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 [de] 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

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 $t i$ :

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

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

{\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 $φ 1$ and $φ 2$ are considered:^[2]

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

Definition

Consider a $d$ -dimensional Riemannian manifold $M$ and a diffusion process $X = {X t : 0 \leq t \leq T}$ on $M$ with infinitesimal generator $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/2⁠ΔM + b$ , where $Δ M$ is the Laplace–Beltrami operator and $b$ is a vector field. For any two smooth curves $φ 1, φ 2 : [0, T] \to M$ ,

\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, $\scriptstyle {\dot {\varphi }}_{1},{\dot {\varphi }}_{2}$ denote the first derivatives of $φ 1, φ 2$ , and $L$ is called the Onsager–Machlup function.

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

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

where $|| \cdot || x$ is the Riemannian norm in the tangent space $T x (M)$ 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]

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

Proof: Let $X = {X t : 0 \leq t \leq T}$ be a Wiener process on $R$ and let $φ : [0, T] \to R$ be a twice differentiable curve such that $φ (0) = X 0$ . Define another process $X φ = {X t φ : 0 \leq t \leq T}$ by $X t φ = X t - φ (t)$ and a measure $P φ$ by

P^{\varphi }=\exp \left(\int _{0}^{T}{\dot {\varphi }}(t)\,dX_{t}^{\varphi }+\int _{0}^{T}{\tfrac {1}{2}}\left|{\dot {\varphi }}(t)\right|^{2}\,dt\right)\,dP.

For every $ε > 0$ , the probability that $| X t - φ (t)| \leq ε$ for every $t \in [0, T]$ satisfies

{\begin{aligned}P\left(\left|X_{t}-\varphi (t)\right|\leq \varepsilon {\text{ for every }}t\in [0,T]\right)&=P\left(\left|X_{t}^{\varphi }\right|\leq \varepsilon {\text{ for every }}t\in [0,T]\right)\\&=\int _{\left\{\left|X_{t}^{\varphi }\right|\leq \varepsilon {\text{ for every }}t\in [0,T]\right\}}\exp \left(-\int _{0}^{T}{\dot {\varphi }}(t)\,dX_{t}^{\varphi }-\int _{0}^{T}{\tfrac {1}{2}}|{\dot {\varphi }}(t)|^{2}\,dt\right)\,dP^{\varphi }.\end{aligned}}

By Girsanov's theorem, the distribution of $X φ$ under $P φ$ equals the distribution of $X$ under $P$ , hence the latter can be substituted by the former:

P(|X_{t}-\varphi (t)|\leq \varepsilon {\text{ for every }}t\in [0,T])=\int _{\left\{\left|X_{t}^{\varphi }\right|\leq \varepsilon {\text{ for every }}t\in [0,T]\right\}}\exp \left(-\int _{0}^{T}{\dot {\varphi }}(t)\,dX_{t}-\int _{0}^{T}{\tfrac {1}{2}}|{\dot {\varphi }}(t)|^{2}\,dt\right)\,dP.

By Itō's lemma it holds that

\int _{0}^{T}{\dot {\varphi }}(t)\,dX_{t}={\dot {\varphi }}(T)X_{T}-\int _{0}^{T}{\ddot {\varphi }}(t)X_{t}\,dt,

where $\scriptstyle {\ddot {\varphi }}$ is the second derivative of $φ$ , and so this term is of order $ε$ on the event where $| X t | \leq ε$ for every $t \in [0, T]$ and will disappear in the limit $ε \to 0$ , hence

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

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]

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]

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

where $|| \cdot ||$ is the Euclidean norm and

(\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

^ Onsager, L. and Machlup, S. (1953)
^ Stratonovich, R. (1971)
^ Takahashi, Y. and Watanabe, S. (1980)
^ Fujita, T. and Kotani, S. (1982)
^ Wittich, Olaf
^ Ikeda, N. and Watanabe, S. (1980), Chapter VI, Section 9
^ Dürr, D. and Bach, A. (1978)
^ Ikeda, N. and Watanabe, S. (1980), Chapter VI, Section 9
^ Zeitouni, O. (1989)
^ Shepp, L. and Zeitouni, O. (1993)
^ Capitaine, M. (1995)
^ Adib, A.B. (2008).
^ Adib, A.B. (2008).
^ 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 (19): 5910–5916. arXiv:0712.1255. doi:10.1021/jp0751458. PMID 17999482. S2CID 16366252.
Capitaine, M. (1995). "Onsager–Machlup functional for some smooth norms on Wiener space". Probab. Theory Relat. Fields. 102 (2): 189–201. doi:10.1007/bf01213388. S2CID 120675014.
Dürr, D. & Bach, A. (1978). "The Onsager–Machlup function as Lagrangian for the most probable path of a diffusion process". Commun. Math. Phys. 60 (2): 153–170. Bibcode:1978CMaPh..60..153D. doi:10.1007/bf01609446. S2CID 41249746.
Fujita, T. & Kotani, S. (1982). "The Onsager–Machlup function for diffusion processes". J. Math. Kyoto Univ. 22: 115–130. doi:10.1215/kjm/1250521863.
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. Bibcode:1953PhRv...91.1505O. doi:10.1103/physrev.91.1505.
Shepp, L. & Zeitouni, O. (1993). "Exponential estimates for convex norms and some applications". Barcelona Seminar on Stochastic Analysis. Vol. 32. Berlin: Birkhauser-Verlag. pp. 203–215. CiteSeerX 10.1.1.28.8641. doi:10.1007/978-3-0348-8555-3_11. ISBN 978-3-0348-9677-1. {{cite book}}: |journal= ignored (help)CS1 maint: location missing publisher (link)
Stratonovich, R. (1971). "On the probability functional of diffusion processes". Select. Transl. In Math. Stat. Prob. 10: 273–286.
Takahashi, Y.; Watanabe, S. (1981). "The probability functionals (Onsager–Machlup functions) of diffusion processes". Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lecture Notes in Mathematics. Vol. 851. Berlin: Springer. pp. 433–463. doi:10.1007/BFb0088735. ISBN 978-3-540-10690-6. MR 0620998.
Wittich, Olaf. "The Onsager–Machlup Functional Revisited". {{cite journal}}: Cite journal requires |journal= (help)
Zeitouni, O. (1989). "On the Onsager–Machlup functional of diffusion processes around non $C 2$ curves". Annals of Probability. 17 (3): 1037–1054. doi:10.1214/aop/1176991255.

External links

Onsager–Machlup function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Onsager-Machlup_function&oldid=22857

[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)

[12] Adib, A.B. (2008).

[13] Adib, A.B. (2008).

[14] Dürr, D. and Bach, A. (1978).

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