THÈSE Koléhè Abdoulaye COULIBALY-PASQUIER

HORIZONTAL DIFFUSION IN PATH SPACE 13To the cost function c we associate the Monge-Kantorovich minimization betweentwo probability measures on M∫(4.2) W c,t (µ, ν) = infη∈Π(µ,ν)c(t, x, y) dη(x, y)M×Mwhere Π(µ, ν) is the set of all probability measures on M × M with marginals µand ν. We denote(4.3) W p,t (µ, ν) = (W ρ p ,t(µ, ν)) 1/pthe Wasserstein distance associated to p > 0.For a probability measure µ on M, the solution of the heat flow equation associatedto L(t) will be denoted by µP t .Define a section (∇ t Z) ♭ ∈ Γ(T ∗ M ⊙ T ∗ M) as follows: for any x ∈ M andu, v ∈ T x M,(∇ t Z) ♭ (u, v) = 1 (g(t)(∇t2u Z, v) + g(t)(u, ∇ t vZ) ) .In case the metric does not depend on t and Z = grad V for some C 2 function Von M, then(∇ t Z) ♭ (u, v) = ∇dV (u, v).Theorem 4.1. We keep notation and assumptions from above.a) Assume(4.4) Ric t −ġ − 2(∇ t Z) ♭ ≥ 0.Then the functionis non-increasing.b) If for some k ∈ R,t ↦→ W c,t (µP t , νP t )(4.5) Ric t −ġ − 2(∇ t Z) ♭ ≥ kg,then we have for all p > 0W p,t (µP t , νP t ) ≤ e −kt/2 W p,0 (µ, ν).Remark 4.2. Before turning to the proof of Theorem 4.1, let us mention that inthe case Z = 0, g constant, p = 2 and k = 0, item b) is due to [19] and [18]. In thecase where g is a backward Ricci flow solution, Z = 0 and p = 2, statement b) isdue to Lott [16] and McCann-Topping [17]. For extensions about L-transportation,see [21].Proof of Theorem 4.1. a) Assume that Ric t −ġ − 2(∇ t Z) ♭ ≥ 0. Then for anyL(t)-diffusion (X t ), we haved ( g(t)(W (X) t , W (X) t ) )= ġ(t) ( W (X) t , W (X) t)dt + 2g(t)(D t W (X) t , W (X) t)= ġ(t) ( )W (X) t , W (X) t dt+ 2g(t)(∇ t W (X) tZ(t, ·) − 1 )2 (Rict ) ♯ (W (X) t ), W (X) t dt(= ġ + 2(∇ t Z) ♭ − Ric t) ( )W (X)t , W (X) t dt ≤ 0.108