Relativistic Diffusions and Schwarzschild Geometry

Transcription

1 Relativistic Diffusions and Schwarzschild Geometry JACQUES FRANCHI Université Louis Pasteur, I.R.M.A. AND YVES LE JAN Université Paris Sud Abstract The purpose of this article is to introduce and study a relativistic motion whose acceleration, in proper time, is given by a white noise. We deal with general relativity and consider more closely the problem of the asymptotic behavior of paths in the Schwarzschild geometry example. c 6 Wiley Periodicals, Inc. Contents 1. Introduction 1. A Relativistic Diffusion in Minkowski Space 4 3. Extension to General Lorentz Manifolds 6 4. The Restricted Schwarzschild Space S The Full Schwarzschild Space S A Relativistic Diffusion for All Positive Proper Times 7. Proofs 3 Appendix. Study of Timelike and Null Geodesics in S 59 Bibliography 63 Corpora cum deorsum rectum per inane feruntur ponderibus propriis, incerto tempore ferme incertisque locis spatio depellere paulum, tantum quod momen mutatum dicere possis. Quod nisi declinare solerent, omnia deorsum imbris uti guttae caderent per inane profundum nec foret offensus natus nec plaga creata principiis: ita nihil umquam natura creasset. Lucretius 1 Introduction The classical theory of Brownian motion is not compatible with relativity; this is clear from the fact that the heat flow propagates instantaneously to infinity. A Communications on Pure and Applied Mathematics, Vol. LIX, c 6 Wiley Periodicals, Inc.

2 J. FRANCHI AND Y. LE JAN Lorentz invariant generalized Laplacian was defined by Dudley cf. [6] on the tangent bundle of the Minkowski space, and it was shown that there is no other adequate definition than this one, 1 as long as Lorentz invariance is assumed. An intuitive description of the associated diffusion i.e., continuous Markov process is that boosts are continuously applied in random directions of space. We show that this process is induced by a left invariant Brownian motion on the Poincaré group. The asymptotic behavior of the paths of this process was studied cf. [8]; see also [7, 9]. Considering the importance of heat kernels in Riemannian geometry and the extensive use that is made of their probabilistic representation via sample paths, it is somewhat surprising that Dudley s first studies were not pursued and extended to the general context, namely, to Lorentz manifolds. It is indeed easy to check that the relativistic diffusion can be defined on any Lorentz manifold using a stochastic development similar to the one used to construct Brownian motion on Riemannian manifolds via the bundle of frames, as done below. The infinitesimal generator is the generator of the geodesic flow perturbed by the vertical Laplacian. But such an extension would have little appeal if some natural questions such as the asymptotic behavior and the nature of harmonic functions could not be solved in some examples of interest. Here we provide a rather complete study of this question in the case of Schwarzschild and Kruskal-Szekeres manifolds, which are used in physics to represent black holes. The specific interest of these manifolds comes from the vanishing of Ricci curvature, their symmetry, and the integrability of the geodesic flow. The picture that comes out in the Kruskal-Szekeres case appears quite remarkable, with paths confined in a neighborhood of the singularity, while their velocity increases. One difficulty of the study and it might explain why Dudley had few followers is that no explicit solution was found. The reason is that, even after reduction using the symmetries, the operator cannot involve fewer than three coordinates even in Minkowski space, instead of one for the Laplacian on Riemann spaces of constant curvature. Estimations and comparison techniques of stochastic analysis are the main tools we use to prove our results. They do not yet include a full determination of the Poisson boundary, but they suggest that for general Lorentz manifolds bounded harmonic functions can be characterized by classes of light rays, i.e., null geodesics. Let us now explain more precisely the content of this article. We consider diffusions, namely, continuous strong Markov processes. We start, in Section, with the flat case of Minkowski space R 1,d, and therefore with the Brownian motion of its unit pseudosphere, integrated then to yield the only true 1 Note, however, that some physical models of diffusion in a relativistic fluid are not Lorentz invariant since the frame of the fluid at rest plays a specific role; cf. [4] and its references. See also [17, ].

3 RELATIVISTIC DIFFUSIONS 3 relativistic diffusion, according to [6]. We get then its asymptotic behavior, somewhat simplifying the point of view of [8]. In Section 3 below, we present an extension of the preceding construction to the framework of general relativity, that is to say, of a generic Lorentz manifold. The process is first defined at the level of pseudo-orthonormal frames, with Brownian noise only in the vertical directions, and projects into a diffusion on the pseudo-unit tangent bundle. The infinitesimal generator we get in Theorem 3. decomposes into the sum of the vertical Laplacian and of the horizontal vector field generating the geodesic flow. In Sections 4 and 5, we deal in detail with the Schwarzschild space, which is the most classical example of a curved Lorentz manifold, used in physics to model the space outside a black hole or a spherical body. Using the symmetry and introducing the energy and the angular momentum, which are constants of the geodesic motion, we reduce the problem to the study of a degenerate three-dimensional diffusion. We then establish in Theorem 4.4 that almost surely the diffusion either hits the hole or wanders out to infinity, both events occurring with positive probability. We prove also in Theorem 4.4 that almost surely, conditionally on the nonhitting of the hole, the relativistic diffusion goes away to infinity in some random asymptotic direction, asymptotically with the velocity of light. Then we prove in Theorem 5.1 that almost surely, conditionally on the hitting of the hole, the relativistic diffusion reaches the essential singularity at the center of the hole within a finite proper time, and we describe the limit. It appears that the Kruskal-Szekeres space has two symmetrical singular boundaries, one being an exit boundary for the relativistic diffusion, and the other an entrance boundary. We show then in Section 6 that the story can be continued further: namely, the Schwarzschild relativistic diffusion, a priori defined until it hits the center of the hole the essential singularity of the so-called Kruskal-Szekeres space, which we also call full Schwarzschild space, can be extended to a diffusion that crosses this singularity. This crossing does not violate global causality if we consider an infinite covering of the full Schwarzschild space. Such hole crossing can then happen again and again, but without accumulation, according to Theorem 6.1 below, so that the extended Schwarzschild relativistic diffusion is well-defined for all positive proper times. We finally study the asymptotic behavior of this extended Schwarzschild relativistic diffusion and show, in Theorem 6.4 below, that there is a unique alternative possibility to the escape to infinity: there is indeed a positive probability that the relativistic diffusion becomes endlessly confined in a spherical neighborhood of the hole with an increasing velocity, and a trajectory becoming asymptotically planar with an asymptotic shape. This implies the existence on the Kruskal- Szekeres space of an infinity of SO 3 -invariant harmonic functions having symmetrical boundary values.

4 4 J. FRANCHI AND Y. LE JAN A Relativistic Diffusion in Minkowski Space Let us consider an integer d and the Minkowski space R 1,d := {ξ = ξ o, ξ R R d }, endowed with the Minkowski pseudometric ξ,ξ := ξ o ξ. Let G denote the connected component of the identity in O1, d, and denote by H d := {ξ R 1,d ξ o > and ξ,ξ = 1} the positive half of the unit pseudosphere. The opposite of the Minkowski pseudometric induces a Riemannian metric on H d, namely the hyperbolic one, so that H d is a model for the d-dimensional hyperbolic space. A convenient parametrization of H d is,θ R + S d 1, given by := argchξ o and θ := ξ/ ξ o 1. In these coordinates the hyperbolic metric is written as d + sh dθ, and the hyperbolic Laplacian is H := + d 1 coth + sh θ, where θ denotes the Laplacian of S d 1 and the associated volume measure is sh d 1 d dθ. Note that G acts isometrically on R 1,d and on H d, and that the Casimir operator on G induces on H d the hyperbolic Laplacian. Fix σ >, and denote by L σ the σ -relativistic Laplacian, defined on R 1,d H d by L σ f ξ, p := p o f ξ oξ, p + d j=1 p j f σ ξ, p + ξ j H p f ξ, p, that is to say, L σ f := p, grad ξ f + σ / H p f. This is a hypoelliptic operator. Given any ξ, p R 1,d H d, there exists a unique in law diffusion process ξ s, p s, s R +, solving the L σ -martingale problem, that is to say, such that for any compactly supported f C R 1,d H d, f ξ s, p s ξ, p L σ f ξ t, p t dt is a martingale. Note that p s is a hyperbolic Brownian motion, and that ξ s = ξ + p t dt. Remark.1. 1 The relativistic trajectories ξ s s R + we get in Minkowski space are fully causal: since their space-time velocities dξ s /ds = p s belong to H d, they are timelike, hence locally causal; moreover, they satisfy dξs o/ds = po s >, which ensures that ts = ξs o increases always strictly. Hence they are globally causal: in the terminology of [14], they satisfy the causality condition they cannot be closed.

5 RELATIVISTIC DIFFUSIONS 5 Note that ξ s is parametrized by its arc length. Mechanically, ξ s describes the trajectory of a relativistic particle of small mass indexed by its proper time submitted to a white noise acceleration in proper time. Its law is invariant under any Lorentz transformation. If we denote by e, e 1,.., e d the canonical base of R 1,d and by e j the dual base with respect to,, the matrices E j := e e j + e j e belong to the Lie algebra of G and generate the boost transformations. Given d independent real Wiener processes ws j, p s = ps o, p s can be defined by p s := s e, where the matrix s G is defined by the following stochastic differential equation: d s = + σ t E j dw j t. j=1 This means, in fact, that the relativistic diffusion process ξ s, p s is the projection of some diffusion process having independent increments, namely a Brownian motion with drift, living on the Poincaré group. This group is the analogue in the present Lorentz-Minkowski setup of the classical group of rigid motions and can be seen as the group of d +, d + real matrices having the form ξ, with 1 G, ξ R 1,d written as a column, and R 1+d written as a row. Its Lie algebra is the set of matrices β x, with β so1, d and x R1,d. The Brownian motion with drift we consider on the Poincaré group solves the stochastic Stratonovitch differential equation s ξ d s s ξ = s βs e d s, 1 1 where β s = σ d j=1 E jws j is a Brownian motion on so1, d. This equation is equivalent to d s = s dβ s and dξ s = s e ds, so that s is a Brownian motion on G. On functions of p = e, its infinitesimal generator d j=1 L E j coincides with a Casimir operator and induces the hyperbolic Laplacian so that p s = s e is a Brownian motion on H d, as required. Then it is well-known that θ s := p s / ps o 1 converges almost surely in S d 1 to some random limit θ and that ps o increases to infinity. See, for example, [1, theorem.3]. Set also s := argchps o. The Euclidian trajectory Zt is defined by ξ st, where st is determined by ξst o = t. Let us note that the Euclidian velocity d Zt/dt = θ st th st has norm < 1 1 here is the velocity of light. Moreover, we have the following: Remark.. The mean Euclidian velocity Zt/t converges almost surely to θ S d 1. PROOF: We have lim tր st = +, so that th st = 1 pst o

6 6 J. FRANCHI AND Y. LE JAN goes to 1. Thus we get almost surely lim t d Zt/dt = θ, and the result follows at once. Remark.3. The scattering amplitude, that is, the law of θ given p, is given by the hyperbolic harmonic measure in the unit ball of R d taken as a model for H d, which has density proportional to Pp, d 1 with respect to the uniform measure of S d 1, P denoting the classical Poisson kernel of the unit ball of R d. See, for example, the case δ = in [1]. 3 Extension to General Lorentz Manifolds Let us now see how the preceding construction can be naturally extended to the framework of manifolds. Let M be a d +1 dimensional manifold, equipped with a pseudo-riemannian metric of signature +,,...,, together with an orientation and a time direction and its Levi-Civita connection. For notational convenience, T 1 M will always denote the positively oriented half of the unit tangent bundle of M. As in the construction of Brownian motion on Riemannian manifolds, we have to use the frame bundle see [1, 15, 19]. So let GM be the bundle of direct pseudo-orthonormal frames, with first element in the positive half of the unit pseudosphere in the tangent space, which has its fibers modeled on the special Lorentz group G. Let V j be the canonical vertical vector field associated with the preceding matrix E j, and H be the first canonical horizontal vector field. Set L := H + σ Let π 1 denote the canonical projection from GM onto the tangent bundle T 1 M, which to each frame associates its first element. The canonical vertical vector fields V kl associated with the matrices E kl := e k el e l ek so1, d, for 1 k < l d, generate an action of SO d on GM, which leaves T 1 M invariant and then allows the identification T 1 M GM / SO d. The Casimir operator is d C = Vkl. j=1 V j d j=1 1 k<l d Note that the matrices {E j, E kl ; 1 j d, 1 k < l d} constitute a pseudoorthonormal base of so1, d endowed with its Killing form. LEMMA 3.1 The operators H, d j=1 V j, C, and L do act on C functions on the pseudo-unit tangent bundle T 1 M, inducing, respectively, the vector field L generating the geodesic flow on T 1 M, the so-called vertical Laplacian v, v again, and the generator G := L + σ / v. More precisely, for any test function F on T 1 M, we have on GM V j. L F π 1 = H F π 1, v F π 1 = CF π 1.

7 RELATIVISTIC DIFFUSIONS 7 Besides, in local coordinates x i, e k j, with e j = e k j / x k, j=1 V j = e k j e k + e k and with g kl denoting in these coordinates the inverse matrix of the pseudo- Riemannian metric of M: d e v F π 1 = Vj F π k 1 = el gkl e k + de k el F π 1. PROOF: Let us observe that for any u = x, e,...,e d GM, if u s denotes the horizontal curve such that u = u and π u = ẋ = e see, for example, [15, p. 38], then π 1 u s is the geodesic generated by π 1 u = x, e. Hence for any differentiable function F on T 1 M, we have e k j, e k H F π 1 u = d o ds F π 1u s = d o ds Fπ 1u s = L Fπ 1 u. Another way of expressing this is to recall that H commutes with the rotation vertical vectors V kl. It is also classical that the Casimir operator C commutes with all vertical vectors V kl, V j. Moreover, since the rotation vectors V kl act trivially on T 1 M, the operators C and d j=1 V j induce the same operator v on T 1 M. Then because e k [etv j x, e] = e k x, e ch t + ek j x, e sh t and we have indeed i.e., V j f x, e = d o dt e k j [etv j x, e] = e k j x, e ch t + ek x, e sh t, f [e tv j x, e] = e k j V j = e k j e k e k + e k f x, e + e k e k j. e k j f x, e, Using that e k el d j=1 ek j el j = g kl, we deduce immediately the expression for d j=1 V j : e k el gkl e k el + d j=1 e k j e k j + de k e k + d j=1 e k el j e l ek j + d j=1 e k el e l, j ek j

8 8 J. FRANCHI AND Y. LE JAN which reduces to the formula of the statement in the particular case of a function depending only on x, e. In accordance with the commutation relations arguments above, d j=1 V j F π 1 is a function depending only on x, e, that is, a function on T 1 M. Now, according to Section, the relativistic motion we will consider lives on T 1 M and admits as infinitesimal generator the operator G = L + σ / v of Lemma 3.1 above. If M is the Minkowski flat space of special relativity, it coincides with the diffusion defined in Section above. To construct this general relativistic diffusion, we use a kind of stochastic development to produce a stochastic flow on the bundle GM, as is classically done to construct the Brownian motion on a Riemannian manifold. But we have now to project on T 1 M and no longer on the base manifold M, and to put the white noises on the acceleration, i.e., on the vertical vectors, and no longer on the velocity, i.e., on the horizontal vectors. To proceed, let us simply fix GM and an R d -valued Brownian motion w = ws j, and let us consider the GM-valued diffusion = s GM solving the following Stratonovitch stochastic differential equation: * s = + H t dt + σ d j=1 V j t dw j t. By Lemma 3.1, the stochastic flow defined by commutes with the action of SO d on GM, and therefore the projection ξ s, ξ s := ξ s, e s = π 1 s defines a diffusion on T 1 M; this is the relativistic diffusion we intended to define and construct. The following theorem defines the relativistic diffusion ξ s, ξ s, possibly until some explosion time. The vector field L denotes the generator of the geodesic flow, which operates on the position ξ-component, and v denotes the vertical Laplacian restriction to T 1 M of the Casimir operator on GM, which operates on the velocity ξ-component. THEOREM 3. i The GM-valued Stratonovitch stochastic differential equation d * d s = H s ds + σ V j s dws j defines a diffusion ξ s, ξ s := π 1 s on T 1 M whose infinitesimal generator is L + σ / v. ii If ξ s : T ξs M T ξ M denotes the inverse parallel transport along the C 1 -curve ξ s s s, then ζ s := ξ s ξ s is a hyperbolic Brownian motion on T ξ M. Therefore the path ξ s is the development of a relativistic diffusion path in the Minkowski space T ξ M. j=1

9 RELATIVISTIC DIFFUSIONS 9 Remark 3.3. In local coordinates x i, e k j, with e j = e k j / x k, we have s = ξ s ; e s,...,e d s, and equation is written as dxs i = ei sds and de k j s = Ŵk il ξ se l j sdxi s + 1 { j }σ e k j s dws d + 1 { j=} σ ei k s dwi s i=1 with Ŵ k il denoting as usual the Christoffel coefficients, or equivalently in the Itô form: dx k s = ek sds, d de k s = Ŵk il ξ se l sdxi s + σ ei k sdwi s + dσ ek sds, i=1 de k j s = Ŵk il ξ se l j sdxi s + σ ek sdw j s + σ ek j sds for j 1, k d. Note that the martingales in the above equations for e s, that is to say, the differentials d Ms k := σ d j=1 ek j sdw s j, k d, have the following quadratic covariation matrix: K kl s := d Mk s, d Ml s ds d = σ e k j sel j s = σ e k sel s gkl ξ s, j=1 i.e., K s = σ e s T e s g 1 ξ s, in accordance with Lemma 3.1. Here T e denotes the transpose of the column vector e, and g 1 denotes the inverse matrix of the pseudometric. K s has rank d: since T e ge = 1, we have indeed K ge = e T e g 1 ge =. Note that K s does not depend on the other frame vectors e j j 1, proving again that the projection ξ s, ξ s is a diffusion on the tangent bundle, as Theorem 3.i asserts. Remark 3.4. The relativistic trajectories ξ s s R + we get on a generic Lorentz manifold M are locally causal, since on one hand they are timelike by construction, and on the other hand M itself is locally causal: about each point p M, there is some neighborhood V p that does not contain any nonspacelike closed C 1 -curve. See [14, prop ]. However, global causality i.e., nonexistence of closed, nonspacelike curves is not necessarily fulfilled. A compact Lorentz manifold cannot obey the global causality or chronology condition. Global causality also fails for the anti de Sitter space [14, sec. 5.], the Kerr solution [14, sec. 5.6], and the Gödel universe [14, sec. 5.7].

10 1 J. FRANCHI AND Y. LE JAN PROOF OF THEOREM 3.: Part i does not need any further proof. For part ii the process ζ s = ξ s ξ s s is continuous and lives on the fixed unit pseudosphere Tξ 1 M. Let ξ s : T ξ M T ξs M denote the parallel transport along the C 1 -curve ξ s s s, and recall that which implies indeed, whence d ξ s l ds j = ξ s k j Ŵl km ξ s ξ s m, d ξ s i ds l = ξ s i q Ŵq lm ξ s ξ s m ; d ξ s i ξ ds l s l j = ξ s i l ξ s k j Ŵkm l ξ s ξ s m = ξ s k j ξ s i q Ŵ q km ξ s ξ m s, d ξ s i ds l = ξ s j l ξ s k j ξ s i q Ŵ q km ξ s ξ s m = ξ s i q Ŵq lm ξ s ξ s m. Recall then from Remark 3.3 that for l d Therefore we get d ξ l s = σ d k=1 e l k s dwk s Ŵl jk ξ s ξ j s dζ i s = d ξ s i l ξ l s + ξ s i l d ξ l s = ξ s i q Ŵq lm ξ s ξ m s ξ l s ds + σ ξ s i l ξ s i l Ŵl jk ξ s ξ s j ξ s k ds d = σ ξ s i l ek l s dwk s = σ k=1 d ẽk i s dwk s, k=1 ξ k s ds. d ek l s dwk s where ẽ k s := ξ se k s for 1 k d and s. Similarly, we have dẽk ls = σ ξ s l ξ j j s dws k, i.e., dẽ ks = σζ s dws k. Observe that, for any s, ζ s, ẽ 1 s,...,ẽ d s constitutes a pseudo-orthonormal basis of the fixed tangent space T ξ M. Hence, as a result of Section and Remark 3.3, we find that the velocity process ζ defines a hyperbolic Brownian motion on the hyperbolic space Tξ 1 M isometric to H d. k=1

11 RELATIVISTIC DIFFUSIONS 11 In the reverse direction, we have of course ξ s = ξ sζ s, meaning indeed that we recover the C 1 -curve ξ as the deterministic development of the flat relativistic diffusion ζ s ds. Remark 3.5. Equation can be expressed intrinsically, in Stratonovitch or in Itô form, by using the covariant differential D, which is defined in local coordinates x i, e k j by De j k := de k j + Ŵli k el j dxi for j, k d. Equation is indeed equivalent to ξ s = e s, De s = σ d j=1 e j s dw j s, De j s = σ e s dw j s for 1 j d. See, for example, [, p. 3] or [11, p. 47]. 4 The Restricted Schwarzschild Space S This space is commonly used in physics to model the complement of a spherical body, star, or black hole; see, for example, [5, 13, 18, 1, ]. We take M = S := {ξ = t, r,θ R [R,+ [ S }, where R R + is a parameter of the central body, endowed with the radial pseudometric 1 R r dt 1 R r 1 dr r dθ. The coordinate t represents the absolute time, and r the distance from the origin. In spherical coordinates θ = ϕ,ψ [,π] R/πZ, dθ = dϕ + sin ϕ dψ. The geodesics are associated with the Lagrangian L ξ,ξ, where L ξ,ξ = 1 R ṫ 1 R 1 ṙ r ϕ r sin ϕ ψ r r and the nonvanishing Christoffel symbols are Ŵrt t = Ŵrr r = R rr R, Rr R Ŵr tt =, r 3 Ŵ r ϕϕ = R r, Ŵr ψψ = R r sin ϕ, Ŵ ϕ rϕ = Ŵψ rψ = r 1, Ŵ ϕ ψψ = sin ϕ cos ϕ, Ŵψ ϕψ = cotg ϕ. The Ricci tensor vanishes, the space S being empty. A theorem of Birkhoff see [1] asserts that there is no other radial pseudometric in S that satisfies this constraint. The limiting case R = is the flat case of special relativity, considered in Section.

12 1 J. FRANCHI AND Y. LE JAN 4.1 The Stochastic Differential System in Spherical Coordinates Let us take as local coordinates the global spherical coordinates ξ ξ,ξ 1,ξ,ξ 3 := t, r,ϕ,ψ. According to Remark 3.3, the system of Itô stochastic differential equations governing the relativistic diffusion ξ s, ξ s can be written here as follows: dt s = e sds, dr s = e 1 sds, dϕ s = e sds, dψ s = e 3 sds, de 3σ s = e sds R r s r s R e se1 sds + d M s, de 1 3σ s = e1 s ds + R r s r s R e1 s ds Rr s R e rs 3 s ds + r s Re s ds + r s Rsin ϕ s e 3 s ds + d Ms 1, de 3σ s = e sds e 1 r se sds + sin ϕ scos ϕ s e 3 s ds + d Ms, s de 3 3σ s = e3 sds e 1 r se3 sds cotg ϕ se se3 sds + d M3 s, s where the martingale M s := Ms, M1 s, M s, M3 s has the following rank 3 quadratic covariation matrix: K s = σ e s T e s g 1 ξ s. 4. Energy and Angular Momentum We shall liberally use the angular momentum b := r θ θ, the energy a := 1 R/rṫ = L/ ṫ, and the norm of b, b := b = r U, with U := θ. Set also T := ṙ, and accordingly T s := ṙ s = e 1 s, U s := θ s = D := min{s > r s = R}. Standard computations yield the following: PROPOSITION 4.1 e s + sin ϕ s e 3 s, i The unit pseudonorm relation which expresses that the parameter s is precisely the arc length, i.e., the so-called proper time is written Ts = as 1 R 1 + b s. r s rs

13 RELATIVISTIC DIFFUSIONS 13 ii The process r s, a s, b s, T s is a degenerate diffusion, with lifetime D, which solves the following system of stochastic differential equations: dr s = T s ds, dt s = d Ms T + 3σ T s ds + r s 3 b R s ds R ds, rs 4 rs da s = d M a s + 3σ a s ds, db s = d M b s + 3σ b s ds + σ r s b s with quadratic covariation matrix of the local martingale M a, M b, M T given by K s := σ a s 1 + R/r s a s b s a s T s a s b s bs + r s b s T s. a s T s b s T s Ts + 1 R/r s We get in particular the following statement, in which the dimension is reduced. COROLLARY 4. The process r s, b s, T s is a diffusion, with lifetime D and infinitesimal generator G := T r + σ b + r + σ T + 1 R r b + σ b 3b + r b + σ bt b T 3σ T + T + ds, r 3 R b r 4 R r We have the following result on the behavior of coordinate a s : T. LEMMA 4.3 There exist a standard real Brownian motion w s and a real process η s, almost surely converging in R as s ր D, such that a s = expσ s + σw s + η s for all s [, D[. In particular, a s almost surely cannot vanish, which means that time t s is always strictly increasing. PROOF: Proposition 4.1 above shows that as 1σ ds d Ms a a s σ ds for s < D so that we have almost surely as s, when D = log a s log a = 3σ s 1 a 1 t σ s + = σ s + o at d Mt a + d M a t at 1 d Mt a a t d M a t = σ s + os. Since 1 R/r s as, this implies D 1 R/r sas ds < almost surely. Consider then the standard real Brownian motion w defined by d Ms a = σ as 1 R dw s, r s

14 14 J. FRANCHI AND Y. LE JAN and the process η defined by the formula in the statement. We have dη s = dlog a s σ ds σ dw s = 1 1 Rrs as σ ds Rrs as 1 σ dw s, and then for any s < D, η s = η + σ 1 Rrt at dt σ 1 R/r t at dw t, R/r t at which converges almost surely to a finite limit as s ր D, since almost surely for all s ], D[, 1 + this last integral being finite. 1 R/r t at 1 1 R/r t at dw t < D 1 Rrt at dt 1 Rrt a t, 4.3 Asymptotic Behavior of the Relativistic Diffusion ξ s, ξ s We see in the appendix that in the geodesic case σ = five types of behavior can occur because of the trajectory of r s ; it can be running from R to + or in the opposite direction, running from R to R in finite proper time, running from + to +, running from R to some R 1 or from R 1 to +, or similarly in the opposite direction, running endlessly in a bounded region away from R. More detailed results can be found in [18] and especially in [3, 1]. A full treatment is given, for future reference, in the appendix below. The stochastic case σ can be seen as a perturbation of the geodesic case σ = ; however, the asymptotic behavior classification is quite different. THEOREM 4.4 i For any initial condition, the radial process r s almost surely reaches R within a finite proper time D or goes to + as s + equivalently, as ts + if a >, and as ts if a <. ii Both events in i occur with positive probability from any initial condition. iii Conditionally on the event {D = } of nonreaching the central body, the Schwarzschild relativistic diffusion ξ s, ξ s goes almost surely to infinity in some random asymptotic direction of R 3, asymptotically with the velocity of light.

15 RELATIVISTIC DIFFUSIONS 15 Note in particular that the relativistic diffusion almost surely cannot explode before the finite proper time D. The proof we give for this theorem is rather long. It is postponed until Section 7. 5 The Full Schwarzschild Space S 5.1 Kruskal-Szekeres and Eddington-Finkelstein Coordinates The full Schwarzschild space S, also known as the Kruskal-Szekeres space see [5, 13, ], and especially [1] and its historical account on page 8, can be defined by extending the previous restricted Schwarzschild space S as follows: On S, set u := r R 1 er/r ch t R and v := r t R 1 er/r sh. R Note that r/r 1 e r/r = u v and that the Schwarzschild pseudometric can be expressed in the Kruskal-Szekeres coordinates u,v,θ as ** 4R 3 e r/r dv du r dθ, r where r = ru v, r denoting the inverse function of [r r/r 1e r/r ] which is an increasing diffeomorphism from R + onto [ 1,+ [. In those Kruskal-Szekeres coordinates, we have S = {u,v,θ R S u > v }. The full Schwarzschild space S is now defined as S := {u,v,θ R S u v > 1} and is equipped with the pseudometric defined by above and by r = ru v. The space S contains S, S isometric to S, two isometric copies of the hole H := {u,v,θ S v > u } and H, and the boundary between ±S and ±H, which is {r = R} = {u = ±v}. It is a Lorentz manifold, to which our general construction of Section 3 applies. The energy and angular momentum are extended to T 1 S by setting a := R r e r/r u v v u and b := r θ θ. As before we set T := ṙ = R /re r/r u u v v and b := b = r U. Recall that a and b are constant along geodesics. The unit pseudonorm relation is written as before: R b a T + r 1 r + 1 =

16 16 J. FRANCHI AND Y. LE JAN or equivalently as u v + re r/r /4R 3 b /r + 1 =, which implies v > u, whence v > u along a timelike path. The following correspondences between a line element ξ, ξ T 1 S and its projection ξ S are easily deduced: ξ, ξ {r > R; a > } ξ S, ξ, ξ {r > R; a < } ξ S, ξ, ξ {r < R; T < } ξ H, ξ, ξ {r < R; T > } ξ H, ξ, ξ {r = R; a = T = } ξ {u = v = }. Now two other coordinates, namely the so-called inward and outward Eddington-Finkelstein coordinates u and u +, prove to be very convenient for performing calculations on S. They are defined, not on the whole S, but for u on S {u+v } and u + on S {u v } by u := R log u + v and u + := R log u v. In those Eddington-Finkelstein coordinates, the metric is expressed as 1 R du ± ± du ± dr r dθ, r and the energy is expressed as a = 1 R u ± ± ṙ = L r u ±. Let us consider the following boundary of the space S above the singularity {r = }: S := + S S, where + S := {u,v,θ R S u v = 1, v > }, S := {u,v,θ R S u v = 1, v < }. The antipodal map J: u,v,θ u,v,θ performs a diffeomorphism between the inward boundary + S and the outward boundary S. Note that r, u +, and u are naturally continued to S S on which it holds that r = v = u + 1 u + = u and that u + + r + R log r R 1 = u r R log r R 1 on { u v }, this quantity being equal to t on ±S. The R-valued absolute time t is continuously extended to S {u = v = } c by setting t = + on {u = v }, t = on {u = v }, t = R argthu/v on ±H, and t = R argthv/u on S as on S. Note that the region A := {u = v = } appears right away as exceptional, as the only part of S S where t cannot be continued, and the only part of S S where both u ± explode.

17 RELATIVISTIC DIFFUSIONS 17 In the Kruskal-Szekeres coordinates, a path u s,v s,θ s is timelike if and only if v s > u s. This implies that any timelike path started in H has to hit + S {r = }, and that any timelike path started in H has to hit {r = R}. In particular, a timelike path started in H and avoiding A has to enter the region {r > R}, either through {t = }, appearing as a particle born and then evolving in S forever, or entering then H through {t = + }, or through {t = + } and then evolving in S, which could be viewed as the case of an antiparticle through S. These dynamics of the full Schwarzschild space S show that the inward Eddington-Finkelstein coordinate u is appropriate to the study of timelike paths started in S H, until they hit the inward boundary + S and even to extend them further, see Section 6.1 below, and that the outward Eddington-Finkelstein coordinate u + is appropriate to the study of timelike paths started in H or even from the outward boundary S, see Section 5.3 below, and entering ±S, until they hit H. 5. Diffusion in the Full Space S: Hitting the Singularity We follow the same route as in the restricted Schwarzschild space to express the relativistic diffusion on T 1 S. Let us proceed, using the Eddington-Finkelstein coordinates u ±. The Lagrangian L ξ,ξ is written as L ξ,ξ = 1 R r u ± ± u ± ṙ r ϕ r sin ϕ ψ. We then apply Remark 3.3 to get the Itô stochastic differential equations of the relativistic diffusion in the full Schwarzschild space: d u ± s = σ d M ± s + 3σ dṙ s = σ d M r s + 3σ u± s ds ± R ṙsds + r s u ± s ds r s ϕ s + sin ϕ s ψ s ds, R R 1 r s + r s R ϕ s + sin ϕ s ψ s ds, r s u ± s ds R r s u ± s ṙs ds d ϕ s = σ d Ms ϕ + 3σ ϕ s ds ṙ s ϕ s ds + sinϕ s cosϕ s ψ s r ds, s d ψ s = σ d Ms ψ + 3σ ψ s ds ṙ s ψ s ds cotg ϕ s ϕ s ψ s ds, r s for some continuous local martingale M ±, M r, Mϕ, Mψ having a quadratic covariation matrix according to Lemma 3.1 and Section 4.1: u ± s u ± s ṙs 1 u ± s ϕ s u ± s ψ s u ± s ṙs 1 ṙs + 1 R r s ṙ s ϕ s ṙ s ψ s u ± s ϕ s ṙ s ϕ s ϕ s + r s ϕ s ψ s. ψ s ṙ s ψ s ϕ s ψ s ψ s + r s sinϕ s u ± s We again have a reduced diffusion r s, b s, T s with minimal dimension, solving the same system of Itô stochastic differential equations as before.

18 18 J. FRANCHI AND Y. LE JAN This system of stochastic differential equations has been derived using Eddington-Finkelstein coordinates, so that it is valid a priori outside {u = v = }. But the smooth functions r, a, b, T of the relativistic diffusion have an Itô decomposition with continuous coefficients so that the formulas involving them hold without restriction. From the pseudonorm relation, we see that T s cannot vanish in the region {r < R}. As r s enters this region necessarily almost surely with derivative T D < indeed T D = a D > by Lemma 4.3, r s is then necessarily strictly decreasing. Specifically, we have the following theorem, which establishes the existence of an exit law of the relativistic diffusion at the inward boundary + S. THEOREM 5.1 The relativistic diffusion in T 1 S either escapes to infinity or enters above H at time D and converges to the singularity within some finite proper time D. Moreover, in the second case we have almost surely the following: i For s D, r s decreases and hits at proper time D, with D < D D + π/r; moreover, lim sրd T s =. ii The variables θ s, u ± s, b s, and a s converge to finite limits as s ր D, and b D cannot vanish. iii As s ր D, we have the following equivalents: [ ] 5 r s b /5 [ ] 5 D R D s and T s b D R b 3/5 D R D s. Remark 5.. The equivalents in Theorem 5.1iii above can be specified further. Indeed, we have almost surely, as s ց, [ ] 5 r D s = b /5 /5 5bD D R s 1 + R s + O s log log s, [ ] 5 T D s = b D R b 3/5 /5 5bD D R s 1 R s s + O log log s. Indeed, using the stochastic differential equation of T s and the iterated logarithm law, together with the equivalents in Theorem 5.1iii, we easily deduce these more precise asymptotic expansions near D. Remark 5.3. We know from Theorem 3. that the relativistic diffusion can start from any initial condition in the full space T 1 S. When it starts above H, the pseudonorm relation forbids any vanishing of T s which then has to remain > until the level {r = R} is hit, which takes a proper time less than π R/ for the very same reason as in Theorem 5.1i. When the diffusion starts above {r = R}, it enters {r R} at once, as does any timelike path. Note that above A {u = v = } {r = R}, we necessarily have T = a =. Moreover, it can be proved that T 1 A is polar for the relativistic diffusion. So, when starting above H, namely in

19 RELATIVISTIC DIFFUSIONS 19 T 1 H, the relativistic diffusion then enters above ±S before possibly entering T 1 H later. We postpone the proof of Theorem 5.1 until Section 7 below. A part of this proof is based on the following proposition, which allows us to recover the whole relativistic diffusion from the reduced relativistic diffusion r s, b s, T s. PROPOSITION 5.4 The spherical coordinate θ s satisfies the following stochastic differential equation conditionally on the reduced relativistic diffusion r s, b s, T s : θ s rs d = σ dβ s θ s θ s bs σ r s θ s ds θ U s b s U s rs s ds bs U s for some standard real Brownian motion β that is independent of r, b, T. Moreover, θ s /U s converges in S as s ր D almost surely. We postpone the proof of this proposition until Section 7 below. COROLLARY 5.5 The curve in the full space S defined by the image of the trajectory {r s, u s,θ s s D } admits almost surely a semitangent at the center of the hole. PROOF: Using the strict monotonicity of r s near the singularity, we see that it is sufficient to verify the left-differentiability of the curve r s u s, r sθ s at s = D. Now using Theorem 5.1iii, as s ր D, on one hand we have u s r s and on the other hand since = u s T s = as T s + 1 r s θ s r s = θ s + rs r s R r s R, θ s r s θ D S, θ s r s = b [ ] s 5 R 1/ r s T s b 1/5 D R D s. T s T s 5.3 Entrance through the Singularity As the inward boundary + S appears as an exit boundary for the relativistic diffusion by Theorem 5.1 and Corollary 5.5, the outward boundary S appears as an entrance boundary, as shown by the following proposition, in which G denotes the generator of the relativistic diffusion acting on C T 1 S: PROPOSITION 5.6 The martingale problem associated with G has a unique continuous solution starting from any point of S. We postpone the proof of this proposition until Section 7.

20 J. FRANCHI AND Y. LE JAN 6 A Relativistic Diffusion for All Positive Proper Times From a probabilistic point of view, it is very natural, as suggested by Proposition 5.6, to extend the relativistic diffusion to all positive proper times, by concatenation of excursion paths, in such a way that the corresponding extended R 3 -valued curve rθ is differentiable at every point recall Corollary 5.5. We point out that such a continuation can be considered as unphysical see [1], end of section 31.6, but it is natural to consider it from the mathematical point of view. Now there are two possibilities for performing the extension through the singularities. The first possibility consists in identifying the inward boundary + S with the outward boundary S, by means of the antipodal map J defined in Section 5.1, thus considering the extended space S ˆ := S S / J. Note that this identification preserves differentiability at every point of the extended R 3 -valued curves rθ. This would not have been the case with the slightly different identification evoked in [1] at the end of section The second possibility is to introduce an infinite covering S of Sˆ indexed by integers: S := {S n S n n Z}/ J, where the S n are copies of S, and J acts as J between + S n and S n+1 for all n. Note that at each crossing of the singularity, a path arriving from the leaf S n continues in the leaf S n+1. The advantage of considering the covering S is to avoid violation of global causality. In fact, going from one construction to the other is completely straightforward, as S is a discrete covering of S. ˆ We actually use Sˆ in the following. 6.1 Extended Relativistic Diffusion As explained just above, let us introduce the following extension of the full Schwarzschild space S, S ˆ := S S / J, meaning that we add to S its boundary S = + S S, identifying pointwise the inward + S and outward S boundaries by means of the antipodal map J defined in Section 5.1. In this identification, a differentiable inward path ending at {r = } can be continued by a differentiable outward path, so that the R 3 -valued curve rθ is differentiable at any point. In particular, geodesics are thus well-defined for any proper time, and there are geodesics that cross endlessly the singularity {r = }, namely those that are described in Case 1. and are also met in cases..1,..,.5., and.6, completed by Remark A.1 of the appendix. For generic values of parameters a, b, k, such geodesics are dense in some disk of R 3 centered at. Theorem 5.1 allows us to extend the relativistic diffusion to a continuous strong Markov process on T 1 S. ˆ Indeed, Proposition 5.6 allows us to prove the first assertion of the following: THEOREM 6.1 There exists a unique, continuous, strong Markov process on T 1 ˆ S with positive lifetime inducing the relativistic diffusion on T 1 S. The lifetime of this extended relativistic diffusion is almost surely infinite.

21 RELATIVISTIC DIFFUSIONS 1 For such a process, we define an increasing sequence of hitting proper times D k as follows: Let D := D [, ] denote the hitting time of {r R} and D 1 := D the hitting time of {r = }, and set by induction, for any n N, D 3n := inf{s > D 3n 1 r s = R}, D 3n+1 := inf{s > D 3n r s = }, D 3n+ := inf{s > D 3n+1 r s = R}. Finally, consider D := sup n D n. This is obviously an increasing sequence of stopping times, strictly increasing as long as it is finite. Section 5.3 extends in a unique way the law of the relativistic diffusion to the proper time interval [, D [. It is clear that the process cannot be extended continuously beyond D. This proves the first assertion of Theorem 6.1, within the lifetime D. We postpone until Section 7 the proof of the second assertion of this theorem: D is almost surely infinite. Remark 6.. We saw in Theorem 5.1i that D 1 D + π/r. Now exactly the same reasoning shows that D 3n+1 D 3n and D 3n+ D 3n+1 are π R/ as long as these times are finite. The time intervals [D 3n 1, D 3n ] correspond to the excursions outside the hole, and the time intervals [D 3n, D 3n+ ] correspond to the excursions inside the hole. Moreover, we see that D k becomes infinite if and only if k = 3n and the process escapes to infinity during its n th excursion outside the hole. Recall that according to Lemma 4.3 during every excursion outside the hole {r R}, the diffusion can have its absolute time coordinate t = t s strictly increasing from to + or strictly decreasing from + to this case can be seen as the antiparticle case, depending on the sign of a at the exit of the hole. Moreover, the R 3 -valued curve s r s θ s is differentiable at any proper time s, whereas the R-valued curves s u ± s present a cusp with half a tangent; this appears in the proof of Theorem 5.1, where we saw that u rt/r ± near D at proper times D 3n+1 and are differentiable at any time s D 3n+1. Remark 6.3. The Liouville measure is invariant for the extended relativistic diffusion on T 1 S. ˆ It induces the invariant measure dr da dt for the subdiffusion r s, a s, T s. 6. Capture of Diffusion by a Neighborhood of the Hole The preceding section naturally leads to the following question: can the extended relativistic diffusion cross the hole infinitely many times, as some geodesics do? Note first that it was clear from Sections 4.3, 5.3, and 6.1 that there is, for any n N and any initial condition, a positive probability that the extended relativistic diffusion crosses the hole exactly n times and thereafter goes away to infinity. So the following theorem essentially asserts that the limiting case of n = crossings of the hole can also happen, thereby completing the picture of all possible asymptotic behaviors of the extended relativistic diffusion. Moreover, it says that this last

22 J. FRANCHI AND Y. LE JAN case corresponds to an asymptotic confinement of the relativistic diffusion in the vicinity of the hole. THEOREM 6.4 Almost surely, from any initial condition, the extended relativistic diffusion can have only two types of asymptotic behavior, each occurring with positive probability: either i r and a go away to infinity and θ converges, or ii := lim sup s r s [R, 3R/], lim inf s r s =, b goes to infinity, b /b converges, and a /b converges to l = ±1/ 1 R/. Moreover, iii For any ε >, if r > 3R/ and if T is large enough, then the probability that the relativistic diffusion goes away to infinity is at least 1 ε. iv For any ε >, if r ]R, 3R/[, T =, and b is large enough, then P r < ε > 1 ε. The proof of this theorem is rather delicate. It will be presented in Section 7 below. The following remark underlines an interesting consequence of the extension of the relativistic diffusion through the singularity: Theorem 6.4 reveals in particular a continuum of harmonic functions of the Schwarzschild space, which would have remained hidden without the extension of Section 6.1. Remark 6.5. For any open interval ]x, y[ ]R, 3R/[, P x < < y is a nontrivial SO 3 -invariant L + σ / v -harmonic function on T 1 S, with J- symmetrical boundary values. The following result describes more precisely what happens when the diffusion is captured by a neighborhood of the hole: while according to Theorem 6.4 they are asymptotically planar, they progressively exhibit another type of regularity. COROLLARY 6.6 In the second case of Theorem 6.4, and more precisely conditionally on the event < 3R/, the times D n of the first maxima of the radius r at each excursion out of the hole are such that lim n r D n = and that D n D 3n+1 dθ s converges as n toward ± dr/ [R r + l r 3 ] r which is well-defined if and only if < 3R/. The proof of this corollary will close Section 7 below. Remark 6.7. The result of Corollary 6.6 concerns the time intervals [D 3n+1, D n ], that is to say, the upcrossings from the singularity to the successive tops of the limiting trajectories. It is very likely that the same result is valid as well for the downcrossings, that is to say, the time intervals [D n 1, D 3n+1] yielding the same angular random limit the sign of T s compensating for the interchange of the bounds D n 1 and D 3n+1 in the integral. So another statement in the spirit

23 RELATIVISTIC DIFFUSIONS 3 of Corollary 6.6, but which demands some more work, should be: almost surely D lim n n dθ D n 1 s = ±, where := dr 1 r[r r + l r 3 ] = dr r[1 r 3 R/ 1 r r] is a strictly increasing continuous function of /R, from [1, 3/[ onto [π/, [. It is thus likely that the shape of the excursions should approach more and more the null geodesics, i.e., the light rays; see the appendix. 7.1 Proof of Theorem Proofs In the proof of this theorem, we shall use the following very simple lemma. LEMMA 7.1 Let M be a continuous local martingale, and A a process such that lim inf s A s / M s > almost surely on { M = }. Then lim s M s + A s = + almost surely on { M = }. PROOF: Writing M s = W M s, for some real Brownian motion W, we find almost surely some ε > and some s such that A s ε M s and M s ε M s for s s, whence M s + A s ε M s for s s. We now successively prove the three assertions of Theorem 4.4. i Almost sure convergence on {D = } of r s to. This proof will be split into six parts. Let us denote by A the set of paths with infinite lifetime D such that the radius r s does not go to infinity. We have to show that it is negligible for any initial condition x = r, b, T = r, b, T to belong to the state space [R, [ R + R. The cylinder {r = 3R/} plays a remarkable role in Schwarzschild geometry. In particular, it contains light lines. We see in the first part of the proof that we have to deal with this cylinder. 1 r s must converge to 3R/, almost surely on A. Observe from the unit pseudonorm relation Proposition 4.1i that T s /a s is bounded by 1. Let us apply Itô s formula to Y s := 1 3R/r s T s /a s : Y s = M s + 3R Tt dt 1 3R/r t bt a t rt + a t rt 3 dt σ 1 R/r t Y s t 1 3R/r t R at dt a t rt dt, with some local martingale M having quadratic variation: d M s = 1 3R 1 1 Rrs T s σ ds σ a r s as as s ds.

24 4 J. FRANCHI AND Y. LE JAN Now Y s is also bounded by 1. Hence Lemma 4.3 implies that the last two terms in the expression of Y s above have almost surely finite limits as s, and similarly for M s, and then for M s. Moreover, the two remaining bounded variation terms in the expression of Y s above increase. As a consequence, we s get that Y s, T t /a t rt dt, and 1 3R/r t bt /a trt 3 dt converge almost surely in R as s. So does dt/a trt. Now using that a r a + R b r ar + 1 r = T a ar + b ar ar by the unit pseudonorm relation, we deduce that almost surely 1 3R d 1 dt = 1 3R T t r t dt r t r t rt dt 1 3R a t dt <. r t This implies the almost sure convergence of 1 3R/r s + 3R /4rs /r s, and thus of 1/r s. Since lim s 1/r s cannot be on A, we have necessarily lim s r s = 3R/ almost surely on A, from the convergence of 1 3R/r t a t /rt dt. b s /a s converges to 3R 3/, and T s /b s goes to, almost surely on A. Indeed, Itô s formula gives for some real Brownian motion w bs as = b a + σ + σ bs as rs as r s b s ds 3σ 1 R/r s dw s a s r t 1 R r s b s a 4 s Since by the unit pseudonorm relation we have bs /a s < rs /1 R/r s, whence bs /a s is bounded on A, the above formula and Lemma 4.3 imply the almost sure convergence of bs /a s on A. Indeed, the bounded variation terms converge, and as bs /a s is positive, the martingale part has to converge also. Using the unit pseudonorm relation again, we deduce that Ts = 1 1 R b s as r s as r + 1 s as has also to converge, necessarily to, since otherwise we would have an infinite limit for T s, which is clearly impossible on A. The value of the limit of b s /a s now follows directly from this and from 1. 3 We have almost surely on A r t 3R bt dt < and ds. T t dt <.

25 RELATIVISTIC DIFFUSIONS 5 Let us write Itô s formula for Z s := r s 3R/T s = 1 d/d sr s 3R/ : Z s Z M s 3σ = 4 r s r r s + r 3R Tt dt + r t 3R bt dt r 4 t R where M is a local martingale having quadratic variation given by M s = σ r t 3R 1 R + Tt dt. r t Note that if M =, then by above lim s Note, moreover, that in this case r t 3R dt s r t 3R/ b t dt/r 4 t M s =. r t r t 3R b t dt r 4 t r t 3R dt b t dt, is also negligible with respect to r t 3R/ bt dt/r t 4. On the other hand, we must have lim inf s Z s = on A. Therefore we deduce from Lemma 7.1 that necessarily M <, and then that M s has to converge, almost surely on A. Using again that lim inf s Z s =, we deduce the almost sure boundedness and convergence on A of Tt dt and of r t 3R/ bt dt/r t 4. 4 r s 3R/ b s and Ts /b s go to as s, almost surely on A. Indeed, on one hand we deduce from 3 that for some real Brownian motion W r s 3R s b s = σ W[ r t 3R 4 ] bt + rt dt + r t 3R T t b t dt + σ r t 3R 3b t + r t dt b t has to converge almost surely on A as s, necessarily to since it is integrable with respect to s. On the other hand, we have for some real Brownian motion W, by Itô s formula, Ts = T [ + σ W b s b + σ + σ T t T 4 t + r t bt T 4 t b 4 t dt + b t 1 R dt + σ r t b t r t 3R Rrt T t T t b t dt r 4 t r t T t b 3 t dt. bt ] dt RT t r t b t dt r t

26 6 J. FRANCHI AND Y. LE JAN Recall from 1 that T t /b t and that b t 3R 3/a t. Thus, using 3 we easily see that all integrals in the above formula converge. Hence we deduce the almost sure convergence of s Ts /b s on A, necessarily to, since it is integrable. 5 It is sufficient to show that r t 3R T t bt dt < and Tt 4 dt <, almost surely on A. Indeed, assuming that these two integrals are finite, Itô s formula shows that we have for some real Brownian motion W T s = T + σ W [ = γ s + ] dt + 4σ Tt + 1 R Tt r t + r t 3R T t b dt s t rt 4 + σ = γ s + σ 1 Rrt T t dt R rt dt [ r t 3R ] dt + R R 3r t r s r = γ s + s 3, 1 Rrt dt R Tt dt T t rt where γ and γ since r t 3R/ = ob 1/ t = oa 1/ t by 4 and are bounded, converging processes on A. Consequently, lim s Ts = almost surely on A, which, with 3 above, implies that A must be negligible. 6 End of the proof of the convergence of r s to on {D = }. By the Schwarz inequality, the first bound in 5 above will follow from T t b t dt < and r t 3R b 3 t dt <. Now these two terms appear in the Itô expression for Zs 1 := r s 3R/T s b s, Zs 1 Z 1 M1 s = σ [ 8 + r t ]Z 1t bt dt + Tt b t dt + r t 3R b 3 dt t rt 4 R r t 3R dt b t rt, with a local martingale M 1 having quadratic variation M 1 s = σ r t 3R bt 1 R + [4 + rt r b t ]Tt dt. t dt