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 who were the first to consider such probability densities.[1]
The dynamics of a continuous stochastic process from time to in one dimension, satisfying a stochastic differential equation
dXt=b(Xt)dt+\sigma(Xt)dWt
where is a Wiener process, can in approximation be described by the probability density function of its value at a finite number of points in time :
p(x1,\ldots,xn)=\left(
| n-1 | |
| \prod | |
| i=1 |
| 1 | |||||||||
|
where
L(x,v)=
| 1 | \left( | |
| 2 |
| v-b(x) | |
| \sigma(x) |
\right)2
and, and . A similar approximation is possible for processes in higher dimensions. The approximation is more accurate for smaller time step sizes, but in the limit the probability density function becomes ill defined, one reason being that the product of terms
| 1 | |||||||||
|
diverges to infinity. In order to nevertheless define a density for the continuous stochastic process, ratios of probabilities of lying within a small distance from smooth curves and are considered:[2]
| P\left(\left|Xt-\varphi1(t)\right|\leq\varepsilonforeveryt\in[0,T]\right) | |
| P\left(\left|Xt-\varphi2(t)\right|\leq\varepsilonforeveryt\in[0,T]\right) |
\to
| T | |
| \exp\left(-\int | |
| 0 |
L\left
| (\varphi | |||
|
1(t)\right)dt+
| T | |
| \int | |
| 0 |
L\left
| (\varphi | |||
|
2(t)\right)dt\right)
as, where is the Onsager–Machlup function.
Consider a -dimensional Riemannian manifold and a diffusion process on with infinitesimal generator, where is the Laplace–Beltrami operator and is a vector field. For any two smooth curves,
\lim\varepsilon\downarrow0
| P\left(\rho(Xt,\varphi1(t))\leq\varepsilonforeveryt\in[0,T]\right) | |
| P\left(\rho(Xt,\varphi2(t))\leq\varepsilonforeveryt\in[0,T]\right) |
=\exp\left(
| T | |
| -\int | |
| 0 |
L\left
| (\varphi | |||
|
1(t)\right)dt
| T | |
| +\int | |
| 0 |
L\left
| (\varphi | |||
|
2(t)\right)dt\right)
where is the Riemannian distance,
\scriptstyle
| \varphi |
1,
| \varphi |
2
The Onsager–Machlup function is given by[3] [4] [5]
L(x,v)=
| 2 | |
| \tfrac{1}{2}\|v-b(x)\| | |
| x |
+\tfrac{1}{2}\operatorname{div}b(x)-\tfrac{1}{12}R(x),
where is the Riemannian norm in the tangent space at, is the divergence of at, and is the scalar curvature at .
The following examples give explicit expressions for the Onsager–Machlup function of a continuous stochastic processes.
The Onsager–Machlup function of a Wiener process on the real line is given by[6]
L(x,v)=\tfrac{1}{2}|v|2.
Proof: Let be a Wiener process on and let be a twice differentiable curve such that . Define another process by and a measure by
P\varphi=\exp\left(
| T | |||
| \int | |||
|
(t)
| \varphi | |
| dX | |
| t |
+
| T | |
| \int | |
| 0\tfrac{1}{2} |
\left|
| \varphi |
(t)\right|2dt\right)dP.
For every, the probability that for every satisfies
\begin{align} P\left(\left|Xt-\varphi(t)\right|\leq\varepsilonforeveryt\in[0,T]\right)&=P\left(\left
| \varphi | |
| |X | |
| t |
\right|\leq\varepsilonforeveryt\in[0,T]\right)
| \\ &=\int | |||||||||
|
By Girsanov's theorem, the distribution of under equals the distribution of under, hence the latter can be substituted by the former:
P(|Xt-\varphi(t)|\leq\varepsilon
| foreveryt\in[0,T])=\int | |||||||||
|
By Itō's lemma it holds that
| T | |||
| \int | |||
|
(t)dXt=
| \varphi |
(T)XT-
| T | |
| \int | |
| 0\ddot{\varphi}(t)X |
tdt,
where
\scriptstyle\ddot{\varphi}
\lim\varepsilon\downarrow
| P(|Xt-\varphi(t)|\leq\varepsilonforeveryt\in[0,T]) | |
| P(|Xt|\leq\varepsilonforeveryt\in[0,T]) |
=\exp\left(
| T | |||
| -\int | |||
|
(t)|2dt\right).
The Onsager–Machlup function in the one-dimensional case with constant diffusion coefficient is given by[7]
| L(x,v)= | 1 | \left| |
| 2 |
| v-b(x) | |
| \sigma |
\right|2+
| 1 | |
| 2 |
| db | |
| dx |
(x).
In the -dimensional case, with equal to the unit matrix, it is given by[8]
| L(x,v)= | 1 |
| 2 |
\|v-b(x)\|2+
| 1 | |
| 2 |
(\operatorname{div}b)(x),
where is the Euclidean norm and
(\operatorname{div}b)(x)=
| d | |
| \sum | |
| i=1 |
| \partial | |
| \partialxi |
bi(x).
Generalizations have been obtained by weakening the differentiability condition on the curve .[9] 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[10] and Hölder, Besov and Sobolev type norms.[11]
The Onsager–Machlup function can be used for purposes of reweighting and sampling trajectories,[12] as well as for determining the most probable trajectory of a diffusion process.[13] [14]