Stochastic processes and Differential geometry
RANDOM WALKS AND HARMONIC FUNCTIONS ON LIE GROUPS
ERGODIC THEORY : ANOSOV DIFFEOMORPHISMS AND HOROCYCLE FLOW
HOMOGENEOUS MANIFOLDS WITH NEGATIVE CURVATURE
DIFFUSION PROCESSES ON DIFFERENTIABLE MANIFOLDS
RANDOM WALKS AND HARMONIC FUNCTIONS ON LIE GROUPS
1966-1977
In 1966, while pursuing my travels into algebraic geometry inexhaustible litterature at Harvard graduate school, I also began exploring applied mathematics, through graduate courses in mathematical economy, econometrics, and probability theory. Besides attending amazing lectures by Nobel prize LEONTIEFF on his celebrated matrix representations in macroeconomics, I discovered William FELLER’s books on random walks and probability, and started learning probability theory on my own, by reading Michel LOEVE, J. DOOB, Paul LEVY, Jacques NEVEU, and the amazing E. B. DYNKINS.
Back in Paris (1968), my enthusiasm for the formalizations of randomness led me to begin a Doctorat d’Etat with Jacques NEVEU at Paris University, supported by a research fellowship at CNRS.
The french probabilistic school in the late sixties, dominated by the towering figures of J.NEVEU and P.A. MEYER, centered on trajectory analysis of continuous time submartingales, formalizations of their stochastic integrals, general Markov processes and their associated formal potential theories, from subharmonic functions to Martin boundaries. I had become quite conversant with these highly technical topics, when J. NEVEU asked me to analyze, for his seminar, a remarkable series of papers by Harry FURSTENBERG, who used powerful probabilistic techniques to study Convolution Equations f * µ = f on semi-simple Lie groups, by analysis of harmonic functions f for the µ-random walk associated to probability µ .
H. FURSTENBERG had shown that for any semi-simple Lie group G, and any absolutely continuous µ-random walk on G, the associated bounded harmonic functions f had a natural representation as integrals of a “Poisson” kernel living on a compact “Poisson” boundary, and that such Poisson boundaries were compact homogenous spaces of G. He also showed that G had only a finite family of Poisson boundaries, all of the form K/H where K is the maximal compact subgroup of G and H a finite subgroup of K/Ko , Ko being the main connected subgroup of K .
Understanding FURSTENBERG’s work was a tough nut to crack, given the width of interacting mathematical domains involved, but an exciting mathematical experience, and I immediately undertook the extension of his results to convolution equations on arbitrary Lie groups.
This became a fascinating three years journey. I first showed that for semi-simple Lie groups, FURSTENBERG’s results still held for all probabilities µ having nonzero absolutely continuous part. I then discovered that the Poisson boundary of the µ-random walk on G only depended on the semi-group Tµ generated by the support of µ , and I clarified the precise link between Tµ and the Weyl chambers of G which enabled the explicit computation of the Poisson boundary.
I then attacked the extension to general Lie groups, and a key point was my intensive use of a crucial first result : large products of identically distributed independent random group elements act on the Poisson boundary by asymptotically shrinking it towards a single point.
This led me to the complete identification of all Lie groups G for which Poisson boundaries [“Poisson spaces”] associated to absolutely continuous random walks are compact homogeneous space of G, and to describe explicitly these homogeneous boundaries.
I was then able to extend these theorems to general locally compact groups, a key step being the exciting discovery that sufficiently “small” compact distinguished subgroups can actually be factored out in this context.
These results constituted my Doctorat d’Etat [director Jacques NEVEU] at University of Paris. Pierre CARTIER was an enthusiastic early reader of my work, and decided to present my main results on this topic at Paris BOURBAKI seminar in 1970.
In collaboration with Pierre CARTIER, we then succeeded in determining the natural isomorphism between the compact Poisson boundary of a random walk on a locally compact group G, and one precise G-orbit within the classical abstract Martin boundary associated to integral representation of unbounded positive harmonic functions.
For me, this raised the difficult question of identifying this intriguing single group orbit within the abstract Martin boundary of random walks, for Lie groups which had no compact Poisson boundary, and in 1973, I began directing Albert RAUGI’s PhD thesis at Paris university, to extend my own work in a quest for non compact Poisson boundaries.
The goal was, for all absolutely continuous random walks on connected Lie groups, to identify a not necessarily compact homogeneous space providing a weaker but still complete integral representation for bounded harmonic functions, and RAUGI’s PhD neatly solved the question.
I was also aware that FURSTENBERG’s further papers on laws of large numbers for cocycle functions associated to random walks on homogeneous spaces of Lie groups could be extended to formulate “central limit laws” for non commutative random walks. This looked like a key step to reach actual “central limit theorems” for large products of independent identically distributed random elements in arbitrary Lie groups. This was an ambitious program, elegantly completed in A. RAUGI’s PhD (1977).
References :
Poisson spaces of locally compact groups
R.Azencott , Lecture Notes Math. vol 148 141 pages Springer-Verlag 1970
Martin boundaries of random walks on locally compact groups
R. Azencott, P.Cartier Proc. 6th Berkeley Symp. Prob.Stat. vol 3 pp 87-129 1970
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ERGODIC THEORY : ANOSOV DIFFEOMORPHISMS AND HOROCYCLE FLOW
1969-1972
Since the early magistral work of Elie CARTAN, non compact Riemannian symmetric spaces are well known to carry natural invariant metrics with non positive curvature.
I had noticed in 1969-70 that Poisson spaces of semi-simple Lie groups all admit as finite coverings a non-compact Riemannian symmetric space, and that there were many analogies between the horocycle flow on such symmetric spaces and the asymptotic shrinking action of “random products” of isometries on such spaces.
This led me to study the celebrated geodesic flow on manifolds with negative curvatures, and to read the astonishing work of I. SINAI on the associated ergodic theory of such flows.
SINAI and altri had shown that the celebrated geodesic flow on Riemannian manifolds of negative curvature is actually an ANOSOV diffeomorphism, which means that the differential of the diffeomorphism splits the tangent space into two complementary subspaces, one subspace being dilated by the differential, and the other contracted by the differential.
SINAI had then shown the existence of two associated smooth transverse “feuilletages” of the underlying manifold for any ANOSOV diffeomorphism, and used them to generate smooth finite partitions of the manifold well adapted to the dynamics.
Simultaneously ORNSTEIN had just shown that entropy was a complete classifier of ergodic type among standard BERNOUILLI schemes, a deep result essentially proved by clever coding schemes. In his Berkeley lectures on this remarkable fact, he recalled a difficult and still open question asked by SINAI : “ was entropy a complete classifer of ergodic type among ANOSOV diffeomorphisms ? ”
I read the proof of another deep ORNSTEIN theorem, showing that another large class, of abstract ergodic systems with vanishing infinitely remote past, namely the so called K-systems, were isomorphic to standard BERNOUILLI schemes, so that entropy was a complete classifier of ergodic type among K-systems.
I then began a meticulous study of SINAI’s foliations in relation to the associated dynamics, which enabled me to analyze the infinitely remote past of these diffeomorphisms, and to deduce, that in fact all ANOSOV diffeomorphisms were isomorphic to K-Systems. Combined with ORNSTEIN theorems, my result showed that all ANOSOV diffeomorphisms were isomorphic to standard BERNOUILLI schemes, and thus solved positively the open SINAI-ORNSTEIN question : entropy was indeed a complete classifier among ANOSOV diffeomorphisms.
This highly visible result for ergodic theory specialists was published in summary form in [1], and opened for me an opportunity for very interesting mathematical discussions on coding schemes with ORNSTEIN and his Stanford University group.
On the strength of this result, I also met William PARRY, a clever ergodic theoretician from Warwick University, and we collaborated on a spectral type analysis of the of the horocycle flow on Riemannian symmetric spaces, using WEYL commutation relations (see [2]).
References :
[1] Anosov Diffeomorphisms and Bernouilli schemes
Comptes Rendus Acad. Sciences, Paris, France, vol 270 ser A, pp 1105-1107 1970
[2] Stability of group representations and Haar spectrum
R. Azencott, W.Parry Trans.Am.Math.Soc. vol 172 pp 317-327 1972
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HOMOGENEOUS MANIFOLDS WITH NEGATIVE CURVATURE
Non compact Riemannian symmetric spaces have non positive curvature, and Elie CARTAN had remarkably used that geometric fact to show that their group of isometries is actually transitive on the symmetric space, a key step to reach his beautiful classification of symmetric spaces. Having noticed that Poisson boundaries of Lie groups are, as homogeneous spaces, generalizations of symmetric spaces, I quite naturally undertook the generalization of CARTAN’s results to a wider class of Riemannian manifolds.
In collaboration with Edouard WILSON at Brandeis University, we achieved a very satisfying and deep result : the complete classification of Riemannian manifolds having non positive curvature, and admitting a transitive group of isometries. This class of Riemannian manifolds, of course includes non compact Riemannian symmetric spaces, but turned out to be a much wider explicit family of homogeneous spaces of Solvable Lie goups.
References :
Homogeneous manifolds with negative curvature
R. Azencott, E.Wilson, Comptes-Rendu sAc.Sci. Paris, vol278 serA, pp 561-562, 1974
Homogeneous manifolds with negative curvature (part 1)
R. Azencott, E.Wilson, Memoirs Am. Math. Soc. vol 8, # 178, pp1-101, 1976
Homogeneous manifolds with negative curvature (part 2 )
R. Azencott, E.Wilson, Trans. Am. Math. Soc. vol 215, pp 323-362 , 1976
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DIFFUSION PROCESSES ON DIFFERENTIABLE MANIFOLDS
1970-1982
Natural continuous time limits of random walks on Lie groups are left invariant diffusion processes on the underlying manifolds. These continuous Markov processes are characterized by the fact that their infinitesimal generator is a left invariant hypo-elliptic 2nd order differential operator, and such invariant operators are obviously determined by a non negative quadratic form on the Lie algebra of the group, plus an eventual arbitrary drift vector. In 1970, the existence of such diffusion processes on differentiable manifolds was known, and implicitly derived from global constructions in DYNKIN’s style, linked to properties of the domain and range of their infinitesimal generator. However the possibility of constructing such diffusion processes by “patching together” local brownian motions living on the tangent spaces of the manifold was not yet clearly and formally understood. I studied this question for generic manifolds, and showed that this probabilistic patching of local images of Brownian motions, stopped at adequate Markov stopping times, was possible and consistent, and proved the existence and uniqueness of the global diffusion associated to any smooth elliptic 2nd order differential operator on arbitrary differentiable manifolds.
I communicated my results in extensive manuscript form to Laurent SCHWARTZ and Daniel STROOCK. They told me that they knew this approach would work, but both of them particularly appreciated my localization formulas for diffusion densities. I had explicited new localization formulas enabling the closed form probabilistic expression of the global density function p(t,x,y) of a diffusion process based only on the local density functions qV(t,x,y) of the local diffusions killed at the exit times from small open sets V. This is equivalent to expressing the global fundamental solution of a 2nd order parabolic differential equation in terms of the local fundamental solutions of the same parabolic differential equation.
This efficient formula, as well as some of the technical patching tricks I had thus introduced, became actually a useful tool for D. STROOCK and S.R.S. VARADHAN, in their massive later joint work on diffusion processes viewed as solutions of “martingale constraints”, allowing them to quickly reduce smoothness results for global diffusion densities to local smoothness estimates. I used again this formula in my later work on MOLCHANOV geometric estimates for diffusion densities on Riemannian manifolds.
My work on horocycle flows and left invariant random walks on symmetric spaces gave me very clear clues to the fact that natural left invariant diffusion processes on a non-compact Riemannian symmetric space S must almost surely have their trajectories converge to a (random) limit point of the maximal Poisson boundary B of the semi-simple Lie group G of isometries of S. This diffusion paths behaviour on symmetric spaces became indeed a direct consequence of explicit computations published by Paul and Marie-Paule MALLIAVIN a decade later, but in 1977, I had a strong feeling that the deep geometric explanation of such paths behaviour must be linked to qualitative effects of the metric curvature tensor on the local “speed” of diffusion paths. This led me to study qualitative behaviour at infinity for diffusion paths on generic Riemannian manifolds, and their links with negativity of the curvature. The reading of beautiful “potential theory” papers by Y. BONY, H. MOKOBODZKY, G. CHOQUET and altri, had initiated me to the strong consequences of the maximum principle for 2nd order elliptic and parabolic differential operators.
I thus realized that it was possible to associate to most generic diffusion processes (Xt) on Riemannian manifolds, a whole family of much “simpler” diffusion processes (Yt) and (Zt), whose Green kernels provided asymptotic lower and upper bounds for the Green kernel of (Xt). To get a tight geometric link with the metric curvature, I deliberately restricted the companion “benchmarking” diffusions (Yt) and (Zt) to have infinitesimal generators depending only on the Riemannian distance.
This apparatus turned out to be an efficient mathematical tool to study the behaviour at infinity of probability densities and Green kernels of generic Riemannian difusions, and to obtain elegant intuitive asymptotic behaviour results for Brownian motions associated to Laplace-Beltrami operators on generic Riemannian manifolds with negative curvature. For instance, a nice qualitative feature is that negative curvature tensors do accelerate local Brownian paths, and all the more so when the spectrum of the negative curvature tensor moves further away from zero.
After reading H. MAC-KEAN original book on multiplicative stochastic integrals and Brownian motion on matrix groups, and playing around with Brownian motion on nilpotent Lie groups, I conjectured that on generic Lie groups G, the local logarithmic map from G onto its Lie algebra LG mapped any left invariant diffusion process (Xt) living on G onto a diffusion process (logXt) living on LG , which had to have an interesting formal stochastic expansion.
I thus asked Gerard BENAROUS, an outstanding young mathematician at ENS Rue d’Ulm, to work on this question in his PhD thesis. He solved the problem quite elegantly and thoroughly, building for (logXt) a universal explicit formal expansions as series of multiple stochastic integrals of euclidean Brownian motion. From then on, G. BENAROUS completed his PhD on his own impetus, at the Courant Institute, by a deep foray into MALLIAVIN calculus, and took off on a remarkable mathematical carreer.
With a team of young PhDs [Paolo BALDI, Andre and Catherine BELLAICHE, Philippe BOUGEROL, Mireille CHALEYAT-MAUREL, Laure ELIE], I then launched a one year intensive seminar at University Paris 7 to study in depth the stunning probabilist work of MOLCHANOV on explicit links between the small time behaviour of fundamental solutions pt(x,y) for 2nd order elliptic operators on Riemannian manifolds and the geodesic structure of the manifold. MOLCHANOV had proved an elegant geometric formula for the 1st term in the small time asymptotics of the diffusion density p(t,x,y), as c(x,y) t-k/2 exp[-d2(x,y)/2t] , where d(x,y) is the geodesic distance between points x and y, and k is the dimension of the manifold. We were able to broaden and clarify MOLCHANOV’s approach, and to extend it neatly to hypo-elliptic differential operators. I clarified the links of these results with the “large deviations” asymptotics of small random perturbations of the deterministic geodesic flow, in the spirit of my simultaneous ongoing work on small random perturbations of dynamic systems ; finally, I edited and coordinated our joint book on the topic.
References:
Methods of localization and diffusions on manifolds
R. Azencott, Pub. Istituto Matematico "Ulisse Dini", Firenze, Italy 1971
Diffusion processes on differentiable manifold
R. Azencott, Comptes Rendus Ac.Sci., Paris, vol 274 ser A, pp 651-654 1972
Diffusion processes on differentiable manifold
R. Azencott, Comptes Rendus Ac.Sci., Paris, vol 276 ser A, pp 363-365 1973
Behaviour of diffusion semi-groups at infinity
R. Azencott, Bull. Soc. Math. France, vol 102 pp 193-240 1974
(Book) Geodesics and small time behaviour of diffusions on manifolds
R.Azencott (editor), P. Baldi, A. Bellaiche, L. Elie, M. Chaleyat-Maurel,
Asterisque, vol 84-85, 250 pages , Paris 1982
Chapters by R. Azencott :
Diffusions processes on differentiable manifolds pp 17-32
Local and Global Estimates for diffusions densities pp 131-150
Cramer Transforms and small time diffusions pp 215-226
Diffusions on the Heisenberg group pp 227-236
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