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
d X t = b ( X t ) d t + σ ( X t ) d W t {\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 xi at a finite number of points in time ti:
p ( x 1 , … , x n ) = ( ∏ i = 1 n − 1 1 2 π σ ( x i ) 2 Δ t i ) exp ( − ∑ i = 1 n − 1 L ( x i , x i + 1 − x i Δ t i ) Δ 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
L ( x , v ) = 1 2 ( v − b ( x ) σ ( x ) ) 2 {\displaystyle L(x,v)={\frac {1}{2}}\left({\frac {v-b(x)}{\sigma (x)}}\right)^{2}}
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
1 2 π σ ( x i ) 2 Δ t i {\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:
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