Onsager–Machlup function

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 Stefan 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 stochastic differential equation

${\displaystyle dX_{t}=b(X_{t})\,dt+\sigma (X_{t})\,dW_{t}}$

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:

${\displaystyle p(x_{1},\ldots ,x_{n})=\left(\prod _{i=1}^{n-1}{\frac {1}{\sqrt {2\pi \sigma (x_{i})^{2}\Delta t_{i}}}}\right)\exp \left(-\sum _{i=1}^{n-1}L\left(x_{i},{\frac {x_{i+1}-x_{i}}{\Delta t_{i}}}\right)\,\Delta t_{i}\right)}$

where

${\displaystyle L(x,v)={\frac {1}{2}}\left({\frac {v-b(x)}{\sigma (x)}}\right)^{2}}$

and Δt_i = t_i+1 − t_i > 0, t₁ = 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 Δt_i, but in the limit Δt_i → 0 the probability density function becomes ill defined, one reason being that the product of terms

${\displaystyle {\frac {1}{\sqrt {2\pi \sigma (x_{i})^{2}\Delta t_{i}}}}}$

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:

${\displaystyle {\frac {P\left(\left|X_{t}-\varphi _{1}(t)\right|\leq \varepsilon {\text{ for every }}t\in [0,T]\right)}{P\left(\left|X_{t}-\varphi _{2}(t)\right|\leq \varepsilon {\text{ for every }}t\in [0,T]\right)}}\to \exp \left(-\int _{0}^{T}L\left(\varphi _{1}(t),{\dot {\varphi }}_{1}(t)\right)\,dt+\int _{0}^{T}L\left(\varphi _{2}(t),{\dot {\varphi }}_{2}(t)\right)\,dt\right)}$

as ε → 0, where L is the Onsager–Machlup function.