Probabilistic Construction of Lagrangean of Diffusion Process and Its Application

Abstract

Lagrangean (Onsager-!'v1achlup function) is rigorously obtained for a diffusion process in Euclidean space by first introducing the probability measure in 'path space' and then evaluat­ ing the measure of neighbourhood of a smooth path. Consequently probabilistic meaning of the detailed balance and the cyclic balance is clarified. On the basis of this probabilistic clarification and the global definitions of the detailed balance and of the cyclic balance, these two balances are successfully characterized with the help of the notions 'vorticity' and 'circula­ tion' in fluid mechanics. Discussed is the transformation property of the Stratonovich type stochastic differential equation as well as of the Lagrangean given by Stratonovich, and it is shown that neither is covariant in general, contrary to the recent argument by Graham.