Onsager–Machlup function – Wikipedia

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

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

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:

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

where

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

and 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

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

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]

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

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

Definition[edit]

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,

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

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 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[edit]

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

Wiener process on the real line[edit]

The Onsager–Machlup function of a Wiener process on the real line R is given by^[6]

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

Proof: Let X = {X_t : 0 ≤ t ≤ T} be a Wiener process on R and let φ : [0, T] → R be a twice differentiable curve such that φ(0) = X₀. Define another process X^φ = {X_t^φ : 0 ≤ t ≤ T} by X_t^φ = X_t − φ(t) and a measure P^φ by

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

For every ε > 0, the probability that |X_t − φ(t)| ≤ ε for every t ∈ [0, T] satisfies

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

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:

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

By Itō’s lemma it holds that

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

where

${displaystyle scriptstyle {ddot {varphi }}}$

is the second derivative of φ, and so this term is of order ε on the event where |X_t| ≤ ε for every t ∈ [0, T] and will disappear in the limit ε → 0, hence

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

Diffusion processes with constant diffusion coefficient on Euclidean space[edit]

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[edit]

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[edit]

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]

See also[edit]

References[edit]

Bibliography[edit]

External links[edit]