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Onsager–Machlup function
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<h2 id="definition">Definition</h2> <p>Consider a d-dimensional <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> M and a <a href="/facts/Diffusion_process/n7u5hjKX">diffusion process</a> <i>X</i> = {<i>Xt</i> : 0 ≤ <i>t</i> ≤ <i>T</i>} on M with <a href="/facts/Infinitesimal_generator_(stochastic_processes)/EUHczpU5">infinitesimal generator</a> 1/2Δ<i>M</i> + <i>b</i>, where Δ<i>M</i> is the <a href="/facts/Laplace%25E2%2580%2593Beltrami_operator/AJmyUpWz">Laplace–Beltrami operator</a> and b is a <a href="/facts/Vector_field/ccFNuPPp">vector field</a>. For any two <a href="/facts/Smooth_function/mffL4ch2">smooth</a> curves <i>φ</i>1, <i>φ</i>2 : [0, <i>T</i>] → <i>M</i>, </p> lim ε ↓ 0 P ( ρ ( X t , φ 1 ( t ) ) ≤ ε for every t ∈ [ 0 , T ] ) P ( ρ ( X t , φ 2 ( t ) ) ≤ ε for every t ∈ [ 0 , T ] ) = exp ( − ∫ 0 T L ( φ 1 ( t ) , φ ˙ 1 ( t ) ) d t + ∫ 0 T L ( φ 2 ( t ) , φ ˙ 2 ( t ) ) d t ) {\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)\,dt+\int _{0}^{T}L\left(\varphi _{2}(t),{\dot {\varphi }}_{2}(t)\right)\,dt\right)} <p>where ρ is the <a href="/facts/Metric_(mathematics)/RkUptSo8">Riemannian distance</a>, φ ˙ 1 , φ ˙ 2 {\displaystyle \scriptstyle {\dot {\varphi }}_{1},{\dot {\varphi }}_{2}} denote the first <a href="/facts/Derivative/7Ito70Dh">derivatives</a> of <i>φ</i>1, <i>φ</i>2, and L is called the Onsager–Machlup function. </p><p>The Onsager–Machlup function is given by<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> L ( x , v ) = 1 2 ‖ v − b ( x ) ‖ x 2 + 1 2 div b ( x ) − 1 12 R ( x ) , {\displaystyle L(x,v)={\tfrac {1}{2}}\|v-b(x)\|_{x}^{2}+{\tfrac {1}{2}}\operatorname {div} \,b(x)-{\tfrac {1}{12}}R(x),} <p>where || ⋅ ||<i>x</i> is the Riemannian norm in the <a href="/facts/Tangent_space/crvpbSeG">tangent space</a> <i>Tx</i>(<i>M</i>) at x, div <i>b</i>(<i>x</i>) is the <a href="/facts/Divergence/zpwwkFoU">divergence</a> of b at x, and <i>R</i>(<i>x</i>) is the <a href="/facts/Scalar_curvature/tfgm3AuK">scalar curvature</a> at x. </p> <h2 id="examples">Examples</h2> <p>The following examples give explicit expressions for the Onsager–Machlup function of a continuous stochastic processes. </p> <h3>Wiener process on the real line</h3> <p>The Onsager–Machlup function of a <a href="/facts/Wiener_process/KPh7vYd7">Wiener process</a> on the <a href="/facts/Real_line/6eq42xX5">real line</a> R is given by<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> L ( x , v ) = 1 2 | v | 2 . {\displaystyle L(x,v)={\tfrac {1}{2}}|v|^{2}.} <p>Proof: Let <i>X</i> = {<i>Xt</i> : 0 ≤ <i>t</i> ≤ <i>T</i>} be a Wiener process on R and let <i>φ</i> : [0, <i>T</i>] → R be a twice differentiable curve such that <i>φ</i>(0) = <i>X</i>0. Define another process <i>Xφ</i> = {<i>Xtφ</i> : 0 ≤ <i>t</i> ≤ <i>T</i>} by <i>Xtφ</i> = <i>Xt</i> − <i>φ</i>(<i>t</i>) and a <a href="/facts/Measure_(mathematics)/yCq7zlI4">measure</a> <i>Pφ</i> by </p> P φ = exp ( ∫ 0 T φ ˙ ( t ) d X t φ + ∫ 0 T 1 2 | φ ˙ ( t ) | 2 d t ) d P . {\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}\,dt\right)\,dP.} <p>For every <i>ε</i> > 0, the probability that |<i>Xt</i> − <i>φ</i>(<i>t</i>)| ≤ <i>ε</i> for every <i>t</i> ∈ [0, <i>T</i>] satisfies </p> P ( | X t − φ ( t ) | ≤ ε for every t ∈ [ 0 , T ] ) = P ( | X t φ | ≤ ε for every t ∈ [ 0 , T ] ) = ∫ { | X t φ | ≤ ε for every t ∈ [ 0 , T ] } exp ( − ∫ 0 T φ ˙ ( t ) d X t φ − ∫ 0 T 1 2 | φ ˙ ( t ) | 2 d t ) d P φ . {\displaystyle {\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}}} <p>By <a href="/facts/Girsanov_theorem/7WWj3pPk">Girsanov's theorem</a>, the distribution of <i>Xφ</i> under <i>Pφ</i> equals the distribution of X under P, hence the latter can be substituted by the former: </p> P ( | X t − φ ( t ) | ≤ ε for every t ∈ [ 0 , T ] ) = ∫ { | X t φ | ≤ ε for every t ∈ [ 0 , T ] } exp ( − ∫ 0 T φ ˙ ( t ) d X t − ∫ 0 T 1 2 | φ ˙ ( t ) | 2 d t ) d P . {\displaystyle 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.} <p>By <a href="/facts/It%25C5%258D%2527s_lemma/NF7tTcgd">Itō's lemma</a> it holds that </p> ∫ 0 T φ ˙ ( t ) d X t = φ ˙ ( T ) X T − ∫ 0 T φ ¨ ( t ) X t d t , {\displaystyle \int _{0}^{T}{\dot {\varphi }}(t)\,dX_{t}={\dot {\varphi }}(T)X_{T}-\int _{0}^{T}{\ddot {\varphi }}(t)X_{t}\,dt,} <p>where φ ¨ {\displaystyle \scriptstyle {\ddot {\varphi }}} is the second derivative of φ, and so this term is of order ε on the event where |<i>Xt</i>| ≤ ε for every <i>t</i> ∈ [0, <i>T</i>] and will disappear in the limit <i>ε</i> → 0, hence </p> lim ε ↓ 0 P ( | X t − φ ( t ) | ≤ ε for every t ∈ [ 0 , T ] ) P ( | X t | ≤ ε for every t ∈ [ 0 , T ] ) = exp ( − ∫ 0 T 1 2 | φ ˙ ( t ) | 2 d t ) . {\displaystyle \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).} <h3>Diffusion processes with constant diffusion coefficient on Euclidean space</h3> <p>The Onsager–Machlup function in the one-dimensional case with constant <a href="/facts/It%25C5%258D_diffusion/vlwAqWSU">diffusion coefficient</a> σ is given by<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> L ( x , v ) = 1 2 | v − b ( x ) σ | 2 + 1 2 d b d x ( x ) . {\displaystyle L(x,v)={\frac {1}{2}}\left|{\frac {v-b(x)}{\sigma }}\right|^{2}+{\frac {1}{2}}{\frac {db}{dx}}(x).} <p>In the d-dimensional case, with σ equal to the unit matrix, it is given by<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> L ( x , v ) = 1 2 ‖ v − b ( x ) ‖ 2 + 1 2 ( div b ) ( x ) , {\displaystyle L(x,v)={\frac {1}{2}}\|v-b(x)\|^{2}+{\frac {1}{2}}(\operatorname {div} \,b)(x),} <p>where || ⋅ || is the <a href="/facts/Euclidean_norm/R2UbzmzM">Euclidean norm</a> and </p> ( div b ) ( x ) = ∑ i = 1 d ∂ ∂ x i b i ( x ) . {\displaystyle (\operatorname {div} \,b)(x)=\sum _{i=1}^{d}{\frac {\partial }{\partial x_{i}}}b_{i}(x).} <h2 id="generalizations">Generalizations</h2> <p>Generalizations have been obtained by weakening the differentiability condition on the curve φ.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> 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<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> and Hölder, Besov and Sobolev type norms.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> <h2 id="applications">Applications</h2> <p>The Onsager–Machlup function can be used for purposes of reweighting and <a href="/facts/Sampling_(statistics)/kIb01xdL">sampling</a> trajectories,<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> as well as for determining the most probable trajectory of a diffusion process.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Lagrangian_(field_theory)/p9BCUz57">Lagrangian</a></li> <li><a href="/facts/Functional_integration/VBGTnDzq">Functional integration</a></li></ul> <h2 id="bibliography">Bibliography</h2> <ul><li>Adib, A.B. (2008). "Stochastic actions for diffusive dynamics: Reweighting, sampling, and minimization". <i>J. Phys. Chem. B</i>. 112 (19): 5910–5916. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0712.1255">0712.1255</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1021%2Fjp0751458">10.1021/jp0751458</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/17999482">17999482</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:16366252">16366252</a>.</li> <li>Capitaine, M. (1995). <a href="https://doi.org/10.1007%2Fbf01213388">"Onsager–Machlup functional for some smooth norms on Wiener space"</a>. <i>Probab. Theory Relat. Fields</i>. 102 (2): 189–201. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01213388">10.1007/bf01213388</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120675014">120675014</a>.</li> <li>Dürr, D. & Bach, A. (1978). <a href="http://projecteuclid.org/euclid.cmp/1103904077">"The Onsager–Machlup function as Lagrangian for the most probable path of a diffusion process"</a>. <i>Commun. Math. Phys</i>. 60 (2): 153–170. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1978CMaPh..60..153D">1978CMaPh..60..153D</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01609446">10.1007/bf01609446</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:41249746">41249746</a>.</li> <li>Fujita, T. & Kotani, S. (1982). <a href="https://doi.org/10.1215%2Fkjm%2F1250521863">"The Onsager–Machlup function for diffusion processes"</a>. <i>J. Math. Kyoto Univ</i>. 22: 115–130. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1215%2Fkjm%2F1250521863">10.1215/kjm/1250521863</a>.</li> <li>Ikeda, N. & Watanabe, S. (1980). <i>Stochastic differential equations and diffusion processes</i>. Kodansha-John Wiley.</li> <li>Onsager, L. & Machlup, S. (1953). "Fluctuations and Irreversible Processes". <i>Physical Review</i>. 91 (6): 1505–1512. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1953PhRv...91.1505O">1953PhRv...91.1505O</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2Fphysrev.91.1505">10.1103/physrev.91.1505</a>.</li> <li>Shepp, L. & Zeitouni, O. (1993). "Exponential estimates for convex norms and some applications". <i>Barcelona Seminar on Stochastic Analysis</i>. Vol. 32. Berlin: Birkhauser-Verlag. pp. 203–215. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.8641">10.1.1.28.8641</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-0348-8555-3_11">10.1007/978-3-0348-8555-3_11</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-0348-9677-1. {{cite book}}: |journal= ignored (help)CS1 maint: location missing publisher (link)</li> <li><a href="/facts/Ruslan_Stratonovich/4xUGIjrm">Stratonovich, R.</a> (1971). "On the probability functional of diffusion processes". <i>Select. Transl. In Math. Stat. Prob</i>. 10: 273–286.</li> <li>Takahashi, Y.; Watanabe, S. (1981). "The probability functionals (Onsager–Machlup functions) of diffusion processes". <i>Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980)</i>. Lecture Notes in Mathematics. Vol. 851. Berlin: Springer. pp. 433–463. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0088735">10.1007/BFb0088735</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-10690-6. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0620998">0620998</a>.</li> <li>Wittich, Olaf. "The Onsager–Machlup Functional Revisited". {{cite journal}}: Cite journal requires |journal= (help)</li> <li>Zeitouni, O. (1989). <a href="https://doi.org/10.1214%2Faop%2F1176991255">"On the Onsager–Machlup functional of diffusion processes around non <i>C</i>2 curves"</a>. <i>Annals of Probability</i>. 17 (3): 1037–1054. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1214%2Faop%2F1176991255">10.1214/aop/1176991255</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li>Onsager–Machlup function. Encyclopedia of Mathematics. URL: <a href="http://www.encyclopediaofmath.org/index.php?title=Onsager-Machlup_function&oldid=22857">http://www.encyclopediaofmath.org/index.php?title=Onsager-Machlup_function&oldid=22857</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Onsager, L. and Machlup, S. (1953) <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Stratonovich, R. (1971) <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Takahashi, Y. and Watanabe, S. (1980) <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Fujita, T. and Kotani, S. (1982) <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Wittich, Olaf <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Ikeda, N. and Watanabe, S. (1980), Chapter VI, Section 9 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Dürr, D. and Bach, A. (1978) <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Ikeda, N. and Watanabe, S. (1980), Chapter VI, Section 9 <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Zeitouni, O. (1989) <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Shepp, L. and Zeitouni, O. (1993) <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Capitaine, M. (1995) <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Adib, A.B. (2008). <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Adib, A.B. (2008). <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Dürr, D. and Bach, A. (1978). <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> </ol>