Penalization of the Wiener Measure and Principal Values

Transcription

1 Penalization of the Wiener Measure and Principal Values by Catherine DONATI-MARTIN and Yueyun HU Summary. Let Q µ be absolutely continuous with respect to the Wiener measure with density D t exp hb sdb s t h B s ds, where h µ x l x and >. When µ /, we show that as +, there exists a penalization effect of the Wiener measure through the densities D t such that the limit law does not charge the paths hitting. The remaining case µ < / together with a more general form of h are also studied. Mathematics Subject Classification. 6F5; 6J55.. Introduction. Let h : R R be a bounded measurable function. On the canonical space Ω CR +, R, we consider the probabilities : W the Wiener measure. Q h the law of the process X h, the unique strong solution of the equation cf. Zvonkin []:. X t B t + hx s ds, t >, where B is a one-dimensional Brownian motion starting from. - Q h the law of the process:. Bh t B t hb s ds, t >. According to Girsanov s theorem: The probability law Q h F t is absolutely continuous with respect to W Ft with density F t being the natural filtration of the canonical process:.3 D h t exp hb s db s h B s ds, t >. Laboratoire de Statistique et Probabilités CNRS UMR-5583, Université Paul Sabatier, 8, route de Narbonne, F-36 Toulouse cedex 4. donati@cict.fr Laboratoire de Probabilités et Modèles Aléatoires CNRS UMR-7599, Université Paris VI, 4 Place Jussieu, F-755 Paris cedex 5. hu@ccr.jussieu.fr

2 Under Q h, B h is a Brownian motion. Moreover,.4 for every t, F Bh t F B t, where Ft X denotes the natural filtration of a process X. see e.g. Revuz and Yor [3, pp. 367] When h is not bounded, these results are no more true and even meaningless. Nevertheless, for some functions h, the process ˆB h makes sense. For example, if h has a singularity at, we can define hb sds def lim + hb sl Bs ds when the limit exists. For instance, this is the case for hx sgnx x α for α < 3/, due to the Hölder continuity of Brownian local times. In that case, we denote the limit by p.v. hb sds as Cauchy s principal values. We refer to Biane and Yor [], Yor [] and Yor [9, Chap. ] for studies and references on principal values. In this paper, we shall concentrate on the case hx µ x, where µ R\{} denotes some fixed parameter. We firstly consider µ. Let ˆB t B t p.v. ds lim B ˆBh t, s where h x x l x. Contrarily to the bounded case see.4, the filtration of ˆB is strictly included in the filtration of B see [,5]. An open problem is to characterize this loss of information see Yor [9, Chap. 7]. Instead of discussing this difficult problem here, we consider the following question: For hx x, what can we say about. and.3? It is well known that. does not admit uniqueness in law, since the law P 3 3 of the 3-dimensional Bessel process BES3 is a solution as well as the law ˆP of the opposite of a BES3. Nevertheless, we can think that there exists a probability Q h singular with respect to W which morally satisfies: Q h t F t exp hb s db s h B s ds W Ft, t >, This density is meaningless, and we shall stu the law Q µ of the solution of. with hx µ x l x, with µ R\{} and >. We obtain as expected! the following Theorem. µ. When, Q converges to P3 + P 3, where P3 denotes the law of a 3-dimensional Bessel process R 3 t, t starting from, and P 3 denotes the distribution of R 3 t, t. Remark M. Yor. Let hx x. The family of the laws of all solutions of. is exactly { ap 3 + a P 3, a }. We give a first proof of Theorem. in Section by using the absolute continuity.3: dq Ft D h t dp Ft, and by stuing carefully the density term D h t. The

3 proof shows that the penalization effect of the density D h t is to prevent the paths from hitting when. This should be compared with the polymer measure which is defined as the limit of the Wiener measure in R d with densities preventing the paths from self intersecting. See [3] and [5] for some related references. Another proof of Theorem. will be given in Section 3 based on the onedimensional diffusion theory, and we shall remark that the penalization parameter µ R plays an important role. More precisely, we summarize the different cases as follows: if µ, the above phenomenon of penalization of holds, see Proposition 3.. if < µ <, Qµ converges to that of a symmetric Bessel process of dimension δ + µ in the terminology of Watanabe [7, 8], see Proposition 3.. if µ, this case is studied in Section 4 by using Krein s spectral theory. See Theorem 4. for the behaviour of the solution of.. if µ <, there exists a stationary measure of the corresponding diffusion, see Proposition 3.3. Finally, we consider some general form of h in Section 5 by using Girsanov s transform. Before closing this introduction, we would like to say that the proof of Theorem. presented in Section is much more complicated than that in Section 3, but this proof, based on the decomposition of Brownian path, clearly shows where the penalization effect comes from, and also brings some by-product such as Lemma.3 and Corollary.. Throughout this paper, we denote by R δ a Bessel process starting from, of dimension δ >, and by P δ its law. For a process X, we denote lt a X its local time process when it exists defined by: lt a t X lim Xs a ds. Acknowledgements: We are very grateful to Professor Marc Yor for helpful discussions and for references.. Proof of Theorem.: Penalization through the densities. We shall prove that for any continuous bounded functional F on C[, t], R, Q F P3 F + 3 P F,. We prove this only for t, the general t > follows by using the same method or from the scaling property. The proof consists of four steps: 3

4 Step : On F, Q is absolutely continuous with respect to W with density:. db s exp l Bs B s B exp l B. D ds Bs l Bs To obtain., we have from the Itô-Tanaka formula that B t + log B t log + d B s l Bs + l t B, db s l Bs B s ds l Bs + l t B. B s And. follows. Step : Decomposition of the Brownian path see [3, Exercise XII.3.8]: Let g def sup{t : B t }. Then, bs def g B gs, s is a standard Brownian bridge, ms def g B g+ gs, s is a standard Brownian meander, b, m, g and sgnb are independent. We shall now express. in terms of b, m, g, sgnb. On one hand, we need to express F B in terms of these processes. This is the content of the following lemma: Lemma.. Let C [, ], R def {ω continuous on [, ] : ω ω }. To any measurable functional F on C[, ], R, we can associate a measurable functional F on C [, ], R [, ] C[, ], R such that: i F ω; ; ω F ω ii The following equality holds :. F B s s F b s s ; g ; sgnb m s s. Moreover, if F is continuous, then: t F ω; t; ω is continuous at t. Proof of Lemma.. Pick up ω, t, ω C [, ], R [, ] C[, ], R, we define F ω, t, ω def F ω t ω, where ω t ω is the continuous path defined by: ω t ω s def { t ω s t, if s t t ω s t t, if t < s. 4

5 Then i and ii follow, and it is not difficult to prove that for fixed ω, ω, t ω t ω is continuous at t. Thus, On the other hand, we have: where Q def,± F g g l B g l m + g l b. g Q m F E[ F bs ; g; sgnb m s g exp l g g ] g b exp l m Q,+ F + Q, F, Q other term Q, follows from symmetry. F sgnb ±. We shall now stu the term Q,+, the Step 3: Conditioning by g: We recall that g is arcsine distributed. Then, Q,+ F dx [ x m E F b; x; m π x x x exp l x x ] x b exp l m [ π y y E exp y l y b ].3 A F y, b, where for any path ω C[, ] R,.4 A F y, ω def E[ F ω; y; m y ] y m exp l y m, where the expectation is taken with respect to m. The asymptotic behaviour of A F is given by: Lemma.. Let y > and ω C[, ], R. Then as, π [ ] [ A F y, ω E F ω; ; R3 E exp ] l R 3 π [ ].5 8 E F R 3. Proof of Lemma.: From Imhof [6] s absolute continuity between the law of m and R 3, π [ A F y, ω E F ω; y; R 3 R 3 y ] y exp l y R 3. 5

6 Let us denote BF y ] y, ω E[ F ω; ; R3 exp l y R 3. Then, A F y, ω This follows from the a.s. convergence of π B F y, ω,. F ω; y; R 3 R 3 y F ω; ; R 3,, see Lemma. and the integrability of /R 3 F is bounded. Now, BF y, ω E[ F ω; ; R 3 exp l y R 3 ] where R 3 t y R 3 y t is a 3-dimensional Bessel process. Since y goes to as, it follows from the ergodicity of the scaling transformation that R 3 and R 3 are asymptotically independent, i.e.: R 3, R 3 d R 3, R 3,, where R 3 is an independent copy of R 3. See e.g. [3, Exercise XIII..7]. It follows that ] [ BF y, ω E[ F ω; ; R3 E exp l R ] 3,, [ ] and.5 follows from F ω; ; R 3 F R 3 and E exp l R 3 /. Step 4: convergence of Q + : Lemma.3. We have [ y E exp y ] l y b <. Proof: We write.3 for F. Then, π y C[,] R y y y P,db exp l y b A F y, b, where P, denotes the law of the Brownian bridge. The function in the integral converges dp, db a.s. to π 8y exp y y b. 6 l

7 By Fatou s lemma, y P, db exp 8π y l y b lim inf Q, C as desired. In order to pass to the limit in the integral.3, we shall decompose the integral in η... + η determined later. Let us denote.6 Φ, η π η..., where η ], [ is a fixed number whose value will be y y E [ exp y ] l y b A F y, b where A F is defined by.4 and for all y, b, A F y, b K for some constant K since E[m ] <. Using Lemmas. and.3, we deduce from the dominated convergence theorem that η being fixed [ ] Φ, η C E F R 3,, where.7 C 8π [ y E exp y ] l y b. It remains to choose η such that the remaining term is arbitrarily small when. Let Φ, η [ y ] π y y E exp l y b A F y, b. η Since y m + y m and F K, we deduce from.4 that { Φ, η K [ y ] ] } E exp η y l y b E [m +. η y y Now, η η y y dz z Let α > be arbitrarily small, we can choose η such that: K η y y α. Thanks to Lemma.3, there exists such that for, K η [ y E exp y ] l y b 7, η. ] E [m α.

8 Thus, we have obtained: Q,+ [ ] F C E F R 3,, where C is given by.7 and necessarily C. In the same way, Q, F E[F R 3 ],, proving the theorem. The following consequence may have an independent interest: Corollary.. We have y E[exp y l y b ] 8π. 3. Convergences in law. In this section and the forthcoming one, we shall consider a family of processes X t, t, which are the unique strong solutions of the following equations: 3. X t Bt + µ ds X s l X s >. We shall simply write Xt X t, t for the solution of 3. corresponding to. By scaling, we have 3. X t, t law Xt/, t We are interested in the convergence in law of X as. When µ, we have shown in Theorem. how the penalization prevents X from hitting the origin, this fact holds for all µ /. We shall also show that when the penalization parameter µ is small, the limit process can hit the origin. Lemma 3.. Assume µ > /. As, we have d X R δ, where R δ denotes a Bessel process of dimension δ def + µ > starting from, and d means the convergence in law on CR + R endowed with the topology of uniform convergence on each compact interval. Proof. Define Y t def X t, for t. Using Itô s formula, the process Y satisfies Y t Y s dw s + ds + µl Y s>, 3.3 t, Y t, 8

9 with a Brownian motion W, which a priori depends on. It is elementary to verify that 3.3 admits the uniqueness in law e.g. by using Zvonkin s method [] and then using Engelbert and Schmidt s criteria, see Karatzas and Shreve [7, Chap. 5.5]. Hence, it suffices to show that the solutions of 3.3 converge in law to R δ. But this can be done in a standard way, see e.g. the proof of Stroock and Varadhan [6, Theorem.3.3]. The only remaining fact to verify is that lim E ds l Y s. To this end, we use the comparison theorem cf. [3] to the two diffusions Y and Rδ which satisfies the following equation: Rδ t R δ s dw s + δ t, t, R δ t, with the dimension < δ < min, + µ. It follows that almost surely for all t, Y t Rδ t. It follows that E ds l Y s E ds l Rδ s,, completing the proof. Remark 3.. To our best knowledge, the comparison theorems for two diffusions often require some regularities of drift terms at least one of the two drift terms, this is why we are not able to deduce the convergence of Y directly from 3.3. Proposition 3.. Assume that µ /. The law of X converges to Q δ + Q δ, where Q δ denotes the law of R δ a Bessel process starting from, of dimension δ def + µ, and Q δ denotes the law of R δ. Proof. Fix a small η >. Since the Bessel process R δ does not visit the origin after time δ, we deduce from Lemma 3. that X s, η s t will keep the same sign as X η, with probability approaching when. This fact together with Lemma 3. imply that the process X s, η s t will converge in law to UR δ s, η s t, as, where U is independent of R δ and U law sgnx η. The symmetry implies that U is a symmetric Bernoulli variable and the rest of the proof is completed by letting η. Remark 3.. Proposition 3. admits a natural generalization to the multidimensional case: By considering a Lipschitz drift term which coincides with µx x when x, we can prove in the same way that the associated diffusion converges in law to U d du Q δ u, where U d denotes the uniform probability measure on the sphere {u R d : u }, and Q δ u the law of the process ur δ. When µ < /, the limit process can hit. Firstly, we have 9

10 Proposition 3. < µ <. Let δ + µ,. As, the law Qµ converges to that of the continuous process U δ t, t, which is determined by the following time-change: 3.4 U δ t α sgnbρ t Bρ t /α, with α def µ, and ρ the inverse of ρ: 3.5 ρt def Bs α ds, t. Remark 3.3. According to the terminology of Watanabe [7, 8], U δ is a symmetric Bessel process of dimension δ. More precisely, U δ is the unique diffusion on R such that U δ and i For every f C R with compact support and vanishing on a neighborhood of, the process fu δ t ds f U δ s + δ ii Almost surely for t >, l {}U δ s ds. f U δ s U δ s is a martingale. iii The function y δ is a scale function of U δ. We can verify that the process U δ defined via 3.4 and 3.5 satisfies the properties i iii. For more details on the skew and bilateral Bessel processes, we refer to Watanabe [7,8]. Proof. The proof follows from Feller s time change for one-dimensional diffusion. Recall 3.. Denote by S the scale function of the diffusion X: 3.6 Sx def We have x S y def 3.7 Zt def SXt x l y + l y > y µ, x R. S XsdBs σzsdbs, where σx def S S x, and S denotes the inverse function of S. Recall that α µ. It is elementary to obtain that 3.8 S z α sgnz α z /α,, z R, 3.9 σx α x /α, x. In view of scaling, we write law X t, t S α α Zt/, t, therefore it suffices to show that 3. α Zt/, t d α α Bρ t, t,.

11 To this end, applying Dubins-Schwarz representation theorem to the continuous martingale Zt shows that for some Brownian motion W, we have where ψ denotes the inverse of ψ, and ψt def Zt W ψ t, R Lt, xσ xdx, with Lt, x the local times of W. Using the scaling property, we obtain: 3. α Zt/ law, t W ψ t, t, where ψ denotes the inverse of ψ, and ψ t def R Lt, x 4α σ x / α dx. Using 3.9, we obtain that uniformly on each interval [, T ] for T >, ψ t α α Applying this observation to 3., we obtain: R Lt, x x α dx, t T. α Zt/ d, t Bρ tα /α, t,, showing 3. by using the scaling property. Proposition 3.3. Assume that µ < /. For t >, X t d + µ 4 µ l x + x µ l x > dx,. Proof. We use again Observe that the diffusion Z is on its natural scale, and its speed measure has a finite mass on R. It turns out see e.g. Rogers and Williams [4, Theorem V.54.5] that Zu d πdx, u, where πdx def σ xdx/ R σ y is the stationary measure of Z. Now, observe that X t law X t/ S Zt/, yielding the desired result after elementary calculations.

12 The case µ / is discussed in the next section. 4. Krein s spectral theory: Application to the case µ /. The main result of this section is the following: Theorem 4.. Let µ / and fix v >. Recall that X denotes the solution of 3.. As, we have log/ sup X s d v s v e, log X v log/ d U, where e and U are respectively exponential distributed with parameter and uniform on [, ]. Before we present the proof, we recall some facts on Krein s correspondence, see Dym and McKean [4], Kotani and Watanabe [] and Kasahara [8] together with their references for details. Let M be the class of functions m : [, ] [, ] which are nondecreasing, right-continuous and m. Put m and let l sup{x : mx < } l if mx. When l >, consider the following integral equations: x φx, λ + λ ψx, λ x + λ hλ def l x y+ y+ dx φ x, λ lim ψx, λ x φx, λ. φz, λdmz, x < l, ψz, λdmz, x < l, When mx, we define hλ. The correspondence mx hλ is called Krein s correspondence and h is called the characteristic function of m. When mx, h and h can be represented as follows: 4.4 hλ c + dνt λ + t, where c denotes the left endpoint of the support of m, and νdt is a nonnegative measure on [, such that dνt +t <. We call dνt the spectral measure. Let H def {h : of form 4.4} {h } {h }.

13 Theorem Krein [], Kasahara [8]. Krein s correspondence m M h H is one to one and onto. Let m n M h n H. We denote by φ n and ψ n the solutions of related to m n. The following three assertions are equivalent: i As n, m n x m x at each continuity point x [, of m x. ii As n, φ n x, λ, ψ n x, λ φ x, λ, ψ x, λ for x [, and λ >. iii As n, h n λ h λ for λ >. We shall make use of the following scaling property: Lemma 4.. Let mx hλ be in Krein s correspondence. For a, b >, we define mx b a m x a, and denote by h, νt, φ, ψ the associated characteristic function, the spectral measure and the fundamental solutions. We have for all x, t, λ, a 4.5 hλ a hbλ, νt b νbt, x φx, λ φ a, bλ, x ψx, λ a ψ a, bλ. When mx <, the functions φx, ξ, ψx, ξ are analytic on ξ, and the scaling property can be extended to all ξ C. Krein s theory together with Kotani and Watanabe s extension are very powerful to treat the real-valued diffusion of generator d d dm dz here, dmz is a Radon measure on R. See Bertoin [] for a nice application to Brownian principal additive functionals. See also Kasahara et al. [9]. Here, we stu a particular example of diffusion, and the method can be applied to a more general class of diffusions. Proposition 4.. Let Z be a diffusion on R of generator σ z d dz with σz z + for z R cf with µ /. As t, we have log t t sup Zs d s t e, log Zt log t d U, where e and U denote respectively the exponential distribution with parameter and the uniform distribution on [, ]. Proof. We consider the two measures m, m M determined by m dx σ xdx and m dx σ xdx x. Since m m by the symmetry of σ, we can only consider m m. Consider Krein s correspondence hλ mx and let φx, λ, ψx, λ and dνt be the fundamental solutions and the spectral measure respectively. We define the extensions of φ, λ and ψ, λ on R by: φz, λ def φ z, λ and ψz, λ sgnz ψ z, λ for z R. Put u + z, λ φz, λ ψz, λ, z R, λ >, hλ u z, λ φz, λ + ψz, λ, hλ g λ z, y def g λ y, z def hλu +z, λu y, λ, z y R. 3

14 We refer to Kotani and Watanabe [] for details. Therefore g λ is the density of the resolvent operator of Z with respect to its speed measure. We denote by pt, y the density of the law of Zt with respect to the speed measure: P Zt dz pt, zσ zdz. We have pt, z pt, z and for x, 4.8 e λt pt, xdt g λ, x hλu + x, λ φx, ξ λ + ξ dνξ, where the last equality follows from the following spectral representation: 4.9 hλu + x y, λφx y, λ φx, ξφy, ξ dνξ, x, y, λ + ξ see Dym and McKean [4, pp. 76]. Inverting the Laplace transform of 4.8, we get 4. pt, x e t ξ φx, ξ dνξ, t >, x. The above integral is absolutely convergent, since we have e.g. φ x, ξ λ+ξ dνξ < by taking x y in 4.9. Now, we are going to prove 4.6 and 4.7. Let Hr def inf{t > : Zt > r} for r >. Observe that e λt φzt, λ, t Hr is a bounded continuous martingale. Recall that φz, λ φ z, λ. The optional stopping time theorem yields 4. E exp λ H r a φ r a, λ, where we take a a def log/ and > is assumed to be small. It remains to stu the behaviour of φ r a, λ when. Define m x a m x a, and h, dν t, φ, ψ in the obvious way. Observe that m x m x, x >,, where m x def l x m. It is immediate to obtain that m x h λ /λ, and φ x, λ + λx, ψ x, λ x for x, λ >. Using Lemma 4. and Kasahara s continuity theorem, we have that for x > and λ >, φ x a, λ φ x, λ + λx,, ψ x a, λ a ψ x, λ x log/,, a hλ h λ λ,. Finally, we apply 4.4 to h and dν with c in 4.4. By considering h λ h, we deduce from 4.4 that as, 4. l s h + s dν s weakly converges to the Dirac measure at, 4

15 see e.g. Dym and McKean [4, pp.79]. Going back to 4., we obtain that for fixed r > and λ > r H log/ d r e,. Inverting the above convergence, we obtain 4.6. To prove 4.7, fix < c < and t >. We have from 4. that 4.3 P Zt / < c c a c dxσ x dxσ x e t ξ/ φx, ξ dνξ dν se st φ ax, s, by using Lemma 4. and the change of variable ξ s, with a def log/. Admitting for the moment that uniformly on y [, ], 4.4 dν se st φ y, s,. Then, it follows from 4.3 that P Zt / < c c dxσ x + o c + o,, a showing the desired convergence in law 4.7 since < c < is arbitrary. It remains us to prove 4.4. Firstly for some large but fixed K >, we deduce from Cauchy-Schwarz inequality that recalling from 4.4 that h dν s +s e st K+ 4.5 φ y, s dν s sup e st + s s K sup s K sup s K e st + s h 3/ φ y, s / / dν s h + s u +, y, φ y,, e st + s h 3/ φ,, where the equality is due to the analogue form of 4.9 for dν, and the last inequality follows from the facts that the function x u +, x, def φ x, h ψ x, is positive and nonincreasing hence is bounded by and the function φ, is increasing. In view of 4.5, it suffices to show that for any fixed K >, when, the following convergence holds uniformly for y [, ], 4.6 K dν se st φ y, s. 5

16 Now consider s K. By using e.g. the integral equation for φ, s and applying 4., it is easy to show that sup φ y, s sy,. s K, y Using the above estimate and applying 4., we obtain 4.6 and complete the whole proof. Proof of Theorem 4.. Follows from 3., with α, and Proposition An example of Girsanov s transform. In the above penalization procedure Sections and 3, the key property of the function h is that hx µ/x as x. Therefore, it is natural to consider hx fx + µ x, x, with f C R R a bounded function such that fx f f x for x and f is bounded. Consider the diffusion 5. Y t Bt + hy s l Y s ds. Using the corresponding result for the case f, we obtain: Theorem 5.. Suppose that µ /. When, the diffusion Y converges in law to Qµ,f + Q µ, f, where Q µ,f and Q µ, f denote respectively the laws of the processes Ξ and Ξ which are the unique solutions in law of: Ξ t Bt + Ξ t Bt + µ + fξ s ds, Ξ s µ f Ξ s ds. Ξ s Proof. The proof is based on Girsanov s transform. Recall that the diffusion X satisfies ds X t Bt + µ X s l X s. 6

17 Define f x fxl x and consider the probability Q defined by 5. dq dp Ft exp f X sdbs f X sds. It follows from Girsanov s transform that under Q, the process X has the same law as Y under P. We need to treat the density in 5.. To this end, define F x x f y for x R. Using Itô s formula, f X sdbs F X t f X s + µ fx s l X s > ds X s fl tx f L t X. It turns out that for any bounded continuous functional Φ, we have 5.3 E Φ Y s, s t E ΦX s, s t exp H t X + K t, where H t is a functional defined by: 5.4 H t X def F Xt f Xs + f Xs + µ fxs ds, Xs with F x def x fy, and K t def F X t F X t + fl tx f L t X f X s + f X s + µ fx s l X s ds. X s Notice that F x F x sup y fy and by assumption, f f. It follows that e K is uniformly bounded and converges in probability to recall that E dsl X s, see Section 3. The functional H t is continuous by the hypothesis on f, the family {exp H t X, > } is uniformly integrable since for all p >, Ee ph tx E exp p pp t exp fx sdbs p sup f x <. x R f X sds Then we apply the convergence result for X to 5.3 and obtain that as, the LHS of 5.3 converges to recalling 5.4 and that R δ denotes a Bessel process of dimension δ + µ E ΦR δ s, s t exp H t R δ + E Φ R s, s t exp H t R δ def Qµ,f Φ + Q µ, f Φ. 7

18 The rest follows from Girsanov s transform in view of the fact that H t R δ F R t fr s dγ s f R s + f R s + µ fr s ds f R s ds, R s if we denote by γ the Brownian driver for the Bessel process R. We give an example: Fix λ > and consider hx λ cothλx. Let V λ be the diffusion taking values in R + and starting from, with infinitesimal generator d dx + λ cothλx d dx. In the terminology of Watanabe [8], V λ is a Bessel diffusion with drift, of dimension 3 and drift parameter λ,. We recall the absolute continuity relation between the law P 3,λ of V λ and P 3 see Pitman and Yor []: 5.6 dp 3,λ sinhλr t exp λ t Ft λr t dp3, t >. Ft In view of Theorems 5. and., we obtain Corollary 5.. Let λ and denote by Q λ the law of the solution of. associated to the function h λ cothλxl x. When +, the sequence Q λ converges in distribution to P3,λ + 3,λ 3,λ P, where P is the distribution of V λ t, t. References: [] Bertoin, J.: Applications de la théorie spectrale des cordes vibrantes aux fonctionnelles additives principales d un brownien réfléchi. Ann. Inst. H. Poincaré Probab. Statist [] Biane, Ph. and Yor, M. Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math [3] Durrett, R.T. and Rogers, L.C.G.: Asymptotic behavior of Brownian polymers. Prob. Th. Rel. Fields [4] Dym, H. and McKean, H.P.: Gaussian Processes, Function Theory and the Inverse Spectral Problem. Academic Press, New York, London [5] van der Hofstad, R., den Hollander, F. and König, W.: Central limit theorem for the Edwards model. Ann. Probab [6] Imhof, J.P.: Density factorizations for Brownian motion and the three dimensional Bessel processes and applications. J. Appl. Probab

19 [7] Karatzas, I. and Shreve, S.E.: Brownian Motion and Stochastic Calculus nd edition Springer, Berlin, 99. [8] Kasahara, Y.: Spectral theory of generalized second order differential operators and its applications to Markov processes. Japan J. Math [9] Kasahara, Y., Kotani, S. and Watanabe, H.: On the Green functions of - dimensional diffusion processes. Publ. Res. Inst. Math. Sci no., [] Kotani, S. and Watanabe, S.: Krein s spectral theory of strings and generalized diffusion processes. In: Functional Analysis in Markov Processes Ed. M. Fukushima Lect. Note Math pp Springer Berlin. [] Krein, M.G.: On a generalization of an investigation of Stieltjes. Dokl. Akad. Nauk SSSR [] Pitman, J. and Yor, M.: Bessel processes and infinitely divisible laws. In: Stochastic integrals Proc. Sympos., Univ. Durham, Durham, 98, pp , Lecture Notes in Math., 85, Springer, Berlin, 98. [3] Revuz, D. and Yor, M. Continuous Martingales and Brownian Motion nd edition, Springer, Berlin, 994. [4] Rogers, L.C.G. and Williams, D.: Diffusions, Markov Processes and Martingales. Vol. II: Itô Calculus. Wiley, Chichester, 987. [5] Ruiz de Chavez, J. Le théorème de Paul Lévy pour des mesures signées. Sém. de Probabilités XVIII. Lecture Notes in Math. 59, 45-55, Springer, Berlin, 984. [6] Stroock, D.W. and Varadhan, S.R.S: Multidimensional Diffusion Processes Springer, Berlin, 979. [7] Watanabe, S.: Generalized arc-sine laws for one-dimensional diffusion processes and random walks. Proceedings of Symposia in Pure Mathematics In: Stochastic Analysis eds: M.C. Cranston and M.A. Pinsky Providence, Rhode Island, 995. [8] Watanabe, S.: Bilateral Bessel diffusion processes with drift and time inversion. 999 Preprint [9] Yor, M.: Some Aspects of Brownian Motion. Part II: Some Recent Martingale Problems. Birkhäuser Verlag, Berlin, 997. [] Yor, M. editor: Exponential Functionals and Principal Values Related to Brownian Motion A collection of research papers. Biblioteca de la Revista Matemática Iberoamericana, Madrid 997. [] Zvonkin, A.K.: A transformation of the phase space of a process that removes the drift. Math. USSR Sbornik