The lagrangian approach to stochastic variational principles on curved manifolds

✍ Scribed by Ettore Aldrovandi; Daniela Dohrn; Francesco Guerra

• Publisher: Springer Netherlands
• Year: 1992
• Tongue: English
• Weight: 787 KB
• Volume: 26
• Category: Article
• ISSN: 0167-8019
• DOI: 10.1007/bf00047204

✦ Synopsis

The classical definition of the action functional, for a dynamical system on curved manifolds, can be extended to the case of diffusion processes. For the stochastic action functional so obtained, we introduce variational principles of the type proposed by Morato. In order to generalize the class of process variations, from the flat case originally given by Morato to general curved manifolds, we introduce the notion of stochastic differential systems. These give a synthetic characterization of the process and its variations as a generalized controlled stochastic process on the tangent bundle of the mazfifold. The resulting programming equations axe equivalent to the quantum Schr~idlnger equation, where the wave function is coupled to an additional vector potential, satisfying a plasma-like equation with a peculiar dissipative behavior.

Mathematics Subject Classifications (1991), 81P20.