Bismut J.-M. Large Deviations and the Malliavin Calculus

Basel: Birkhäuser, 1984. — 232 p.This book is a unique and profound contribution to the investigation of diffusion processes and stochastic analysis on manifolds. It employs the Malliavin calculus and large deviation techniques to study the asymptotics of the conditional probabilities of bridges associated with certain hypoelliptic diffusions. The program is fully completed in the elliptic case. In the hypoelliptic case, a deterministic Malliavin calculus is developed which exhibits the importance of the corresponding Malliavin covariance matrix in studying the curves of minimal action, and their relations to bicharacteristic curves. Two conjectures are formulated in the hypoelliptic case. A case study is done for the Heisenberg group.
An important volume for research mathematicians working in probability, diffusion theory, geometry, and mathematical physics.