# 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 S. Machlup who were the first to consider such probability densities.[1]

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 xi at a finite number of points in time ti:

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

where

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

and Δti = ti+1ti > 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

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

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:[2]

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

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

## Definition

Consider a d-dimensional Riemannian manifold M and a diffusion process X = {Xt : 0 ≤ tT} 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,

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

where ρ is the Riemannian distance, ${\displaystyle \scriptstyle {\dot {\varphi }}_{1},{\dot {\varphi }}_{2}}$ denote the first derivatives of φ1, φ2, and L is called the Onsager–Machlup function.

The Onsager–Machlup function is given by[3][4][5]

${\displaystyle L(x,v)={\tfrac {1}{2}}\|v-b(x)\|_{x}^{2}+{\tfrac {1}{2}}\operatorname {div} \,b(x)-{\tfrac {1}{12}}R(x),}$

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[6]

${\displaystyle L(x,v)={\tfrac {1}{2}}|v|^{2}.}$

### 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[7]

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

In the d-dimensional case, with σ equal to the unit matrix, it is given by[8]

${\displaystyle L(x,v)={\frac {1}{2}}\|v-b(x)\|^{2}+{\frac {1}{2}}(\operatorname {div} \,b)(x),}$

where || ⋅ || is the Euclidean norm and

${\displaystyle (\operatorname {div} \,b)(x)=\sum _{i=1}^{d}{\frac {\partial }{\partial x_{i}}}b_{i}(x).}$

## Generalizations

Generalizations have been obtained by weakening the differentiability condition on the curve φ.[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]

## Applications

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]

## References

1. ^ Onsager, L. and Machlup, S. (1953)
2. ^ Stratonovich, R. (1971)
3. ^ Takahashi, Y. and Watanabe, S. (1980)
4. ^ Fujita, T. and Kotani, S. (1982)
5. ^ Wittich, Olaf
6. ^ Ikeda, N. and Watanabe, S. (1980), Chapter VI, Section 9
7. ^ Dürr, D. and Bach, A. (1978)
8. ^ Ikeda, N. and Watanabe, S. (1980), Chapter VI, Section 9
9. ^ Zeitouni, O. (1989)
10. ^ Shepp, L. and Zeitouni, O. (1993)
11. ^ Capitaine, M. (1995)
14. ^ Dürr, D. and Bach, A. (1978).

## Bibliography

• Adib, A.B. (2008). "Stochastic actions for diffusive dynamics: Reweighting, sampling, and minimization". J. Phys. Chem. B. 112: 5910–5916. doi:10.1021/jp0751458.
• Capitaine, M. (1995). "Onsager–Machlup functional for some smooth norms on Wiener space". Probab. Theory Relat. Fields. 102: 189–201. doi:10.1007/bf01213388.
• Dürr, D. & Bach, A. (1978). "The Onsager–Machlup function as Lagrangian for the most probable path of a diffusion process". Commun. Math. Phys. 60: 153–170. doi:10.1007/bf01609446.
• Fujita, T. & Kotani, S. (1982). "The Onsager–Machlup function for diffusion processes". J. Math. Kyoto Univ. 22: 115–130.
• Ikeda, N. & Watanabe, S. (1980). Stochastic differential equations and diffusion processes. Kodansha-John Wiley.
• Onsager, L. & Machlup, S. (1953). "Fluctuations and Irreversible Processes". Physical Review. 91 (6): 1505–1512. doi:10.1103/physrev.91.1505.
• Shepp, L. & Zeitouni, O. (1993). "Exponential estimates for convex norms and some applications". Progress in Probability. Berlin: Birkhauser-Verlag. 32: 203–215. doi:10.1007/978-3-0348-8555-3_11.
• Stratonovich, R. (1971). "On the probability functional of diffusion processes". Select. Transl. in Math. Stat. Prob. 10: 273–286.
• Takahashi, Y. & Watanabe, S. (1980). "The probability functionals (Onsager–Machlup functions) of diffusion processes". Lecture Notes in Mathematics. Springer. 851: 432–463.
• Wittich, Olaf. "The Onsager–Machlup Functional Revisited".
• Zeitouni, O. (1989). "On the Onsager–Machlup functional of diffusion processes around non C2 curves". Annals of Probability. 17 (3): 1037–1054. doi:10.1214/aop/1176991255.