Diffusion on the Total Space of a Vector Bundle

In recent years the study of the differential geometry of the total space E, of a vector bundle π : R → M , initiated by R.Miron [11], [12] has been developed by many people (see [13] and the references therein). If we take a horizontal complement of the vertical subbundle V E, we can express the geometrical objects defined on E in a more simplified form and new geometric objects can be obtained. Recently P.L.Antonelli and T.Zastawniak in a series of papers [2], [3], [4] extended the Riemannian theory of diffusion processes and stochastic development to the case of Finsler manifolds, the extension being motivated by important problems in Biology [3], [5]. In this paper we extend their formalism to study some geometric problems of the theory of the diffusion process and the stochastic development on E, related to these new geometric objects on E. We thereby obtain further generalization and geometric meaning for certain results of [2], [3]. But few probabilistic calculations are given here, for they are given in [2], [3], [4]. In a forthcoming publication, as a particular case, the theory of diffusion and stochastic development on Lagrange manifolds will be discussed [9]. Mathematics Subject Classification: 60J60, 58G32