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The '''Onsager–Machlup function''' is a function that summarizes the dynamics of a [[stochastic process#Continuous time and continuous state space|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 [[S. Machlup]] who were the first to consider such probability densities.<ref>Onsager, L. and Machlup, S. (1953)</ref>

The dynamics of a continuous stochastic process {{math|''X''}} from time {{math|''t'' {{=}} 0}} to {{math|''t'' {{=}} ''T''}} in one dimension, satisfying a [[stochastic differential equation]]

:<math>
dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t
</math>

where {{math|''W''}} is a [[Wiener process]],
can in approximation be described by the [[probability density function]] of its value {{math|''x''''i''}} at a finite number of points in time {{math|''t''''i''}}:

:<math>
p(x_1,\ldots,x_n) =
\left(
\prod^{n-1}_{i=1} \frac{1}{\sqrt{2\pi\sigma(x_i)^2\Delta t_i}}
\right)
\exp\left(
- \sum^{n-1}_{i=1} L\left(x_i,\frac{x_{i+1}-x_i}{\Delta t_i}\right) \, \Delta t_i
\right)
</math>

where

:<math>
L(x,v) = \frac{1}{2}\left(\frac{v - b(x)}{\sigma}\right)^2
</math>

and {{math|Δ''t''''i'' {{=}} ''t''''i''+1 − ''t''''i'' > 0}}, {{math|''t''1 {{=}} 0}} and {{math|''t''''n'' {{=}} ''T''}}. A similar approximation is possible for processes in higher dimensions. The approximation is more accurate for smaller time step sizes {{math|Δ''t''''i''}}, but in the limit where the time step sizes go to zero the probability density function becomes ill defined, one reason being that the product of terms <math>1/\sqrt{2\pi\sigma(x_i)^2\Delta t_i}</math> [[limit of a sequence#Infinite limits|diverges to infinity]]. In order to nevertheless define a density for the continuous stochastic process {{math|''X''}}, [[ratio]]s of probabilities of {{math|''X''}} lying within a small distance {{math|ε}} from [[smooth function|smooth]] curves {{math|φ1}} and {{math|φ2}} are considered:<ref>Stratonovich, R. (1971)</ref>

:<math>
\frac{
P\left( |X_t - \phi_1(t)| \leq \varepsilon \text{ for every }t\in[0,T] \right)
}{
P\left( |X_t - \phi_2(t)| \leq \varepsilon \text{ for every }t\in[0,T] \right)
}
</math>

[[Convergence (mathematics)|converges]] to

:<math>
\exp\left(
- \int^T_0 L(\phi_1(t),\dot{\phi}_1(t)) \, dt
+ \int^T_0 L(\phi_2(t),\dot{\phi}_2(t)) \, dt
\right)
</math>

as {{math|''ε''}} goes to zero, where {{math|''L''}} is the '''Onsager–Machlup function'''.

==Definition==

Consider a {{math|''d''}}-dimensional [[Riemannian manifold]] {{math|''M''}}
and a [[diffusion process]] {{math|''X'' {{=}} {''Xt'' : 0 ≤ ''t'' ≤ ''T'' }}}
on {{math|''M''}}
with [[infinitesimal generator (stochastic processes)|infinitesimal generator]]
{{math|<math>\textstyle\frac{1}{2}</math>Δ''M'' + ''b''}},
where {{math|Δ''M''}} is the [[Laplace–Beltrami operator]]
and {{math|''b''}} is a [[vector field]].
For any two [[smooth function|smooth]] curves
{{math|''φ''1:[0,''T''] → ''M''}} and {{math|''φ''2:[0,''T''] → ''M''}},

:<math>
\begin{align}
&
\lim_{\varepsilon\downarrow0}
\frac{
P\left( \rho(X_t,\phi_1(t)) \leq \varepsilon \text{ for every }t\in[0,T] \right)
}{
P\left( \rho(X_t,\phi_2(t)) \leq \varepsilon \text{ for every }t\in[0,T] \right)
}
\\
&=
\exp\left(
-\int^T_0 L(\phi_1(t),\dot{\phi}_1(t)) \, dt
+\int^T_0 L(\phi_2(t),\dot{\phi}_2(t)) \, dt
\right)
\end{align}
</math>

where ρ is the [[metric (mathematics)|Riemannian distance]], {{math|<math>\scriptstyle \dot{\phi}</math>1}} and {{math|<math>\scriptstyle \dot{\phi}</math>2}}
denote the first [[derivative]]s of {{math|''φ''1}} and {{math|''φ''2}},
and {{math|''L''}} is called the '''Onsager–Machlup function'''.

The Onsager–Machlup function is given by
<ref>Takahashi, Y. and Watanabe, S. (1980)</ref>
<ref>Fujita, T. and Kotani, S. (1982)</ref>
<ref>Wittich, Olaf</ref>

:<math>
L(x,v)
= \tfrac{1}{2}\|v-b(x)\|_x^2
+\tfrac{1}{2}\operatorname{div}\, b(x)
- \tfrac{1}{12}R(x),
</math>

where <math>\scriptstyle \|\,\cdot\,\|</math>{{math|''x''}}
is the Riemannian norm in the [[tangent space]] {{math|''T''''x''(''M'')}} at {{math|''x''}}, {{math|div ''b''(''x'')}} is the [[divergence]] of {{math|''b''}} at {{math|''x''}}, and {{math|''R''(''x'')}} is the [[scalar curvature]] at {{math|''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<ref>Ikeda, N. and Watanabe, S. (1980), Chapter VI, Section 9</ref>
:<math>
L(x,v)=\tfrac{1}{2}|v|^2.
</math>

<div class="NavFrame collapsed">
<div class="NavHead">[Proof]</div>
<div class="NavContent" style="text-align:left">

Let {{math|''X'' {{=}} {''X''''t'' : 0 ≤ ''t'' ≤ ''T'' }}} be a Wiener process on '''R''' and let {{math|''φ'':[0,''T''] → '''R'''}} be a twice differentiable curve such that {{math|''φ''(0) {{=}} ''X''0}}.

Define another process
{{math|''X''''φ'' {{=}} {''X''''t''''φ'' : 0 ≤ ''t'' ≤ ''T'' }}}
by {{math|''X''''t''''φ'' {{=}} ''Xt'' − ''φ''(''t'')}}
and a [[measure (mathematics)|measure]] {{math|''P''''φ''}} by

:<math>
P^\phi =
\exp\left(
\int^T_0\dot{\phi}(t) \, dX^\phi_t
+ \int^T_0\tfrac{1}{2}|\dot{\phi}(t)|^2 \, dt
\right) \, dP.
</math>

For every {{math|''ε'' > 0}}, the probability that {{math|{{!}}''X''''t'' − ''φ''''t''{{!}} ≤ ''ε''}} for every {{math|''t'' ∈ [0,''T'']}}
satisfies

:<math>
\begin{align}
&P(|X_t-\phi(t)|\leq\varepsilon \text{ for every }t\in[0,T])
\\[6pt]
&=P(|X^\phi_t|\leq\varepsilon \text{ for every }t\in[0,T])
\\[6pt]
&=\int_{\{|X^\phi_t|\leq\varepsilon\text{ for every }t\in[0,T]\}}
\exp\left(
-\int^T_0\dot{\phi}(t) \, dX^\phi_t
-\int^T_0\tfrac{1}{2}|\dot{\phi}(t)|^2 \, dt
\right) \, dP^\phi.
\end{align}
</math>

By [[Girsanov theorem|Girsanov's theorem]], the distribution of {{math|''Xφ''}} under {{math|''Pφ''}} equals the distribution of {{math|''X''}} under {{math|''P''}}, hence the latter can be substituted by the former:

:<math>
\begin{align}
&
P(|X_t-\phi(t)|\leq\varepsilon \text{ for every }t\in[0,T])
\\
&=\int_{\{|X_t|\leq\varepsilon\text{ for every }t\in[0,T]\}}
\exp\left(
-\int^T_0\dot{\phi}(t) \, dX_t
-\int^T_0\tfrac{1}{2}|\dot{\phi}(t)|^2 \, dt
\right) \, dP.
\end{align}
</math>

By [[Itō's lemma]] it holds that

:<math>
\int^T_0\dot{\phi}(t) \, dX_t = \dot{\phi}(T)X_T - \int^T_0\ddot{\phi}(t)X_t \, dt,
</math>

where <math>\scriptstyle \ddot{\phi}</math> is the second derivative of {{math|''φ''}},
and so this term is of order {{math|ε}} on the event
where {{math|{{!}}''Xt''{{!}} ≤ ε}} for every {{math|''t'' ∈ [0, ''T'']}}
and will disappear in the limit where {{math|ε}} goes to zero, hence
:<math>
\lim_{\varepsilon\downarrow 0}
\frac{
P(|X_t-\phi(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^T_0\tfrac{1}{2}|\dot{\phi}(t)|^2 \, dt
\right).
</math>
</div>
</div>

===Diffusion processes with constant diffusion coefficient on Euclidean space===

The Onsager–Machlup function in the one-dimensional case with constant [[Itō diffusion|diffusion coefficient]] {{math|σ}} is given by<ref>Dürr, D. and Bach, A. (1978)</ref>

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

In the {{math|''d''}}-dimensional case,
with {{math|σ}} equal to the unit matrix,
it is given by<ref>Ikeda, N. and Watanabe, S. (1980), Chapter VI, Section 9</ref>

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

where <math>\scriptstyle \|\,\cdot\,\|</math> is the [[Euclidean norm]] and
{{math|
(div ''b'')(''x'') {{=}} ∑''d''''i''{{=}}1
<math>
\textstyle\frac{1}{2}\frac{\partial}{\partial x_i}
</math>
''b''''i''(''x'')
}}.

==Generalizations==

Generalizations have been obtained by weakening the differentiability condition
on the curve {{math|''φ''}}.<ref>Zeitouni, O. (1989)</ref>
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<ref>Shepp, L. and Zeitouni, O. (1993)</ref> and Hölder, Besov and Sobolev type norms.<ref>Capitaine, M. (1995)</ref>

==Applications==

The Onsager–Machlup function can be used for purposes of reweighting and [[Sampling (statistics)|sampling]] trajectories,<ref>Adib, A.B. (2008).</ref>
as well as for determining the most probable trajectory of a diffusion process.<ref>Adib, A.B. (2008).</ref><ref>Dürr, D. and Bach, A. (1978).</ref>

==See also==

* [[Lagrangian]]
* [[Functional integration]]

==References==
{{Reflist|30em}}

==Bibliography==
{{refbegin|30em}}
* {{Cite journal | author = Adib, A.B. | year = 2008 | title = Stochastic actions for diffusive dynamics: Reweighting, sampling, and minimization | journal = J. Phys. Chem. B | volume = 112 | pages = 5910–5916}}
* {{Cite journal | author = Capitaine, M. | year = 1995 | title = Onsager–Machlup functional for some smooth norms on Wiener space | journal = Probab. Theory Relat. Fields | volume = 102 | pages = 189–201}}
* {{Cite journal | author = Dürr, D. and Bach, A. | year = 1978 | title = The Onsager–Machlup function as Lagrangian for the most probable path of a diffusion process | journal = Commun. Math. Phys. | volume = 60 | pages = 153–170}}
* {{Cite journal | author = Fujita, T. and Kotani, S. | year = 1982 | title = The Onsager–Machlup function for diffusion processes | journal = J. Math. Kyoto Univ. | volume = 22 | pages = 115–130}}
* {{Cite book | author = Ikeda, N. and Watanabe, S.| year = 1980 | title = Stochastic differential equations and diffusion processes | publisher = Kodansha-John Wiley}}
* {{Cite journal | author = Onsager, L. and Machlup, S. | year = 1953 | title = Fluctuations and Irreversible Processes | journal = Physical Review | volume = 91| number = 6 | pages = 1505–1512}}
* {{Cite journal | author = Shepp, L. and Zeitouni, O. | year = 1993 | title = Exponential estimates for convex norms and some applications | journal = Progress in Probability | volume = 32 | pages = 203–215 | location = Berlin. Birkhauser-Verlag}}
* {{Cite journal | author = [[Ruslan Stratonovich|Stratonovich, R.]] | year = 1971 | title = On the probability functional of diffusion processes | journal = Select. Transl. in Math. Stat. Prob. | volume = 10 | pages = 273–286}}
* {{Cite journal | author = Takahashi, Y. and Watanabe, S. | year = 1980 | title = The probability functionals (Onsager–Machlup functions) of diffusion processes | journal = Springer Lecture Notes in Math. | volume = 851| pages = 432–463}}
* {{Cite journal | author = Wittich, Olaf | title = The Onsager–Machlup Functional Revisited}}
* {{Cite journal | author = Zeitouni, O. | year = 1989 | title = On the Onsager–Machlup functional of diffusion processes around non {{math|''C''2}} curves | journal = Annals of Probability | volume = 17 | number = 3 | pages = 1037–1054}}
{{refend}}

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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

{{DEFAULTSORT:Onsager-Machlup function}}
[[Category:Functional analysis]]
[[Category:Functions and mappings]]
[[Category:Stochastic processes]]

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