It is shown that in a recent work the program proposed cannot be carried out in curved space, while in flat space the result is a consequence of the restriction to differentiable paths. The use made of Fourier series is also shown to require modifications.
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.
The two different forms of stochastic Lagrangian in the literature which stem from a common deterministic model L(x\ifmmode \dot{}\else \.{}\fi{},x,t)=(1/2)mx${\ifmmode \dot{}\else \.{}\fi{}}^{2}$+A (x,t)x\ifmmode \dot{}\else \.{}\fi{}-V(x,t) are analyzed referring to the question of Nelson raised in his recent book Quantum Fluctuations, and a resolution is presented. It is argued that there exists a self-contained framework of stochastic mechanics, different from Nelson's original proposal, which is prescribed by (i) a=(1/2)(DD+DlstarDlstar)x for the stochastic acceleration, (ii) the same procedure as Nelson of connecting both sides of the stochastic differential equation for the process x(t) with the current and osmotic velocities but with entirely real valued wave functions, and (iii) a special relation between V and A which conforms the above L to the Onsager-Machlup Lagrangian.