Onsager–Machlup function

The Onsager–Machlup function is a function that summarizes the dynamics of a . 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.

The dynamics of a continuous stochastic process $X$ from time $t = 0$ to $t = T$ in one dimension, satisfying a

where $W$ is a Wiener process, can in approximation be described by the probability density function of its value $x i$ at a finite number of points in time $t i$ :

where

and $Δ t i = t i +1 - t i > 0$ , $t 1 = 0$ and $t n = T$ . A similar approximation is possible for processes in higher dimensions. The approximation is more accurate for smaller time step sizes $Δ t i$ , but in the limit $Δ t i \to 0$ 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 $φ 1$ and $φ 2$ are considered:

as $ε \to 0$ , where $L$ is the Onsager–Machlup function.

Consider a $d$ -dimensional Riemannian manifold $M$ and a diffusion process $X = {X t : 0 \leq t \leq T}$ on $M$ with $1 / 2 Δ M + b$ , where $Δ M$ is the Laplace–Beltrami operator and $b$ is a vector field. For any two smooth curves $φ 1, φ 2 : [0, T] \to M$ ,

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