[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/onsager-machlup-function-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/onsager-machlup-function-wikipedia\/","headline":"Onsager\u2013Machlup function – Wikipedia","name":"Onsager\u2013Machlup function – Wikipedia","description":"The Onsager\u2013Machlup function is a function that summarizes the dynamics of a continuous stochastic process. It is used to define","datePublished":"2022-05-03","dateModified":"2022-05-03","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5d8ae2f4700784168c4ccba5f99906b0dfd19220","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5d8ae2f4700784168c4ccba5f99906b0dfd19220","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/onsager-machlup-function-wikipedia\/","about":["Wiki"],"wordCount":10164,"articleBody":"The Onsager\u2013Machlup 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\u00a0[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 equationdXt=b(Xt)dt+\u03c3(Xt)dWt{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 xi at a finite number of points in time ti:p(x1,\u2026,xn)=(\u220fi=1n\u2212112\u03c0\u03c3(xi)2\u0394ti)exp\u2061(\u2212\u2211i=1n\u22121L(xi,xi+1\u2212xi\u0394ti)\u0394ti){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)}whereL(x,v)=12(v\u2212b(x)\u03c3(x))2{displaystyle L(x,v)={frac {1}{2}}left({frac {v-b(x)}{sigma (x)}}right)^{2}}and \u0394ti = ti+1 \u2212 ti > 0, t1 = 0 and tn = T. A similar approximation is possible for processes in higher dimensions. The approximation is more accurate for smaller time step sizes \u0394ti, but in the limit \u0394ti \u2192 0 the probability density function becomes ill defined, one reason being that the product of terms12\u03c0\u03c3(xi)2\u0394ti{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 \u03b5 from smooth curves \u03c61 and \u03c62 are considered:[2]P(|Xt\u2212\u03c61(t)|\u2264\u03b5\u00a0for every\u00a0t\u2208[0,T])P(|Xt\u2212\u03c62(t)|\u2264\u03b5\u00a0for every\u00a0t\u2208[0,T])\u2192exp\u2061(\u2212\u222b0TL(\u03c61(t),\u03c6\u02d91(t))dt+\u222b0TL(\u03c62(t),\u03c6\u02d92(t))dt){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 \u03b5 \u2192 0, where L is the Onsager\u2013Machlup function.Table of ContentsDefinition[edit]Examples[edit]Wiener process on the real line[edit]Diffusion processes with constant diffusion coefficient on Euclidean space[edit]Generalizations[edit]Applications[edit]See also[edit]References[edit]Bibliography[edit]External links[edit]Definition[edit]Consider a d-dimensional Riemannian manifold M and a diffusion process X = {Xt\u00a0: 0 \u2264  t \u2264 T}  on M with infinitesimal generator 1\/2\u0394M + b, where \u0394M is the Laplace\u2013Beltrami operator and b is a vector field. For any two smooth curves \u03c61, \u03c62\u00a0: [0, T] \u2192 M,lim\u03b5\u21930P(\u03c1(Xt,\u03c61(t))\u2264\u03b5\u00a0for every\u00a0t\u2208[0,T])P(\u03c1(Xt,\u03c62(t))\u2264\u03b5\u00a0for every\u00a0t\u2208[0,T])=exp\u2061(\u2212\u222b0TL(\u03c61(t),\u03c6\u02d91(t))dt+\u222b0TL(\u03c62(t),\u03c6\u02d92(t))dt){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 \u03c1 is the Riemannian distance, \u03c6\u02d91,\u03c6\u02d92{displaystyle scriptstyle {dot {varphi }}_{1},{dot {varphi }}_{2}} denote the first derivatives of \u03c61, \u03c62, and L is called the Onsager\u2013Machlup function.The Onsager\u2013Machlup function is given by[3][4][5]L(x,v)=12\u2016v\u2212b(x)\u2016x2+12divb(x)\u2212112R(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),}where || \u22c5 ||x is the Riemannian norm in the tangent space Tx(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\u2013Machlup function of a continuous stochastic processes.Wiener process on the real line[edit]The Onsager\u2013Machlup function of a Wiener process on the real line R is given by[6]L(x,v)=12|v|2.{displaystyle L(x,v)={tfrac {1}{2}}|v|^{2}.}Proof: Let X = {Xt\u00a0: 0 \u2264 t \u2264 T}  be a Wiener process on R and let \u03c6\u00a0: [0, T] \u2192 R be a twice differentiable curve such that \u03c6(0) = X0. Define another process X\u03c6 = {Xt\u03c6\u00a0: 0 \u2264 t \u2264 T}  by Xt\u03c6 = Xt \u2212 \u03c6(t) and a measure P\u03c6 byP\u03c6=exp\u2061(\u222b0T\u03c6\u02d9(t)dXt\u03c6+\u222b0T12|\u03c6\u02d9(t)|2dt)dP.{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 \u03b5 > 0, the probability that |Xt \u2212 \u03c6(t)| \u2264 \u03b5 for every t \u2208 [0, T] satisfiesP(|Xt\u2212\u03c6(t)|\u2264\u03b5\u00a0for every\u00a0t\u2208[0,T])=P(|Xt\u03c6|\u2264\u03b5\u00a0for every\u00a0t\u2208[0,T])=\u222b{|Xt\u03c6|\u2264\u03b5\u00a0for every\u00a0t\u2208[0,T]}exp\u2061(\u2212\u222b0T\u03c6\u02d9(t)dXt\u03c6\u2212\u222b0T12|\u03c6\u02d9(t)|2dt)dP\u03c6.{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\u03c6 under P\u03c6 equals the distribution of X under P, hence the latter can be substituted by the former:P(|Xt\u2212\u03c6(t)|\u2264\u03b5\u00a0for every\u00a0t\u2208[0,T])=\u222b{|Xt\u03c6|\u2264\u03b5\u00a0for every\u00a0t\u2208[0,T]}exp\u2061(\u2212\u222b0T\u03c6\u02d9(t)dXt\u2212\u222b0T12|\u03c6\u02d9(t)|2dt)dP.{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 It\u014d’s lemma it holds that\u222b0T\u03c6\u02d9(t)dXt=\u03c6\u02d9(T)XT\u2212\u222b0T\u03c6\u00a8(t)Xtdt,{displaystyle int _{0}^{T}{dot {varphi }}(t),dX_{t}={dot {varphi }}(T)X_{T}-int _{0}^{T}{ddot {varphi }}(t)X_{t},dt,}where \u03c6\u00a8{displaystyle scriptstyle {ddot {varphi }}} is the second derivative of \u03c6, and so this term is of order \u03b5 on the event where |Xt| \u2264 \u03b5 for every t \u2208 [0, T] and will disappear in the limit \u03b5 \u2192 0, hencelim\u03b5\u21930P(|Xt\u2212\u03c6(t)|\u2264\u03b5\u00a0for every\u00a0t\u2208[0,T])P(|Xt|\u2264\u03b5\u00a0for every\u00a0t\u2208[0,T])=exp\u2061(\u2212\u222b0T12|\u03c6\u02d9(t)|2dt).{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\u2013Machlup function in the one-dimensional case with constant diffusion coefficient \u03c3 is given by[7]L(x,v)=12|v\u2212b(x)\u03c3|2+12dbdx(x).{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 \u03c3 equal to the unit matrix, it is given by[8]L(x,v)=12\u2016v\u2212b(x)\u20162+12(divb)(x),{displaystyle L(x,v)={frac {1}{2}}|v-b(x)|^{2}+{frac {1}{2}}(operatorname {div} ,b)(x),}where || \u22c5 || is the Euclidean norm and(divb)(x)=\u2211i=1d\u2202\u2202xibi(x).{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 \u03c6.[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 H\u00f6lder, Besov and Sobolev type norms.[11]Applications[edit]The Onsager\u2013Machlup 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]^ Onsager, L. and Machlup, S. (1953)^ Stratonovich, R. (1971)^ Takahashi, Y. and Watanabe, S. (1980)^ Fujita, T. and Kotani, S. (1982)^ Wittich, Olaf^ Ikeda, N. and Watanabe, S. (1980), Chapter VI, Section 9^ D\u00fcrr, D. and Bach, A. (1978)^ Ikeda, N. and Watanabe, S. (1980), Chapter VI, Section 9^ Zeitouni, O. (1989)^ Shepp, L. and Zeitouni, O. (1993)^ Capitaine, M. (1995)^ Adib, A.B. (2008).^ Adib, A.B. (2008).^ D\u00fcrr, D. and Bach, A. (1978).Bibliography[edit]Adib, A.B. (2008). “Stochastic actions for diffusive dynamics: Reweighting, sampling, and minimization”. J. Phys. Chem. B. 112 (19): 5910\u20135916. arXiv:0712.1255. doi:10.1021\/jp0751458. PMID\u00a017999482. S2CID\u00a016366252.Capitaine, M. (1995). “Onsager\u2013Machlup functional for some smooth norms on Wiener space”. Probab. Theory Relat. Fields. 102 (2): 189\u2013201. doi:10.1007\/bf01213388. S2CID\u00a0120675014.D\u00fcrr, D. & Bach, A. (1978). “The Onsager\u2013Machlup function as Lagrangian for the most probable path of a diffusion process”. Commun. Math. Phys. 60 (2): 153\u2013170. Bibcode:1978CMaPh..60..153D. doi:10.1007\/bf01609446. S2CID\u00a041249746.Fujita, T. & Kotani, S. (1982). “The Onsager\u2013Machlup function for diffusion processes”. J. Math. Kyoto Univ. 22: 115\u2013130. doi:10.1215\/kjm\/1250521863.Ikeda, N. & Watanabe, S. (1980). Stochastic differential equations and diffusion processes. Kodansha-John Wiley.Onsager, L. & Machlup, S. (1953). “Fluctuations and Irreversible Processes”. Physical Review. 91 (6): 1505\u20131512. Bibcode:1953PhRv…91.1505O. doi:10.1103\/physrev.91.1505.Shepp, L. & Zeitouni, O. (1993). Exponential estimates for convex norms and some applications. Progress in Probability. Vol.\u00a032. Berlin: Birkhauser-Verlag. pp.\u00a0203\u2013215. CiteSeerX\u00a010.1.1.28.8641. doi:10.1007\/978-3-0348-8555-3_11. ISBN\u00a0978-3-0348-9677-1.Stratonovich, R. (1971). “On the probability functional of diffusion processes”. Select. Transl. In Math. Stat. Prob. 10: 273\u2013286.Takahashi, Y.; Watanabe, S. (1981). “The probability functionals (Onsager\u2013Machlup functions) of diffusion processes”. Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lecture Notes in Mathematics. Vol.\u00a0851. Berlin: Springer. pp.\u00a0433\u2013463. doi:10.1007\/BFb0088735. MR\u00a00620998.Wittich, Olaf. “The Onsager\u2013Machlup Functional Revisited”. Zeitouni, O. (1989). “On the Onsager\u2013Machlup functional of diffusion processes around non C2 curves”. Annals of Probability. 17 (3): 1037\u20131054. doi:10.1214\/aop\/1176991255.External links[edit]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/onsager-machlup-function-wikipedia\/#breadcrumbitem","name":"Onsager\u2013Machlup function – Wikipedia"}}]}]