THÃSE KolÃ©hÃ¨ Abdoulaye COULIBALY-PASQUIER

More documents

Recommendations

Info

Brownian motion with respect to time-changingRiemannian metrics, applications to Ricci flowK.A. CoulibalyAbstractWe generalize Brownian motion on a Riemannian manifold to the case ofa family of metrics which depends on time. Such questions are natural forequations like the heat equation with respect to time dependent Laplacians(inhomogeneous diffusions). In this paper we are in particular interested inthe Ricci flow which provides an intrinsic family of time dependent metrics.We give a notion of parallel transport along this Brownian motion, and establisha generalization of the Dohrn-Guerra or damped parallel transport,Bismut integration by part formulas, and gradient estimate formulas. One ofour main results is a characterization of the Ricci flow in terms of the dampedparallel transport. At the end of the paper we give a canonical definition ofthe damped parallel transport in terms of stochastic flows, and derive an intrinsicmartingale which may provide information about singularities of theflow.1 g(t)-Brownian motionLet M be a compact connected n-dimensional manifold which carries a family oftime-dependent Riemannian metrics g(t). In this section we will give a generalizationof the well known Brownian motion on M which will depend on the family ofmetrics. In other words, it will depend on the deformation of the manifold. Suchfamily of metrics will naturally come from geometric flows like mean curvature flowor Ricci flow. The compactness assumption for the manifold is not essential. Let∇ t be the Levi-Civita connection associated to the metric g(t), ∆ t the associatedLaplace-Beltrami operator. Let also (Ω, (F t ) t≥0 , F, P) be a complete probabilityspace endowed with a filtration (F t ) t≥0 satisfying ordinary assumptions like rightcontinuity and W be a R n -valued Brownian motion for this probability space.Definition 1.1 Let us take (Ω, (F t ) t≥0 , F, P) and a C 1,2 -family g(t) t∈[0,T [ of metricsover M. An M-valued process X(x) defined on Ω × [0, T [ is called a g(t)-Brownian motion in M started at x ∈ M if X(x) is continuous, adapted, and if361
for every smooth function f,is a local martingale.f(X s (x)) − f(x) − 1 2∫ s0∆ t f(X t (x)) dtWe shall prove existence of this inhomogeneous diffusion and give a notion ofparallel transport along this process.Let (e i ) i∈[1..d] be an orthonormal basis of R n , F(M) the frame bundle over M, πthe projection to M. For any u ∈ F(M), let L i (t, u) = h t (ue i ) be the ∇ t horizontallift of ue i and L i (t) the associated vector field. Further let V α,β be the canonicalbasis of vertical vector fields over F(M) defined by V α,β (u) = Dl u (E α,β ) where E α,βis the canonical basis of M n (R) and wherel u : GL n (R) → F(M)is the left multiplication. Finally let (O(M), g(t)) be the g(t) orthonormal framebundle.Proposition 1.2 Assume that g(t) t∈[0,T [ is a C 1,2 (t, x)-family of metrics over M,andA : [0, T [ × F(M) → M n (R)(t, U) ↦→ (A α,β (t, U)) α,βis locally Lipschitz in U unformly in all compact of t. Consider the Stratonovichdifferential equation in F(M):{ ∗dUt = ∑ ni=1 L i(t, U t ) ∗ dW i + ∑ α,β A α,β(t, U t )V α,β (U t ) dt(1.1)U 0 ∈ F(M) such that U 0 ∈ (O(M), g(0)).Then there is a unique symmetric choice for A such that U t ∈ (O(M), g(t)). Moreover:A(t, U) = − 1 2 ∂ 1G(t, U),where (∂ 1 G(t, U)) i,j = 〈Ue i , Ue j 〉 ∂tg(t) .Proof : Let us begin with curves. Let I be a real interval, π : T M → M theprojection, V and C two curves in C 1 (I, T M) such thatx(t) := π(V (t)) = π(C(t)), for all t ∈ IWe want to compute:d()〈V (t), C(t)〉dt g(t,x(t))| t=0237
Page 1: Université de PoitiersTHÈSEpour o
Page 4 and 5: AbstractIn the first part of this t
Page 6 and 7: ivTABLE DES MATIÈRES3 Kendall-Cran
Page 8 and 9: 8 CHAPITRE 1. INTRODUCTIONdifféren
Page 10 and 11: 10 CHAPITRE 1. INTRODUCTIONdonne se
Page 12 and 13: 12 CHAPITRE 1. INTRODUCTIONHamilton
Page 14 and 15: 14 CHAPITRE 1. INTRODUCTIONx et ell
Page 16 and 17: 16 CHAPITRE 1. INTRODUCTIONCette é
Page 18 and 19: 18 CHAPITRE 1. INTRODUCTION- Exempl
Page 20 and 21: 20 CHAPITRE 1. INTRODUCTION20
Page 22 and 23: 22CHAPITRE 2. INTRODUCTION À L’A
Page 24 and 25: 24CHAPITRE 2. INTRODUCTION À L’A
Page 26: 26CHAPITRE 2. INTRODUCTION À L’A
Page 29 and 30: 2. ÉQUATIONS DIFFÉRENTIELLES STOC
Page 31 and 32: 2. ÉQUATIONS DIFFÉRENTIELLES STOC
Page 33 and 34: 3. QUELQUES APPLICATIONS DU CALCUL
Page 35: Chapitre 3Brownian motion with resp
Page 39 and 40: For the solution U t of (1.1) we ge
Page 41 and 42: Remark : Recall that in the compact
Page 43 and 44: 2 Local expression, evolution equat
Page 45 and 46: The last equality comes from Green
Page 47 and 48: Theorem 3.2 For every solution f(t,
Page 49 and 50: Consequently:d(df(T − t, .) X Tt
Page 51 and 52: Proof : The first remark after theo
Page 53 and 54: Remark : Hamilton gives a proof of
Page 55 and 56: Corollary 3.7 For χ(M) < 0, there
Page 57 and 58: We also have:D S,T −t dπ ˜// 0,
Page 59 and 60: Proof : By differentiation under x
Page 61 and 62: where we have used in the second eq
Page 63 and 64: [10] K. D. Elworthy and M. Yor. Con
Page 65 and 66: Chapter 4Some stochastic process wi
Page 67 and 68: We will just look at the smooth sol
Page 69 and 70: that is to say:d(Y T,it ) = − ∂
Page 71 and 72: 2 Tightness, and first example on t
Page 73 and 74: proof : It is clear that F is smoot
Page 75 and 76: Proposition 2.6 Let g(t) be a famil
Page 77 and 78: v) ˜g(∞) is a metric such that (
Page 79 and 80: Then:for all ɛ > 0 , there exists
Page 81 and 82: Finally, we obtain:∂∂t | t=t 0
Page 83 and 84: where Ut 3 is the horizontal lift o
Page 85 and 86: √πWe can choose ɛ, ɛ 2 such th
Page 87 and 88:
We will now show that the coupling
Page 89 and 90:
HenceWe get:√√ n∑1 − ɛI t
Page 91 and 92:
By uniqueness in law of such proces
Page 95 and 96:
Chapter 5Horizontal diffusion in pa
Page 97 and 98:
2 M. ARNAUDON, A. K. COULIBALY, AND
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
10 M. ARNAUDON, A. K. COULIBALY, AN
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Chapter 6Compléments de calculs113
Page 115 and 116:
d’avoir une famille de connexion,
Page 117 and 118:
Dans le calcul de ligne 8 à ligne
Page 119 and 120:
On utilise le fait que W (.) t est
Page 121 and 122:
Chapter 7Appendix121121
Page 123 and 124:
774 M. Arnaudon et al. / C. R. Acad
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Bibliography[ABT02]Marc Arnaudon, R
Page 131 and 132:
BIBLIOGRAPHY 131[DeT83][Dri92]Denni
Page 133 and 134:
BIBLIOGRAPHY 133[Jos84][Jos05][JS03
Page 135 and 136:
135
show all

THÃSE KolÃ©hÃ¨ Abdoulaye COULIBALY-PASQUIER

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?

THÃSE KolÃ©hÃ¨ Abdoulaye COULIBALY-PASQUIER