The Onsager–Machlup function is a function that summarizes the dynamics of a continuous stochastic process. 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 S. Machlup who were the first to consider such probability densities.
Contents
- Definition
- Examples
- Wiener process on the real line
- Diffusion processes with constant diffusion coefficient on Euclidean space
- Generalizations
- Applications
- References
The dynamics of a continuous stochastic process X from time t = 0 to t = T in one dimension, satisfying a stochastic differential equation
where W is a Wiener process, can in approximation be described by the probability density function of its value xi at a finite number of points in time ti:
where
and Δti = ti+1 − ti > 0, t1 = 0 and tn = T. A similar approximation is possible for processes in higher dimensions. The approximation is more accurate for smaller time step sizes Δti, but in the limit Δti → 0 the probability density function becomes ill defined, one reason being that the product of terms
diverges to infinity. In order to nevertheless define a density for the continuous stochastic process X, ratios of probabilities of X lying within a small distance ε from smooth curves φ1 and φ2 are considered:
as ε → 0, where L is the Onsager–Machlup function.
Definition
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. For any two smooth curves φ1, φ2 : [0, T] → M,
where ρ is the Riemannian distance,
The Onsager–Machlup function is given by
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.
Examples
The following examples give explicit expressions for the Onsager–Machlup function of a continuous stochastic processes.
Wiener process on the real line
The Onsager–Machlup function of a Wiener process on the real line R is given by
Diffusion processes with constant diffusion coefficient on Euclidean space
The Onsager–Machlup function in the one-dimensional case with constant diffusion coefficient σ is given by
In the d-dimensional case, with σ equal to the unit matrix, it is given by
where || ⋅ || is the Euclidean norm and
Generalizations
Generalizations have been obtained by weakening the differentiability condition on the curve φ. Rather than taking the maximum distance between the stochastic process and the curve over a time interval, other conditions have been considered such as distances based on completely convex norms and Hölder, Besov and Sobolev type norms.
Applications
The Onsager–Machlup function can be used for purposes of reweighting and sampling trajectories, as well as for determining the most probable trajectory of a diffusion process.