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We introduce the probability distributions describing quantum observables in conventional quantum mechanics and clarify their relations to the tomographic probability distributions describing quantum states. We derive the evolution equation…
One of the most important problems in system identification and statistics is how to estimate the unknown parameters of a given model. Optimization methods and specialized procedures, such as Empirical Minimization (EM) can be used in case…
We calculate crossing probabilities and one-sided last exit time densities for a class of moving barriers on an interval $[0,T]$ via Schwartz distributions. We derive crossing probabilities and first hitting time densities for another class…
In trajectory forecasting tasks for traffic, future output trajectories can be computed by advancing the ego vehicle's state with predicted actions according to a kinematics model. By unrolling predicted trajectories via time integration…
This work is a numerical experiment of stochastic motion of conservative Hamiltonian system or weakly damped Brownian particles. The objective is to prove the existence of path probability and to compute its values. By observing a large…
One key issue in the probability density function (PDF) approach for disperse two-phase turbulent flows is to close the diffusion term in the phase space. This study aimed to derive a kinetic equation for particle dispersion in turbulent…
We propose an approach to compute the boundary crossing probabilities for a class of diffusion processes which can be expressed as piecewise monotone (not necessarily one-to-one) functionals of a standard Brownian motion. This class…
This paper presents a mesoscopic traffic flow model that explicitly describes the spatio-temporal evolution of the probability distributions of vehicle trajectories. The dynamics are represented by a sequence of factor graphs, which enable…
The theory of slow manifolds is an important tool in the study of deterministic dynamical systems, giving a practical method by which to reduce the number of relevant degrees of freedom in a model, thereby often resulting in a considerable…
Modeling and simulating movement of vehicles in established transportation infrastructures, especially in large urban road networks is an important task. It helps with understanding and handling traffic problems, optimizing traffic…
In a physical system, changing parameters such as temperature can induce a phase transition: an abrupt change from one state of matter to another. Analogous phenomena have recently been observed in large language models. Typically, the task…
A method yielding simple relationships among bilateral birth-and-death processes is outlined. This allows one to relate birth and death rates of two processes in such a way that their transition probabilities, first-passage-time densities…
Markov cohort state-transition models have been the standard approach for simulating the prognosis of patients or, more generally, the life trajectories of individuals over a time period. Current approaches for estimating the variance of a…
This paper establishes a Transition Path Theory (TPT) for L\'{e}vy-type processes, addressing a critical gap in the study of the transition mechanism between meta-stabile states in non-Gaussian stochastic systems. A key contribution is the…
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…
In a general class of one dimensional random differential equation the convergence of the distribution function of the solution to stationary state distribution is studied. In particular it is proved the boundedness respectively the…
We investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions. We accomplish such a program in the case of two-dimensional…
We consider a class of stochastic control problems which has been widely used in optimal foraging theory. The state processes have two distinct dynamics, characterized by two pairs of drift and diffusion coefficients, depending on whether…
The master equation approach is proposed to describe the evolution of passengers in a subway system. With the transition rate constructed from simple geographical consideration, the evolution equation for the distribution of subway…
In this letter we present an analytic method for calculating the transition probability between two random Gaussian matrices with given eigenvalue spectra in the context of Dyson Brownian motion. We show that in the Coulomb gas language, in…