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}}}}

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

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

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]