The Onsager-Machlup (OM) functional is well-known for characterizing the most probable transition path of a diffusion process with non-vanishing noise. However, it suffers from a notorious issue that the functional is unbounded below when the specified transition time $T$ goes to infinity. This hinders the interpretation of the results obtained by minimizing the OM functional. We provide a new perspective on this issue. Under mild conditions, we show that although the infimum of the OM functional becomes unbounded when $T$ goes to infinity, the sequence of minimizers does contain convergent subsequences on the space of curves. The graph limit of this minimizing subsequence is an extremal of the abbreviated action functional, which is related to the OM functional via the Maupertuis principle with an optimal energy. We further propose an energy-climbing geometric minimization algorithm (EGMA) which identifies the optimal energy and the graph limit of the transition path simultaneously. This algorithm is successfully applied to several typical examples in rare event studies. Some interesting comparisons with the Freidlin-Wentzell action functional are also made.
The Numerical Onsager-Machlup action functional for Euler discretized SDEs driven by fractional Brownian motion
School of Sciences, Great Bay University（大湾大学理学院）；School of Mathematics, University of Science and Technology of China（中国科学技术大学数学系）；School of Mathematics, Sun Yat-sen University（中山大学数学学院）；Department of Mathematics and Department of Physics, Great Bay University（大湾大学数学系和物理系）；School of Mathematics, Southeast University（东南大学数学学院）
Yet another proof of Onsager-Machlup formula for diffusion processes on a Riemannian manifold after Takahashi-Watanabe, Fujita-Kotani. The proof is purely probabilistic and contains a precise study on an ergodic effect for a key Wiener functional.
This work is devoted to deriving the Onsager-Machlup action functional for stochastic partial differential equations with (non-Gaussian) Levy process as well as Gaussian Brownian motion. This is achieved by applying the Girsanov transformation for probability measures and then by a path representation. This enables the investigation of the most probable transition path for infinite dimensional stochastic dynamical systems modeled by stochastic partial differential equations, by minimizing the Onsager-Machlup action functional.
This paper is devoted to deriving the Onsager-Machlup action functional for Mckean-Vlasov stochastic differential equations in a class of norms that dominate $L^2([0,1], \mathbb{R}^d)$, such as supremum norm $\|\cdot\|_{\infty}$, H$\mathrm{\ddot{o}}$lder norms $\|\cdot\|_α$ with $α<\frac{1}{4}$ and $L^p$-norms with $p>4$ are included. Moreover, the corresponding Euler-Lagrange equation for Onsager-Machlup action functional is derived and a example is given.
The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a MAP estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager-Machlup functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory fo…
We derive Onsager-Machlup functionals for countable product measures on weighted $\ell^p$ subspaces of the sequence space $\mathbb{R}^{\mathbb{N}}$. Each measure in the product is a shifted and scaled copy of a reference probability measure on $\mathbb{R}$ that admits a sufficiently regular Lebesgue density. We study the equicoercivity and $Γ$-convergence of sequences of Onsager-Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussi…
We consider the solution of a stochastic differential equation with additive multidimensional fractional noise. In the case $\frac14<H<\frac12$, we compute the Onsager-Machlup functional (with respect to the driving fractional Brownian motion) for the supremum norm and the Hölder norms with exponent $α\in \left(0,H-\frac14\right)$ for any element of the Cameron-Martin space $\mathcal H_H$, extending a previous result of Moret and Nualart. In the more general case $H<\frac12$ and $α\in \left(0,H\right)$, we formulate a condition on $h\in\mathcal H_H$ under which the computation of the Onsager-M…
In this paper we will make the computation of the Onsager-Machlup functional of an inhomogeneous uniformly elliptic diffusion process. This functional will have formally the same picture as in the homogeneous case, the only difference come from the infinitesimal variation of the volume. For example in the Ricci flow case, we find some functional which is not so far to the $\mathcal L_{0} $ distance used by Lott to study this flow \cite{Lott:08}. We finish by a application to small ball probability for weighted sup norm, for inhomogeneous diffusion.
The definition and manipulation of Langevin equations with multiplicative white noise require special care (one has to specify the time discretisation and a stochastic chain rule has to be used to perform changes of variables). While discretisation-scheme transformations and non-linear changes of variable can be safely performed on the Langevin equation, these same transformations lead to inconsistencies in its path-integral representation. We identify their origin and we show how to extend the well-known Itō prescription ($dB^2=dt$) in a way that defines a modified stochastic calculus to be u…
Chemical reactions can be modelled via diffusion processes conditioned to make a transition between specified molecular configurations representing the state of the system before and after the chemical reaction. In particular the model of Brownian dynamics - gradient flow subject to additive noise - is frequently used. If the chemical reaction is specified to take place on a given time interval, then the most likely path taken by the system is a minimizer of the Onsager-Machlup functional. The Gamma limit of this functional is determined in the case where the temperature is small and the trans…
The emergence of transition phenomena between metastable states induced by noise plays a fundamental role in a broad range of nonlinear systems. The computation of the most probable paths is a key issue to understand the mechanism of transition behaviors. Shooting method is a common technique for this purpose to solve the Euler-Lagrange equation for the associated action functional, while losing its efficacy in high-dimensional systems. In the present work, we develop a machine learning framework to compute the most probable paths in the sense of Onsager-Machlup action functional theory. Speci…
