Onsager–Machlup function

Template:Short description 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 Template:Interlanguage link who were the first to consider such probability densities.

The dynamics of a continuous stochastic process Template:Mvar from time Template:Math to Template:Math in one dimension, satisfying a stochastic differential equation

$d{X}_t=b(X_t)dt+\sigma (X_t)d{W}_t$

where Template:Mvar is a Wiener process, can in approximation be described by the probability density function of its value Template:Math at a finite number of points in time Template:Math:

$p(x_1,\dots ,x_n)=\left(\prod _{i=1}^{n-1}\frac{1}{\sqrt{2\pi \sigma (x_i{)}^2\mathrm{\Delta }{t}_i}}\right)\exp (-\sum _{i=1}^{n-1}L(x_i,\frac{x_{i+1}-x_i}{\mathrm{\Delta }{t}_i})\mathrm{\Delta }{t}_i)$

where

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

and Template:Math, Template:Math and Template:Math. A similar approximation is possible for processes in higher dimensions. The approximation is more accurate for smaller time step sizes Template:Math, but in the limit Template:Math the probability density function becomes ill-defined, one reason being that the product of terms

$\frac{1}{\sqrt{2\pi \sigma (x_i{)}^2\mathrm{\Delta }{t}_i}}$

diverges to infinity. In order to nevertheless define a density for the continuous stochastic process Template:Mvar, ratios of probabilities of Template:Mvar lying within a small distance Template:Mvar from smooth curves Template:Math and Template:Math are considered:

$\frac{P(|X_t-{\phi }_1(t)|\le \epsilon \text{ for every }t\in [0,T])}{P(|X_t-{\phi }_2(t)|\le \epsilon \text{ for every }t\in [0,T])}\to \exp (-{\int }_0^{T}L({\phi }_1(t),{\dot{\phi }}_1(t))dt+{\int }_0^{T}L({\phi }_2(t),{\dot{\phi }}_2(t))dt)$

as Template:Math, where Template:Mvar is the Onsager–Machlup function.

Definition

Consider a Template:Mvar-dimensional Riemannian manifold Template:Mvar and a diffusion process Template:Math on Template:Mvar with infinitesimal generator Template:Math, where Template:Math is the Laplace–Beltrami operator and Template:Mvar is a vector field. For any two smooth curves Template:Math,

$\underset{\epsilon \downarrow 0}{lim}\frac{P(\rho (X_t,{\phi }_1(t))\le \epsilon \text{ for every }t\in [0,T])}{P(\rho (X_t,{\phi }_2(t))\le \epsilon \text{ for every }t\in [0,T])}=\exp (-{\int }_0^{T}L({\phi }_1(t),{\dot{\phi }}_1(t))dt+{\int }_0^{T}L({\phi }_2(t),{\dot{\phi }}_2(t))dt)$

where Template:Mvar is the Riemannian distance, ${\dot{\phi }}_1,{\dot{\phi }}_2$ denote the first derivatives of Template:Math, and Template:Mvar is called the Onsager–Machlup function.

The Onsager–Machlup function is given by

$L(x,v)=\frac{1}{2}\Vert v-b(x){\Vert }_x^2+\frac{1}{2}\mathrm{div}b(x)-\frac{1}{12}R(x),$

where Template:Math is the Riemannian norm in the tangent space Template:Math at Template:Mvar, Template:Math is the divergence of Template:Mvar at Template:Mvar, and Template:Math is the scalar curvature at Template:Mvar.

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 Template:Math is given by

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

Proof: Let Template:Math be a Wiener process on Template:Math and let Template:Math be a twice differentiable curve such that Template:Math. Define another process Template:Math by Template:Math and a measure Template:Math by

$P^{\phi }=\exp ({\int }_0^T\dot{\phi }(t)d{X}_t^{\phi }+{\int }_0^T\frac{1}{2}{|\dot{\phi }(t)|}^{2}dt)dP.$

For every Template:Math, the probability that Template:Math for every Template:Math satisfies

$\begin{array}{cc}P(|X_t-\phi (t)|\le \epsilon \text{ for every }t\in [0,T])& =P(\left|X_t^{\phi }\right|\le \epsilon \text{ for every }t\in [0,T])\\ & ={\int }_{\{\left|X_t^{\phi }\right|\le \epsilon \text{ for every }t\in [0,T]\}}\exp (-{\int }_0^T\dot{\phi }(t)d{X}_t^{\phi }-{\int }_0^T\frac{1}{2}|\dot{\phi }(t){|}^{2}dt)d{P}^{\phi }.\end{array}$

By Girsanov's theorem, the distribution of Template:Math under Template:Math equals the distribution of Template:Mvar under Template:Mvar, hence the latter can be substituted by the former:

$P(|X_t-\phi (t)|\le \epsilon \text{ for every }t\in [0,T])={\int }_{\{\left|X_t^{\phi }\right|\le \epsilon \text{ for every }t\in [0,T]\}}\exp (-{\int }_0^T\dot{\phi }(t)d{X}_t-{\int }_0^T\frac{1}{2}|\dot{\phi }(t){|}^{2}dt)dP.$

By Itō's lemma it holds that

${\int }_0^T\dot{\phi }(t)d{X}_t=\dot{\phi }(T)X_T-{\int }_0^T\ddot{\phi }(t)X_{t}dt,$

where $\ddot{\phi }$ is the second derivative of Template:Mvar, and so this term is of order Template:Mvar on the event where Template:Math for every Template:Math and will disappear in the limit Template:Math, hence

$\underset{\epsilon \downarrow 0}{lim}\frac{P(|X_t-\phi (t)|\le \epsilon \text{ for every }t\in [0,T])}{P(|X_t|\le \epsilon \text{ for every }t\in [0,T])}=\exp (-{\int }_0^T\frac{1}{2}|\dot{\phi }(t){|}^{2}dt).$

Diffusion processes with constant diffusion coefficient on Euclidean space

The Onsager–Machlup function in the one-dimensional case with constant diffusion coefficient Template:Mvar is given by

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

In the Template:Mvar-dimensional case, with Template:Mvar equal to the unit matrix, it is given by

$L(x,v)=\frac{1}{2}\Vert v-b(x){\Vert }^2+\frac{1}{2}(\mathrm{div}b)(x),$

where Template:Math is the Euclidean norm and

$(\mathrm{div}b)(x)=\sum _{i=1}^d\frac{\partial }{\partial x_i}{b}_i(x).$

Generalizations

Generalizations have been obtained by weakening the differentiability condition on the curve Template:Mvar. 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 and Hölder, Besov and Sobolev type norms.

Applications

The Onsager–Machlup function can be used for purposes of reweighting and sampling trajectories, as well as for determining the most probable trajectory of a diffusion process.

See also

• Lagrangian
• Functional integration

Bibliography

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External links

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