Functional integration and the Onsager-Machlup Lagrangian for continuous Markov processes in Riemannian geometries

Abstract

The Onsager-Machlup Lagrangian defined by the functional-integral representation of general continuous Markov processes is derived explicitly and concisely. Attention is given both to the mathematics and its physical interpretation. The method is based on (i) the consideration of continuous and differentiable paths connecting fixed end points in the convariant propagator, (ii) the application of a recently developed Fourier-series analysis of these trajectories in locally flat spaces, and (iii) the use of the proper transformations between locally Euclidean and globally Riemannian geometries. The spectral analysis allows for arbitrary paths rather than an a priori straight line even in the short-time propagator and avoids any ad hoc discretization rule. Specialized to globally flat spaces the result agrees with formulas given by Stratonovich, Horsthemke and Bach, Graham, and Dekker. It is demonstrated that a unique covariant path integral is equivalent to a whole class of stochastically equivalent lattice expressions.