By Lucien Guillou, Alexis Marin

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**Sample text**

Vq_l). 3. e. 5) and Pc the induced semigroup on measurable forms defined by = when the expectation exists. 3), as seen by It6's formula. g. [Elw92]. 1 Let M be a Riemannian manifold. Assume non-explosion and Esups

Take ~ C F L ( T M , T ' M ) . Then tr~'r = Z i Vx'r r + Z i -))" Thus for 0 a 1-form, / = ,, I On the other hand = - (~, t r V . } (W, -} for vector fields U, V and W. ,,J and = /M E o(xJ)r <0, ~ / ~ ) ( x i ) ( T ( X i , - ) ) > i,j z,J So and V is torsion skew symmetric. 2. (1). Assume I. Let f E C~(M; R). Then taking a suitable sequence {g~} in [0, 1]) converging to 1 we have C~(M; MA(f)d# = O. e. (gradEf) + (A, gradEf}X) d# = 0. 9) The infinitesimal generators and associated operators 54 which is symmetric in A and f .

2) using a finite set of vector fields X 1 , . . , X m. C. The rest of this section will be on the comparison of 5 and the usual divergence 5. 1 Let V be a metric connection on T M with defining map X . 1. Assume Z V x J X j = O. Then 5 = 5, the usual divergence, on differential 1-forms. g. this holds in the gradient Brownian system case, or if V is torsion skew symmetric ). 2. Assume y~j V x J X j = ~ h for some smooth h : M -~ IR. Then ~ i Lx~Lx~ = A + Lvh on differential 1-forms. Here A is the Laplace-Beltrami operator.