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

${\displaystyle dX_{t}=b(X_{t}),円dt+\sigma (X_{t}),円dW_{t}}${\displaystyle dX_{t}=b(X_{t}),円dt+\sigma (X_{t}),円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:

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

where

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

and Δt_i = t_i+1 − t_i> 0, t₁ = 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 Δt_i, but in the limit Δt_i → 0 the probability density function becomes ill-defined, one reason being that the product of terms

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

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

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

Consider a d-dimensional Riemannian manifold M and a diffusion process X = {X_t: 0 ≤ t ≤ T} on M with infinitesimal generator ⁠1/2⁠Δ_M + b, where Δ_M is the Laplace–Beltrami operator and b is a vector field. For any two smooth curves φ₁, φ₂: [0, T] → M,

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

where ρ is the Riemannian distance, ${\displaystyle \scriptstyle {\dot {\varphi }}_{1},{\dot {\varphi }}_{2}}${\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]}

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

where || ⋅ ||_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.

The following examples give explicit expressions for the Onsager–Machlup function of a continuous stochastic processes.

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]

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]}

• Lagrangian
• Functional integration

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