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Analytical (rational) mechanics is the mathematical structure of Newtonian deterministic dynamics developed by D'Alembert, Langrange, Hamilton, Jacobi, and many other luminaries of applied mathematics. Diffusion as a stochastic process of…
The trajectories of diffusion processes are continuous but non-differentiable, and each occurs with vanishing probability. This introduces a gap between theory, where path probabilities are used in many contexts, and experiment, where only…
For diffusive stochastic dynamics, the probability to observe any individual trajectory is vanishingly small, making it unclear how to experimentally validate theoretical results for ratios of path probabilities. We provide the missing link…
We develop an information-theoretic formulation of stochastic dynamics in which the fundamental stochastic variable is the total action connecting spacetime points, rather than individual paths. By maximizing Shannon entropy over a joint…
The path probability of a particle undergoing stochastic motion is studied by the use of functional technique, and the general formula is derived for the path probability distribution functional. The probability of finding paths inside a…
Transition of a system between two states is an important but difficult problem in natural science. In this article we study the transition problem in the framework of transition path ensemble. Using the overdamped Langevin method, we…
The escape probability is a deterministic concept that quantifies some aspects of stochastic dynamics. This issue has been investigated previously for dynamical systems driven by Gaussian Brownian motions. The present work considers escape…
We introduce a stochastic optimal transport for the Langevin dynamics with positive mass and study its zero--mass limit. The new aspect of this paper is that we only fix the initial and terminal probability distributions of the positions of…
A statistical model of quantization based on an exponential distribution of infinitesimal action is proposed. Trajectory which does not extremize the action along an infinitesimal short segment of path is allowed to occur with a very small…
The evaluation of the path-integral representation for stochastic processes in the weak-noise limit shows that these systems are governed by a set of equations which are those of a classical dynamics. We show that, even when the noise is…
In physics, there is a scalar function called the action which behaves like a cost function. When minimized, it yields the "path of least action" which represents the path a physical system will take through space and time. This function is…
Cell state determination is the outcome of intrinsically stochastic biochemical reactions. Tran- sitions between such states are studied as noise-driven escape problems in the chemical species space. Escape can occur via multiple possible…
This work is devoted to the investigation of the most probable transition time between metastable states for stochastic dynamical systems. Such a system is modeled by a stochastic differential equation with non-vanishing Brownian noise, and…
Recently, progress has been made in the theory of turbulence, which provides a framework on how a deterministic process changes to a stochastic one owing to the change in thermodynamic states. It is well known that, in the framework of…
We present certain mathematical aspects of an information method which was formulated in an attempt to investigate diffusion phenomena. We imagine a regular dynamical hamiltonian systems under the random perturbation of thermal (molecular)…
Flow matching trains a neural velocity field by regression against a target velocity associated with a prescribed probability path connecting a simple initial distribution to the data distribution. A central design choice is the path…
Machine Learning algorithms are typically regarded as appropriate optimization schemes for minimizing risk functions that are constructed on the training set, which conveys statistical flavor to the corresponding learning problem. When the…
We propose a generalization of classical statistical mechanics which describes the behavior of dissipative systems placed in contact with a heat bath. In contrast to conventional statistical mechanics, which assigns probabilities to the…
By optimal fluctuation method, we study short-time distribution $P(\mathcal{A}=A)$ of the functionals, $\mathcal{A}=\int_{0}^{t_f} x^n(t) dt$, along constrained trajectories of random acceleration process for a given time duration $t_f$,…
Maximum entropy (maxEnt) inference of state probabilities using state-dependent constraints is popular in the study of complex systems. In stochastic dynamical systems, the effect of state space topology and path-dependent constraints on…