THÈSE Koléhè Abdoulaye COULIBALY-PASQUIER

10 M. ARNAUDON, A. K. COULIBALY, AND A. THALMAIEROn the other hand, using (2.9) we have[](2.19) E sup ‖Y t (u) − X t (u)‖ 2 ≤ Ct≤τ 0∫ u0[]E sup ‖Y t (v) − X t (v)‖ 2 dvt≤τ 0from which we deduce that almost surely, for all t ∈ [0, τ 0 ], X t (u) = Y t (u). Consequently,exploiting the fact that the two processes are continuous in (t, u), theymust be indistinguishable.□3. Horizontal diffusion along non-homogeneous diffusionIn this Section we assume that the elliptic generator is a C 1 function of time:L = L(t) for t ≥ 0. Let g(t) be the metric on M such thatL(t) = 1 2 ∆t + Z(t)where ∆ t is the g(t)-Laplacian and Z(t) a vector field on M.Let (X t ) be an inhomogeneous diffusion with generator L(t). Parallel transportP t (X) t along the L(t)-diffusion X t is defined analogously to [5] as the linear mapwhich satisfiesP t (X) t : T X0 M → T Xt M(3.1) D t P t (X) t = − 1 2 ġ♯ (P t (X) t ) dtwhere ġ denotes the derivative of g with respect to time; the covariant differential D tis defined in local coordinates by the same formulas as D, with the only differencethat Christoffel symbols now depend on t.Alternatively, if J is a semimartingale over X, the covariant differential D t J maybe defined as ˜D(0, J) = (0, D t J), where (0, J) is a semimartingale along (t, X t ) in˜M = [0, T ] × M endowed with the connection ˜∇ defined as follows: ifs ↦→ ˜ϕ(s) = (f(s), ϕ(s))is a C 1 path in ˜M and s ↦→ ũ(s) = (α(s), u(s)) ∈ T ˜M is C 1 path over ˜ϕ, then(˜∇ũ(s) = ˙α(s), ( ∇ f(s) u ) )(s)where ∇ t denotes the Levi-Civita connection associated to g(t). It is proven in [5]that P t (X) t is an isometry from (T X0 M, g(0, X 0 )) to (T Xt M, g(t, X t )).The damped parallel translation W t (X) t along X t is the linear mapsatisfying(3.2) D t W t (X) t =W t (X) t : T X0 M → T Xt M(∇ t W t (X) tZ(t, ·) − 1 )2 (Rict ) ♯ (W t (X) t ) dt.If Z ≡ 0 and g(t) is solution to the backward Ricci flow:(3.3) ġ = Ric,then damped parallel translation coincides with the usual parallel translation:(see [5] Theorem 2.3).P t (X) = W t (X),105