THÈSE Koléhè Abdoulaye COULIBALY-PASQUIER

Proposition 4.2 For all v ∈ T x M we have:d((// T 0,t) −1 T X t v) = 1 2 (//T 0,t) −1 (( ∂ ∂ tg(T − t)) − Ric T −t ) #,T −t (T X t v) dt.Proof :have:For a triple of tangent vectors (L t , L), (A t , A), (Z t , Z) ∈ T I × T M, we˜R((L t , L), (A t , A))(Z t , Z) = (0, R T −t (L, A)Z).Hence, according to the relation dY (x 0 ) = (dt, ∗dX t ) = (dt, // T 0,te i ∗ dW i ) and thedefinition of the Ricci tensor:−1˜// 0,td ˜//0,t T Y t(0, v) = − 1 2 (0, Ric#T −t (T X t v)) dt. (4.5)In order to compute in R n , we write:(// T 0,t) −1 T X t v = ((// T 0,t) −1 dπ ˜//−10,t)( ˜//0,t T Y t(0, v)). (4.6)−1By (4.5), we have d ˜//0,t T Y t(0, v) ∈ dA where A is the space of finite variationprocesses. We get:d((// T 0,t) −1 T X t v) = d((// T 0,t) −1 dπ ˜//−10,t)( ˜//0,t T Y t(0, v))+((// T 0,t) −1 dπ ˜//−10,t)d( ˜//0,t T Y t(0, v)).By (4.6) and lemma 4.1 we get:d((// T 0,t) −1 T X t v) = ∗d((// T 0,t) −1 dπ ˜//−10,t)( ˜//0,t T Y t(0, v))+ ((// T 0,t) −1 dπ ˜//−10,t) ∗ d( ˜//0,t T Y t(0, v))= ∗d((// T 0,t) −1 dπ ˜//−10,t)( ˜//0,t T Y t(0, v))−1 2 ((//T 0,t) −1 dπ)(0, Ric #,T −t (T X t v) dt= 1 2 (//T 0,t) −1 ( ∂ ∂ tg(T − t)) #T −t (T X t v) dt−1 2 (//T 0,t) −1 Ric #,T −t (T X t v) dt.For all f 0 ∈ C ∞ (M), and f(t, .) a solution of (3.3), f(T − t, X T t (x)) is a martingalefor all x ∈ M.Corollary 4.3 For all v ∈ T x M:is a martingale.df(T − t, X T t (x))v = df(T − t, .) X Tt (x)// T 0,tv,5823