UDС 519.216.2;514.764.3 ON RELATIONS BETWEEN INFINITESIMAL GENERATORS AND MEAN DERIVATIVES OF STOCHASTIC PROCESSES ON MANIFOLDS* Yu. <...> E. Gliklikh Voronezh State University Поступила в редакцию 01.08.2008 г. <...> We describe the relation between the forward (backward) infinitesimal generator on the one hand and forward (backward, respectively) mean derivative on the other hand for a stochastic process on manifold. <...> The relation is formulated in terms of a mapping from the second order tangent bundle to the first order one, generated by a given connection. <...> It is also shown that the so-called quadratic mean derivative can be obtained from the generator by another morphism between the corresponding bundles. <...> Key words: Stochastic processes; infinitesimal generators; mean derivatives; connections on manifolds; second order tangent vectors. <...> There are two types of differential operators associated to a stochastic process: the infinitesimal generators (forward and backward) and mean derivatives (forward, backward and quadratic). <...> Recall that the generators are determined invariantly as the so-called second order tangent vectors, quadratic mean derivatives are invariant as well and take values in (2,0) -tensors while the forward and backward mean derivatives on a manifold are well defined for a given connection and then take values in first order vectors. <...> In this paper we show that given a connection, one can construct forward (backward) mean derivative from forward (backward, respectively) generator by application of a natural mapping from second order tangent bundle to the first order one, generated by the connection. <...> The quadratic mean derivative can be obtained from the generator by another special operator between the corresponding bundles that is independent of the choice of connection. <...> Let Ua ,, … ∂ the local coordinates in Ua and by ∂ ∂ º ∂ qqn 1 ,, (we do not distinguish in notations the tangent vectors and the corresponding first order differential operators). <...> Consider a differential operator in Ua the corresponding basis vectors in tangent spaces to Ua of order no greater than 2 without constant term of the form Bt m b qq q (, )= matrix (). i ∂ + b ∂∂ , i ∂ where the coefficients bij bij ij ∂ ij 2 (1) form a symmetric Definition 1. <...> A second order tangent vector to a manifold M at a point mM ope ra <...>