Onsager–Machlup function

It is used to define a probability density for a stochastic process, and it is similar to the Lagrangian of a dynamical system.

It is named after Lars Onsager and Stefan Machlup [de] who were the first to consider such probability densities.

Consider a d-dimensional Riemannian manifold M and a diffusion process X = {Xt : 0 ≤ t ≤ T} on M with infinitesimal generator ⁠1/2⁠ΔM + b, where ΔM is the Laplace–Beltrami operator and b is a vector field.

The Onsager–Machlup function is given by[3][4][5] where || ⋅ ||x is the Riemannian norm in the tangent space Tx(M) at x, div b(x) is the divergence of b at x, and R(x) is the scalar curvature at x.

Define another process Xφ = {Xtφ : 0 ≤ t ≤ T} by Xtφ = Xt − φ(t) and a measure Pφ by For every ε > 0, the probability that |Xt − φ(t)| ≤ ε for every t ∈ [0, T] satisfies By Girsanov's theorem, the distribution of Xφ under Pφ equals the distribution of X under P, hence the latter can be substituted by the former: By Itō's lemma it holds that where