Continuity of Zero-Hitting Times of Bessel Processes and Welding Homeomorphisms of SLE κ
CContinuity of Zero-Hitting Times of Bessel Processes andWelding Homeomorphisms of SLE κ Dmitry Beliaev ∗ Atul Shekhar † Vlad Margarint ‡ April 23, 2020
Abstract
We consider a family of Bessel Processes that depend on the starting point x anddimension δ , but are driven by the same Brownian motion. Our main result is that almostsurely the first time a process hits 0 is jointly continuous in x and δ , provided δ ≤
0. Asan application, we show that the SLE( κ ) welding homeomorphism is continuous in κ for κ ∈ [0 , κ in κ . The main tool in our proofs is random walks with increments distributedas infinite mean Inverse-Gamma laws. Keywords:
Bessel Processes, Schramm-Loewner-Evolution, Welding Homeomorphism.
AMS 2010 Subject Classification:
In this article we prove the joint continuity of level zero hitting times of Bessel processes w.r.t.its starting point and its dimension. For a real δ , the Bessel process of dimension δ startedfrom x ∈ R \ { } is defined as the solution to the stochastic differential equation (SDE) dZ t = dB t + δ −
12 1 Z t dt, Z = x, (1.1)where B t is a standard Brownian motion. Let ζ xδ := inf { t > | Z t ( x ) = 0 } . Also, set ζ δ = 0.It is well known that ζ xδ < ∞ almost surely if and only if δ <
2. For a fixed starting point x ,the random variable ζ xδ is very well understood. There is an extensive literature covering thesubject, see e.g. [8]. We are interested in ζ = { ζ xδ } x,δ considered as a stochastic process indexedby x and δ . Our main result is the following theorem: ∗ University of Oxford. Email: [email protected] † University Lyon 1. Email: [email protected] ‡ NYU Shanghai. Email: [email protected] a r X i v : . [ m a t h . P R ] A p r heorem 1.1. The function ( x, δ ) (cid:55)→ ζ xδ is almost surely jointly continuous in x ∈ R and δ ≤ . Remark 1.
The continuity of ζ xδ w.r.t. x is not expected to hold for δ > . For example, when δ = 1 , ζ x is a L´evy subordinator process which in particular has jumps. However, the almostsure continuity of ζ δ at a fixed x follows easily for all δ < using a Laplace transform compu-tation, see [1, Lemma 5] for details. The continuity of ζ for δ ≤ also implies that x (cid:55)→ ζ xδ isa continuous increasing bijection of [0 , ∞ ) . For δ ∈ (0 , / , this function is injective, but notsurjective (or equivalently continuous). For δ ∈ (3 / , , this will not be injective with positiveprobability, see [8, Proposition 2.11]. The process ζ is very closely related to Schramm-Loewner-Evolutions (SLEs). We providean application of Theorem 1.1 to the continuity in κ for the welding homeomorphism of SLE κ for κ ∈ [0 , H = { x + iy | y > } be the upper half plane. Given a simple curve γ : [0 , T ] → H ∪ { } such that γ = 0 and γ t ∈ H for all t >
0, the welding homeomorphism associated to γ is definedas follows. Let f : H → H \ γ [0 , T ] be the (unique) conformal map such that lim z → f ( z ) = γ T and f ( z ) = z + O (1) as z → ∞ . The map f extends continuously to H (see Chapter 2 in [10]).For some real numbers x − T < < x + T , f maps both (cid:2) x − T , (cid:3) and (cid:2) , x + T (cid:3) to γ [0 , T ] . The intervals (cid:0) −∞ , x − T (cid:3) and (cid:2) x + T , + ∞ (cid:1) are similarly mapped under f to ( −∞ ,
0] and [0 , ∞ ) respectively. Thewelding homeomorphism φ = φ γ : [0 , ∞ ) → [0 , ∞ ) associated to γ is defined by the relation f ( x ) = f ( − φ ( x )) , i.e. for x ∈ (cid:2) , x + T (cid:3) , φ ( x ) is the unique point such that f ( x ) = f ( − φ ( x )),and for x ∈ (cid:2) x + T , ∞ (cid:1) , φ ( x ) is the unique point such that f ( − φ ( x )) = − f ( x ) . The homeomor-phism φ contains information about the curve γ . For example, when φ is quasisymmetric, ituniquely characterizes γ , see [6].For κ ∈ [0 , κ is almost surely a simple curve, call it γ κ . Wewill write φ κ for the associated welding homeomorphism. We ask ourselves whether these home-omorphisms φ κ are continuous in κ . Our motivation to ask this is to study the related problemof continuity of γ κ in κ . To best of our knowledge it is an open problem for the full range of κ ∈ [0 , ∞ ) or even for κ ∈ [0 , κ ∈ [0 , − √ ∪ (8(2 + √ , ∞ ) and[3] for a recent progress for κ < /
3. Our approach to this problem is based on the followingheuristic argument.It follows from the results of [12] and [4] that SLE κ , for κ ∈ [0 , φ κ almost surely characterize the curve γ κ uniquely. In otherwords, the homeomorphism φ κ contain all the information about the curve γ κ . Heuristicallyspeaking, this suggests that the continuity of φ κ in κ should imply the continuity of γ κ in κ for κ <
4. Note however that this roadmap is as of now incomplete. This is because φ κ are notquasisymmetric (otherwise this would imply that γ k is a quasislit, and then a result of Rohde-Marshall [9] would imply that Loewner driving function of γ k , which is √ kB , is 1 / φ κ which are satisfied uniformly in κ and which2ecovers γ κ uniquely. We plan to address this in our future projects. For the purpose of presentarticle we only prove the continuity of φ κ in κ .Asking for the continuity of φ κ in κ is not yet well posed if we work with the above definitionof φ κ . This is because it is a priori not known whether γ κ are curves (let alone simple curves)simultaneously for all κ ∈ [0 ,
4] (we will often say that that a collection of events { A α } α occursimultaneously in α if P [ ∩ α A α ] = 1). This indeed is itself very closely related to the continuity of γ κ in κ , which is the problem we want to address in the first place. The correct way to formulatethis problem is to ask for a continuous modification of the stochastic field { φ κ ( x ) } x ≥ ,κ ∈ [0 , . Ourfollowing theorem answers it. Theorem 1.2.
There exists a random field ψ ( κ, x ) : [0 , × [0 , ∞ ) → [0 , ∞ ) such that(a) Almost surely, ψ is jointly continuous in ( κ, x ) ∈ [0 , × [0 , ∞ ) .(b) Simultaneously for all κ ∈ [0 , , ψ ( κ, · ) is a homeomorphism of [0 , ∞ ) .(c) P [ φ κ = ψ ( κ, · )] = 1 , ∀ κ ∈ [0 , . Remark 2.
We believe that there is an alternative approach to Theorem 1.2 based on Sheffield’sQuantuam Zipper. It was proven in [13] that welding homeomorphism can be constructed byidentifying points with same quantum length. This is also a promising approach, but it doesrequire some additional work to give a rigorous proof. For example, we will need continuity ofthe quantum measure µ γ with respect to the quantum parameter γ . For γ < (corresponds to κ < ), this was done in [5]. For κ = 4 or γ = 2 these measures converge to , so one has toconsider an appropriate scaling limit (see [2]). Another issue is that we need all measures µ γ tobe ‘nice’ simultaneously for all γ , so that we can invert the map x (cid:55)→ µ γ ([0 , x ]) simultaneouslyfor all γ . All this requires some additional work. We believe that this could be done, but thisapproach is highly technical for proving the above Theorem which is relatively simple. We thusgive a self contained proof of this result using the simpler approach based on Bessel processes. The paper is organized as follows. In the Section 2, we recall some basic facts on Loewnertheory and Bessel processes. Some technical lemmas are proved in Section 3. In the Section 4we give the construction of function ψ using an intermediary result Proposition 4.1, and proveTheorem 1.1 and Theorem 1.2. Finally, we prove the Proposition 4.1 in the Section 5. Acknowledgments:
D.B. was supported by the Engineering & Physical Sciences ResearchCouncil (EPSRC) Fellowship EP/M002896/1. A.S. acknowledges the financial support fromthe European Research Council (ERC) through a project grant LIKO. V.M. acknowledges thesupport of NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai.3
Preliminaries
We recall some basic facts from the Loewner theory. Given a curve γ as described in theintroduction, one can choose a parametrization of γ such that ∀ t (cid:62)
0, the half plane capacityof γ [0 , t ] is 2 t , i.e. lim z →∞ z ( g t ( z ) − z ) = 2 t , where g t : H \ γ [0 , t ] → H is the unique conformalmap such that g t ( z ) − z →
0, as | z | → ∞ . We will assume that γ is defined for t ∈ [0 ,
1] in thisparametrization. The Loewner transform of γ is the real-valued continuous function U definedby U t := lim z → γ t ; z ∈ H \ γ [0 ,t ] g t ( z ). For each z ∈ H , g t ( z ) satisfies the ordinary differential equation(ODE) given by ∂ t g t ( z ) = 2 g t ( z ) − U t , g ( z ) = z. (2.1)We refer to the equation (2.1) as the Loewner differential equation (LDE).This process could be reversed. Given a driving function U t one can solve LDE (2.1). Theresulting map g t is a conformal map from the set of points H \ K t where the solution exists upto time t onto the upper half-plane. It is a standard fact that for U = √ κW where κ ∈ [0 , W is a standard Brownian motion, there is a continuous simple curve γ = γ κ such that K t = γ [0 , t ]. The curves γ κ are known as SLE κ (curves). From now on we assume that thedriving function U t is of this form.To recover the curve γ (when it exists) from U , it is beneficial to look at the flow associatedto reverse-time LDE as follows. Let ˆ U t = U − U − t be the time reversal of U . For each fixed s ∈ [0 , t ≥ s and z ∈ H , let h ( s, t, z ) denote the solution of the reverse time stochastic LDEgiven by dh ( s, t, z ) = d ˆ U t − h ( s, t, z ) dt, h ( s, s, z ) = z ∈ H . (2.2)The map h : { (cid:54) s (cid:54) t (cid:54) } × H → H is called the flow associated with the equation (2.2)and it satisfies the so called flow property: h ( s, t, z ) = h ( u, t, h ( s, u, z )) , ∀ s (cid:54) u (cid:54) t. We will need the following Lemma from [11].
Lemma 2.1. If z = iy , y > , then | Re h ( s, t, z )) | (cid:54) r ∈ [ s,t ] | ˆ U r − ˆ U s | , and Im( h ( s, t, z )) ≤ (cid:112) y + 4( t − s ) . The following lemma is a rewriting of Lemma 2 . Lemma 2.2. If f t : H → H \ γ [0 , t ] is the conformal map such that lim z → f t ( z ) = γ t and f t ( z ) = z + O (1) as | z | → ∞ , then f t ( z ) = h (1 − t, , z ) . In particular f ( z ) = h (0 , , z )4he welding homeomorphism of a simple γ [0 ,
1] defined in the Section 1 can thus be con-structed using the continuous extension of h (0 , , · ) to H . It is therefore natural to considersolution h ( s, t, x ) of (2.2) started from x ∈ R \ { } . Note however that in this case the solutionmight hit zero in finite time and we will consider h ( s, t, x ) only up to this hitting time.When U = √ κW, we will denote the time reverse Brownian motion ˆ W by B and write h κ ( s, t, z ) for the flow obtained when ˆ U = √ κB . Note that if Z δ ( s, t, x ) denote the solution to(1.1) with the initial value Z δ ( s, s, x ) = x , then for κ (cid:54) = 0 and δ = 1 − κ ( δ and κ are henceforthalways related as such), h κ ( s, t, √ κx ) √ κ = Z δ ( s, t, x ) . (2.3)It follows that T κ ( s, √ κx ) = ζ δ ( s, x ), where T κ ( s, x ) := inf (cid:8) t > s | h k ( s, t, x ) = 0 (cid:9) ζ δ ( s, x ) := inf (cid:8) t > s | Z δ ( s, t, x ) = 0 (cid:9) . Also set T κ ( s,
0) = ζ δ ( s,
0) = s . To simplify some notations we will use T κ ( x ) and ζ δ ( x ) todenote T κ (0 , x ) and ζ δ (0 , x ).The following result in well known, see e.g. Proposition 2 . .
11 of [8].Recall the Inverse-Gamma( α, β ) distribution has a density proportional to t − − α exp {− β/t } . Lemma 2.3. (a) For δ < , the ζ δ (1) has the Inverse-Gamma (1 − δ , ) law.(b) If δ ≤ , then for all < x < y < ∞ , P [ ζ δ ( x ) < ζ δ ( y )] = 1 . In this section we prove some technical lemmas we will need to prove our main results.
Lemma 3.1 (Gronwall inequality) . Let x > and M t and N t satisfy M t (cid:54) x + ˆ U t − (cid:90) t M r dr ( respectively ≥ ) and N t = x + ˆ U t − (cid:90) t N r dr. Then M t ≤ N t (resp. M t ≥ N t ). In particular, for z = x + iy , x, y > , Re ( h κ (0 , t, z )) ≥ h κ (0 , t, x ) for all t ≤ T κ ( x ) . (3.1) Proof.
Note that M t − N t (cid:54) (cid:90) t M r − N r ) M r N r dr, h κ (0 , t, z ) = X t + iY t , then dX t = √ κdB t − X t X t + Y t dt ≥ √ kdB t − X t dt, which implies (3.1). Lemma 3.2. As κ → , √ κ sup t ≤ T κ ( x ) | B t | → uniformly over x in compact sets.Proof. Using monotonicity and scaling property of T κ ( x ) w.r.t. x , it suffices to consider x = 1.Note that T κ ( x ) = ζ δ (1 / √ κ ). Let κ n = 2 − n and κ n +1 ≤ κ ≤ κ n , then ζ δ (cid:18) √ κ (cid:19) (cid:54) ζ δ (cid:18) √ κ n +1 (cid:19) (cid:54) ζ δ n (cid:18) √ κ n +1 (cid:19) . So, √ κ sup t (cid:54) T κ (1) | B t | (cid:54) √ κ n sup t (cid:54) ζ δn ( / √ κ n +1 ) | B t | . Now, using Chebyshev inequality and Burkholder-Davis-Gundy inequality we obtain that P (cid:32) √ κ n sup t (cid:54) ζ δn (1 / √ κ n +1 ) | B t | ≥ κ / n (cid:33) ≤ κ n E (cid:16) sup t (cid:54) ζ δn (1 / √ κ n +1 ) | B t | (cid:17) √ κ n = O (cid:18) √ κ n E (cid:20) ζ δ n (cid:18) √ κ n +1 (cid:19)(cid:21)(cid:19) . Note that ζ δ n (1) ∼ Inverse-Gamma (cid:16) κ n + , (cid:17) , hence E [ ζ δ n (1)] = 12 1 κ n − = O ( κ n ) , which implies P (cid:32) √ κ n sup t (cid:54) ζ δn (1 / √ κ n +1 ) | B t | ≥ κ / n (cid:33) = O ( √ κ n ) . Borel-Cantelli Lemma implies that for n large enough, √ κ n sup t (cid:54) ζ δn (1 / √ κ n +1 ) | B t | ≤ κ / n , and the conclusion follows.We will also use the following lemma on random walks with Inverse-Gamma(1 , ) incre-ments. Let { T n } n ≥ be an i.i.d. sequence of random variables each distributed as Inverse-Gamma(1 , / E [ T ] = + ∞ and the strong law of large numbers implies thatalmost surely T + ... + T n n −→ + ∞ , i.e. T + T + · · · + T n tends to infinity faster than linear func-tion. The following lemma gives that the precise speed of convergence is n log n . The additionallog n factor will be crucial for our proofs. 6 emma 3.3. If { S n } n ≥ is a sequence of random variables such that ∀ n ≥ , S n d = T + · · · + T n ,then S n n log n p −→ . Proof.
We show that the Laplace transforms E (cid:20) exp (cid:18) − tS n n log n (cid:19)(cid:21) → e − t/ , (3.2)as n → ∞ . Then, the L´evy continuity Theorem implies the claim. To prove (3.2), note that E (cid:20) exp (cid:18) − tS n n log n (cid:19)(cid:21) = (cid:18) E (cid:20) exp (cid:18) − tT n log n (cid:19)(cid:21)(cid:19) n = (cid:18)(cid:114) tn log n K (cid:18)(cid:114) tn log n (cid:19)(cid:19) n , (3.3)where K ( x ) is the modified Bessel function of the second kind. We have used the fact that E (cid:2) e − tT (cid:3) = √ tK ( √ t ) . Finally, note thatlim x → + log ( xK ( x )) x (log x + 1) = 12 , and plugging this asymptotics in (3.3) gives (3.2). ψ ( κ, x ) . In this section we give the construction of ψ ( κ, x ). This will be based on the following Propo-sition. Proposition 4.1. (a) Almost surely for all κ ∈ [0 , simultaneously, the function x (cid:55)−→ T κ ( x ) is a strictly increasing continuous bijection [0 , ∞ ) → [0 , ∞ ) , and it is a strictlydecreasing continuous bijection ( −∞ , → [0 , ∞ ) . (b) Almost surely for all κ ∈ [0 , simultaneously, the function κ (cid:55)→ T κ ( · ) ∈ C ([0 , ∞ ) , [0 , ∞ )) or C (( −∞ , , [0 , ∞ )] is continuous. The proof of this proposition is postponed until the next Section.Proposition 4.1 has a simple corollary. To state it we will need the following notations. Weassume that κ ∈ [0 ,
4] and t ∈ [0 ,
1] . For x (cid:62) h κ, + t ( x ) = (cid:26) h κ (0 , t, x ) , if t ≤ T κ ( x ) T κ ( x ) − t, if t (cid:62) T κ ( x )Similarly, for x ≤ h κ, − t ( x ) = (cid:26) h κ (0 , t, x ) , if t ≤ T κ ( x ) t − T κ ( x ) , if t (cid:62) T κ ( x )7he definition of the ˜ h κ, ± t is a bit artificial for t > T κ ( x ), but it will help us represent the weldinghomeomorphisms in a neat way. An immediate corollary to Proposition 4.1 is the following. Corollary 4.2. (a) Almost surely for all κ ∈ [0 , simultaneously, maps x (cid:55)−→ ˜ h κ, +1 ( x ) and x (cid:55)−→ ˜ h κ, − ( x ) are strictly increasing continuous bijections [0 , ∞ ) → [ − , ∞ ) and ( −∞ , → ( −∞ , respectively.(b) Furthermore, the functions κ (cid:55)→ ˜ h κ, +1 ∈ C ([0 , ∞ ) , [ − , ∞ )) ,κ (cid:55)→ ˜ h κ, − ∈ C (( −∞ , , ( −∞ , are continuous. We now define the continuous field ψ . Set ψ ( κ,
0) = 0. For x ∈ (0 , ∞ ), let ψ κ ( x ) be theunique point such that ˜ h κ, − ( − ψ κ ( x )) = − ˜ h κ, +1 ( x ) . Note that this definition is designed so that h κ (0 , t, · ) started at x and − ψ κ ( x ) either hit zeroat the same time or h κ (0 , , x ) = − h k (0 , , − ψ κ ( x )). This is consistent with the definition of φ given in the Section 1. The proof of Theorem 1.1 and Theorem 1.2-( a ) , ( b ) is immediate fromProposition 4.1. Proof of Theorem 1.2-(c).
To prove φ κ = ψ ( κ, · ), using Lemma 2.2, it suffices to verify thatlim z → x h κ (0 , , z ) = lim z →− ψ κ ( x ) h κ (0 , , z ) . (4.1)If T κ ( x ) >
1, then by definition, T κ ( − ψ ( κ, x )) >
1. This implies that h κ (0 , , z ) → h κ (0 , , x ) as z → x, and h κ (0 , , z ) → h κ (0 , , − ψ ( κ, x )) as z → ψ ( κ, x ) . Then, the (4.1) follows by the definition of ψ ( κ, x ).If T κ ( x ) ≤
1, then T κ ( − ψ ( κ, x )) = T κ ( x ) ≤
1. Let T κ ( − ψ ( κ, x )) = T κ ( x ) = 1 − t , t ≥ h κ (0 , , z ) = h κ (1 − t , , h κ (0 , − t , z )). We claim that h κ (0 , − t , z ) → z → x. (4.2)Then, using Lemma 2.2, it follows that h κ (0 , , z ) → γ κt as z → x . Similarly, h κ (0 , , z ) → γ κt as z → − ψ ( κ, x ) as well, establishing (4.1).To prove (4.2), note that as z → x , Re ( z ) is arbitrarily close to x . Then, using Lemma 3.1and the continuity of T κ ( x ) in x , it follows that Re ( h κ (0 , t, z )) > t ≤ T κ ( x ) − (cid:15) ( z ),where (cid:15) ( z ) → z → x . Then it easily follows that h κ (0 , T κ ( x ) − (cid:15) ( z ) , z )) is arbitrarily small.Finally, the (4.2) follows from the Lemma 2.1. 8 Proof of Proposition 4.1.
Proof of Proposition 4.1-(a).
We first claim that almost surely simultaneously for all κ ∈ [0 , s ∈ [0 ,
1] (or equivalently for all s ≥ x → T κ ( s, x ) − s = 0 . (5.1)When κ = 0, it follows from an explicit computation that h ( s, t, x ) = (cid:112) x − t − s ) , whichimplies T ( s, x ) − s = x and (5.1) easily follows. For κ ∈ (0 , , it suffices to consider x → <κ < κ ≤
4, then T κ ( s, √ κ x ) − s ≤ T κ ( s, √ κ , x ) − s (cid:54) T ( s, x ) − s. It suffices to prove that almost surely for all s ∈ [0 , x → T ( s, x ) − s = 0 . Note that T k ( s, x ) is monotonic increasing in x and the limit T ( s, − s := lim x → T ( s, x ) − s always exists. We now prove that this limit is zero for all s ∈ [0 , x n = (cid:114) e − n n , k n = e n , λ n = 1 n . Figure 1: Random walk construction of zero-hitting times.For each n ≥
1, define a sequence { s nk } k ≥ by s n = 0, and s nk +1 = T ( s nk , x n ), see Figure 1.Note that by scaling, Strong Markov Property of the Brownian motion and Lemma 2.3, { s nk } k ≥
9s a random walk with the increments distributed according to x n × Inverse-Gamma(1 , / s nk n x n k n log k n p −→ . Note that x n k n log k n >
2, and since convergence in probability implies almost sure convergencealong a subsequence, we obtain that, almost surely, s nk n > n. (5.2)Next, consider the event A n := k n − (cid:91) k =0 (cid:8) s nk +1 − s nk > λ n (cid:9) . Then, using independence and the fact that P (cid:20) Inverse-Gamma (cid:18) , (cid:19) ≤ λ (cid:21) = exp (cid:18) − λ (cid:19) , we get P [ A n ] = 1 − exp (cid:18) − k n x n λ n (cid:19) (cid:54) k n x n λ n . Note that ∞ (cid:88) n =1 k n x n λ n < ∞ , and the Borel-Cantelli Lemma implies, almost surely, for all n large enough s nk +1 − s nk (cid:54) λ n , ∀ k = 0 , , . . . k n − . Now, for any s ∈ [0 , n such that for some0 ≤ k ≤ k n − s ∈ [ s nk , s nk +1 ]. Using the flow property and the monotonicity, we obtain that T ( s, − s (cid:54) T (cid:0) s, h ( s nk , s, x n ) (cid:1) − s = T ( s nk , x n ) − s (cid:54) s nk +1 − s nk (cid:54) λ n , which implies that T ( s, − s = 0 . The fact that x (cid:55)→ T κ ( x ) on [0 , ∞ ) is strictly increasing follows easily from Lemma 2.3-(b). As for its continuity, we first prove the left continuity. For any x ∈ (0 , ∞ ) , if y ↑ x , let T κ ( x − ) := lim y ↑ x T κ ( y ). If T κ ( x − ) < T κ ( x ), then by taking the monotonic limit of h κ (0 , t, y )as y ↑ x , we obtain a solution to the (2.2) starting from x which hits zero before time T κ ( x ).Since (2.2) has a unique solution, this gives a contradiction. Thus, T κ ( x − ) = T κ ( x ).For the right-continuity of T κ ( x ), for any 0 ≤ x < y < ∞ , using again the flow property, wehave that that T κ ( y ) − T κ ( x ) = T κ ( T κ ( x ) , h κ (0 , T κ ( x ) , y )) − T κ ( x ) . Also, as y ↓ x , a similar monotonicity argument as above implies that h κ (0 , T κ ( x ) , y ) → y → x + T κ ( y ) = T κ ( x ), finishing the proof.10 roof of Proposition 4.1-(b). We first check the continuity in κ at κ = κ ∈ (0 , T κ ( x ) = ζ δ ( x/ √ κ ), it suffices to check the continuity of ζ δ ( · ) in δ . Note that if 0 < κ < κ ≤ ζ δ ( x ) ≤ ζ δ ( x ). If either κ ↓ κ or κ ↑ κ , we will establish the pointwise convergence ζ δ ( x ) → ζ δ ( x ). Then, by Dini’s Theorem, we obtain the uniform convergence on compact sets.For pointwise convergence, let κ ↑ κ first. Note that Z δ (0 , t, x ) is monotonically increasingwith δ (or κ ). If lim κ ↑ κ ζ δ ( x ) < ζ δ ( x ), then by taking the limit function lim κ ↑ κ Z δ (0 , t, x ) , onecan construct a solution to (1.1) with δ = 1 − κ started from x which hits zero before ζ δ ( x ),which is a contradiction.For κ ↓ κ , using the flow property, ζ δ ( x ) − ζ δ ( x ) = ζ δ ( ζ δ ( x ) , Z δ (0 , ζ δ ( x ) , x )) − ζ δ ( x ) . Again, using a similar argument as before, it is easy to check that Z δ (0 , ζ δ ( x ) , x ) → κ ↓ κ . Using (5.1) again implies lim κ ↓ κ ζ δ ( x ) = ζ δ ( x ).The continuity in κ at κ = 0 requires a different argument. Note that for t ≤ T κ ( x ), h κ (0 , t, x ) = x + √ κB t − (cid:90) t h κ (0 , r, x ) dr ≤ x + √ κ sup t ≤ T κ ( x ) B t − (cid:90) t h κ (0 , r, x ) dr. Then, Lemma 3.1 implies that h κ (0 , t, x ) ≤ (cid:114) ( x + √ κ sup t ≤ T κ ( x ) B t ) − t. Thus, T κ ( x ) ≤ ( x + √ κ sup t ≤ T κ ( x ) B t ) . Similarly, T κ ( x ) ≥ ( x + √ κ inf t ≤ T κ ( x ) B t ) . Using Lemma 3.2, we conclude that T κ ( x ) → x uniformly on compact sets as κ → References [1] Henri Elad-Altman, Bismut-Elworthy-Li formulae for Bessel processes,
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