THÃSE KolÃ©hÃ¨ Abdoulaye COULIBALY-PASQUIER

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Let f ∈ C ∞ (M) then we have:for even k:for odd k:df(1 [T Nk ,T N k+1 ] (t) ∗ dZ 3 t ) dM ≡ 1 2 1 [T N k ,T N k+1 ] (t)∆˜g(t) f(Z 3 t )dtdf(1 [T Nk ,T N k+1 ] (t) ∗ dZ 3 t ) = 1 2 1 [T N k ,T N k+1 ] ∆˜g(t) f(Z 3 t )dtSo Z 3 and Z 2 are two diffusions with the same starting distribution and thesame generator, hence they are equal in law. For the gluing with Z 1 after C N thisis just the strong Markov property for (t, Z).Proposition 3.6 There exists α > 0 such that:P(T N 1− N < 1 2 ) > αproof :By the C ∞ -convergence of the metric we get:∀t < T, |∆˜g(t) f − ∆˜g(−∞) f| ≤˜Ceδtwhere the constant comes from Theorem 3.1, and the derivative of f up to ordertwo. We also obtain, by lemma 3.4, for a constant ɛ 2 that will be fixed below:|ρ t − ρ −∞ | ≤ ɛ 2 .Over the sphere (M, ˜g(−∞)), we have by ordinary comparison theorem:∆˜g(−∞) ρ −∞ (x) ≤ (n)cot(ρ −∞ (x)).We can suppose after normalization that the radius of the sphere (M, ˜g(−∞)) isone, Radius −∞ (M) = 1 (i.e. cst = 1) in 3.4. We deduce from above that:In [N, T N 1 [, we have ρ t (Z 1 t , Z 3 t ) >∆˜g(t) ρ −∞ (x) ≤ (n)cot(ρ −∞ (x)) + ˜Ce δt .√ π1+ɛ∧ Cn(d,K,cst−ɛ)2so:4√ π1+ɛ∧ Cn(d,K,cst−ɛ)24− ɛ 2 ≤ ρ t (Z 1 t , Z 3 t ) − ɛ 2 ≤ ρ −∞ (Z 1 t , Z 3 t ) ≤ π8419
√πWe can choose ɛ, ɛ 2 such that,1+ɛ∧ Cn(d,K,cst−ɛ)2− ɛ4 2 ≥ β > 0. We obtain:cot(ρ −∞ (Zt 1 , Zt 3 )) ≤ cot(β),and∆˜g(t) ρ −∞ (Z 1 t , .)(Z 3 t ) ≤ (n)cot(β) + ˜Ce δT ,(recall T
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Université de PoitiersTHÈSEpour o
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AbstractIn the first part of this t
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ivTABLE DES MATIÈRES3 Kendall-Cran
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8 CHAPITRE 1. INTRODUCTIONdifféren
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10 CHAPITRE 1. INTRODUCTIONdonne se
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12 CHAPITRE 1. INTRODUCTIONHamilton
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14 CHAPITRE 1. INTRODUCTIONx et ell
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16 CHAPITRE 1. INTRODUCTIONCette é
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18 CHAPITRE 1. INTRODUCTION- Exempl
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20 CHAPITRE 1. INTRODUCTION20
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22CHAPITRE 2. INTRODUCTION À L’A
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2. ÉQUATIONS DIFFÉRENTIELLES STOC
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2. ÉQUATIONS DIFFÉRENTIELLES STOC
Page 33 and 34: 3. QUELQUES APPLICATIONS DU CALCUL
Page 35 and 36: Chapitre 3Brownian motion with resp
Page 37 and 38: for every smooth function f,is a lo
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: where Ut 3 is the horizontal lift o
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 105 and 106: 10 M. ARNAUDON, A. K. COULIBALY, AN
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 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
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THÃSE KolÃ©hÃ¨ Abdoulaye COULIBALY-PASQUIER

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THÃSE KolÃ©hÃ¨ Abdoulaye COULIBALY-PASQUIER