 0 is bounded and result follows from this last inequality and (3.3).Proposition 3.1 enables us to pull out the factor ρ in the inequality (3.1).Next we address the problem of how to pull out the factor 2 k in (3.1).The factor occurs due to the effect of the gradient in the expression ∇(P E k g).Suppose y = (y 1 , y 2 , y 3 ) ∈ R 3 .We shall want to show that where h i is an L 2 function on ∂B 1/4 with norm comparable to g. Furthermore we shall need to show that h i is approximately concentrated in Fourier space on the range of E k .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 Let Y ,m (θ, ϕ) be the spherical harmonics on the unit sphere.Thus is a nonnegative integer, m is an integer satisfying − ≤ m ≤ and Thus Let P ,m (z), = 0, 1, 2, . . ., 0 ≤ m ≤ be the associated Legendre functions.Then one has [M, p. 495], for m ≥ 0, where |σ ±m, | = 1.If we use the relations to be found in [H, p. 289,290], we may conclude that Hence we have shown that where a, b are 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 g be square integrable on the sphere ∂B 1/4 , such that E k g = g for some k ≥ 1.Then for any i, 1 ≤ i ≤ 3, there exist functions h + , h − on ∂B 1/4 such that and the functions h + , h − satisfy where C is a universal constant.
Proof of Theorem 3.1: Consider the function h k on B 1/4 defined by In view of Lemma 3.1 and Proposition 2.1 the function h k is in L 2 (B 1/4 ) and where A ρ is the operator (3.2).The result follows now from the last two inequalities and Proposition 3.1.
Next we wish to prove a version of Theorem 1.5 which takes account of both the location of f and g in Fourier space.
Theorem 3.2.Suppose f , g are in L 2 (S).Then there exists ε > 0 and constants η, C > 0 depending only on p > 1 such that b 3,p < ε implies The basic fact we want to use is that the function P (E k g)(y), y ∈ B 1/4 falls off rapidly from the boundary, |y| = 1/4.In fact a simple computation shows that the function is essentially concentrated on the shell 1 4 (1 − 2 −k ) < |y| < 1 4 .We need to define a norm which is sensitive to this fact.For h : B 1/4 → C, k = 0, 1, 2, . . .and δ > 0 we define a norm h 2,k,δ by , where U r,k are spherical shells given by Observe that if k = 0 then one just gets back the L 2 norm of h.
where the operator P r acts on functions with domain |y| = 1 4 (1−2 r−1 2 −k ) and gives the solution of the Dirichlet problem.Thus u(y It is easy to see that the function g r has L 2 norm bounded as g r 2 ≤ exp[−c2 r ] g 2 for some positive constant c > 0. Now applying Theorem 1.4 again we have that The result follows now from this last inequality and (3.5).
Next we need to show that the operator T sym is a bounded operator on the space determined by 2,k,δ .As before we shall define two spaces associated with this norm.First the space ) is defined as all h such that the function The weighted norm of h is then given by h 2,k,δ,weight = f 2,k,δ .We have now two theorems analagous to Propositions 2.2, 2.3.Proposition 3.2.There exists δ > 0 such that the operator T sym is bounded on the space L 2 k,δ,weight (B 1/4 ) and there exists a constant C > 0 depending only on δ, p > 1 such that T sym ≤ C b 3,p .Proposition 3.3.There exists δ > 0 such that the operator T sym is bounded on the space L 2 k,δ (B 1/4 ) and there exists a constant C > 0 depending only on δ, p > 1 such that T sym ≤ C b 3,p .
Proof: We shall first prove Proposition 3.3.Let us suppose x ∈ U k,k .Then where C is a universal constant and By the Minkowski inequality we have then Evidently h 1 2 ≤ 2 −(k−2)δ h 2,k,δ .On the other hand . Now from Proposition 2.3 and the last four inequalities we conclude that where the constant C depends only on p > 1 and δ > 0.
We can easily deal with the case of the integral of T sym h over U 0,k by observing that by Proposition 2.3.Thus we are left to deal with integrals over U r,k with 1 ≤ r ≤ k − 1. Define the function h 1 (y) by Next for s ≤ r − 2 and integer n satisfying 2 ≤ n ≤ k − r + 4 let g s,n (x) be the function Then we have the inequality It follows by the Minkowski inequality that . By Proposition 2.3 we have Observe next that for any z, From Holder's inequality we have Similarly we have The last three inequalities imply then that It follows then from the previous inequality and (3.6), (3.7) that if for some constant C depending only on δ and p > 1.This completes the proof of Proposition 3.3.Proposition 3.2 follows in an exactly analogous way from Proposition 2.2.
Next let us consider the operator K ρ defined by (2.12).Let f be a function in L 2 (∂B (1−ρ)/4 ).We define the function for h ∈ L 2 weight (B 1/4 ).Explicitly we have In view of Proposition 2.4 we see that M ρ is a bounded operator from L 2 (∂B (1−ρ)/4) ) to L 2 (B 1/4 ).Furthermore there is a constant C depending only on p > 1 such that provided 0 < ρ < 1/2.We shall need to more accurately estimate the effect of M ρ on a function f which is concentrated in a band of Fourier space.In particular we have the following: Proposition 3.4.Suppose k, k are nonnegative integers and 0 < ρ < 1/2.Then there exists δ > 0 and a constant C depending only on p > 1 such that for f ∈ L 2 (∂B (1−ρ)/4 ), Proof: In view of the inequality (3.8) we may assume that k > k .We shall first show that we may also assume 2 −k > ρ.This will follow from the inequality (3.9) Observe that it is sufficient to prove (3.9) under the condition 2 −k < ρ/2.
In that case we write where for some universal constants C, C 1 by the Schwarz inequality.This last inequality implies we can conclude that (3.12) for some universal constant C 1 .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 f k (x) be the function where ∆ S is the Laplacian on the unit sphere.Evidently there is a Again we write for some universal constant C 2 .Hence if n ≥ 1 we have the inequality We can now bound the integrals of |g n (y)| 2 over U 0,k exactly as in (3.12) by using the inequalities (3.13), (3.14).There is therefore a universal constant C 1 such that Proof of Theorem 3.2: Let h(y) = |b(y)| 1/2 n(y)•∇P E k g(y), y ∈ B 1/4 .Then from the Harnack principle and Lemma 3.2 it is easy to see that h is in the space L 2 k,δ,weight (B 1/4 ) for every δ > 0 and where the constant C δ,p depends only on δ, p > 1.Now by Proposition 3.2 one sees that if b 3,p is sufficiently small then the function for some suitable constant C δ,p .Observe next that If we use now Proposition 3.4 we have that for constants C 1 , C 2 depending only on p > 1 provided we choose 0 < δ < δ appropriately.
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 > ρ we proceed again as in Proposition 3.4.The functions g n are now defined by Now, using these last two estimates the proof of the theorem is identical to the proof of Proposition 3.4.We can analyse the operator P ρ precisely since we know its eigenfunctions.In fact if Y l,m (θ, φ), 0 ≤ θ < π, 0 ≤ φ < 2π, is a spherical harmonic and we take g = Y l,m then P g(y) = (4|y|) l Y l,m (θ, φ), y ∈ B 1/4 .Hence (4.1)

Proof of Theorem 3.3: Let
It follows in particular that P ρ is selfadjoint, whence P ρ = P * ρ .We also have that P ρ 1 = P ρ Y 0,0 = 1.Hence for any f ∈ L 2 (S) we have that (4.2) Theorem 1.2 follows from (4.2) and the fact that Q ρ ≤ C b 3,p where C depends only on p, ρ.In fact Theorem 1.3 follows by a similar argument from Theorem 1.5.Since Y l,m is an eigenfunction of −∆ S with eigenvalue l(l + 1), l = 0, 1, 2, . . ., it follows from (4.1) that there is a universal constant c > 0 with (4.3) 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 Theorem 1.5 yields an appropriate estimate on E k Q * ρ f 2 , which when combined with (4.3) proves Theorem 1.3.To see this let us assume f − Avf 2,ν ≤ δ|Avf |.Then (4.4) E k f 2 ≤ δ|Avf |/2 νk , k = 1, 2, . . . .Now, for some g satisfying E k g 2 = 1, (4.5) Hence from Theorem 1.5 and (4.4) we have, 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 > ν we have that We conclude that where the constant C(δ) depends only on δ.Combining this last inequality with (4.3) we have that The theorem follows now by choosing b 3,p sufficiently small so that C(δ) b 3,p < c.
where the constant C δ,p depends only on δ, p > 1.Now by Proposition 3.3 one sees that if b 3,p is sufficiently small then the functionξ(y) = (I − T sym ) −1 h(y) is in L 2 k,δ (B 1/4 ) and (3.16) ξ 2,k,δ ≤ C δ,p b 2 k E k g 2 ,for some suitable constant C δ,p .Observe next thatE k f, Q ρ E k g = ρ B 1/4 ξ(y)A ρ E k f (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 1.3We first define the density f ρ,b in terms of the density f .To do this we consider the Dirichlet problem(∆ + b(y) • ∇)v(y) = 0, y∈ B 1/4 , v(y) = g(y), y ∈ ∂B 1/4 .Formally v(y) is given by the formulav(y) = P g(y) + Qg(y), y ∈ B 1/4 ,where P is the Poisson integral (1.4) and Qg is defined by (1.5).Now v can be represented in terms of the diffusion process X b (t) by the expressionv(y) = E y [g(X b (τ ))], y ∈ B 1/4 ,where τ is the first hitting time on ∂B 1/4 for the process started at X b (0) = y.It is clear then that if we regard the density f as a function on ∂B (1−ρ)/4 and the density f ρ,b as a function on ∂B 1/4 , then∂B (1−ρ)/4 f (y)v(y) dy /normalisation = ∂B 1/4 f ρ,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 f and f ρ,b as functions on the unit sphere S thatf ρ,b , g = f, P ρ g + Q ρ g , g ∈ L 2 (S),where P ρ g(y) = P g(y), y ∈ ∂B (1−ρ)/4 .Hence f ρ,b = P * ρ f + Q * ρ f , where P * ρ and Q * ρ are the formal adjoints of P ρ , Q ρ respectively.