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The exact closed equation of motion for microscopic distribution function of classical many-body system with account of interactions retardation between particles is derived. It is shown that interactions retardation leads to irreversible…
This note considers fairly general quasi-homogeneous systems of first-order nonlinear ODEs and homogeneous systems of second-order nonlinear ODEs that contain arbitrary functions of several arguments. It presents several exact solutions to…
The long-time behavior of stochastic Hamilton-Jacobi equations is analyzed, including the stochastic mean curvature flow as a special case. In a variety of settings, new and sharpened results are obtained. Among them are (i) a…
In finite-dimensional dynamical systems, stochastic stability provides the selection of physical relevant measures from the myriad invariant measures of conservative systems. That this might also apply to infinite-dimensional systems is the…
Stochastic processes are proposed whose master equations coincide with classical wave, telegraph, and Klein-Gordon equations. Similar to predecessors based on the Goldstein-Kac telegraph process, the model describes the motion of particles…
We study the instability properties of Nesterov's ODE in non-conservative settings, where the driving term is not necessarily the gradient of a potential function. While convergence properties under Nesterov's ODE are well-characterized for…
In this paper, the successive approximation method is applied to investigate the existence and uniqueness of solutions to the stochastic differential equations (SDEs) driven by L\'evy noise under non-Lipschitz condition which is a much…
We introduce and analyse a continuum model for an interacting particle system of Vicsek type. The model is given by a non-linear kinetic partial differential equation (PDE) describing the time-evolution of the density $f_t$, in the single…
This article develops a stochastic differential equation (SDE) for modeling the temporal evolution of queue length dynamics at signalized intersections. Inspired by the observed quasiperiodic and self-similar characteristics of the queue…
This paper deals with stability of a certain class of fractional order linear and nonlinear systems. The stability is investigated in the time domain and the frequency domain. The general stability conditions and several illustrative…
A cyclic random motion at finite velocity with orthogonal directions is considered in the plane and in $\mathbb{R}^3$. We obtain in both cases the explicit conditional distributions of the position of the moving particle when the number of…
ODE solvers with randomly sampled timestep sizes appear in the context of chaotic dynamical systems, differential equations with low regularity, and, implicitly, in stochastic optimisation. In this work, we propose and study the stochastic…
For system of two ordinary differential equations of the second order representing autonomous non-conservative holonomic mechanical system, in case of dynamics such as one-frequency periodical oscillations, is found integrated invariant of…
The technique of stochastic solutions, previously used for deterministic equations, is here proposed as a solution method for partial differential equations driven by distribution-valued noises.
In this paper, we study the problem of how to optimally steer the state covariance of a general continuous-time linear stochastic system over a finite time interval subject to additive noise. Optimality here means reaching a target state…
A central challenge in physics is to describe non-equilibrium systems driven by randomness, such as a randomly growing interface, or fluids subject to random fluctuations that account e.g. for local stresses and heat fluxes not related to…
A stochastic model of a continuous nondemolition observation of a free quantum Brownian motion is presented. The nonlinear stochastic wave equation describing the posterior dynamics of the observed quantum system is solved in a Gaussian…
In this paper, the problem of stability in terms of two measures is considered for a class of stochastic partial differential delay equations with switching. Sufficient conditions for stability in terms of two measures are obtained based on…
We consider systems of n particles that move with constant velocity between collisions. Their total momentum but not necessarily their kinetic energy is preserved at collisions. As there are no further constraints, these systems are…
It is well-known that the circulation of the velocity field of a fluid along a closed material curve is conserved for any solution of the Euler equation. We offer a slightly more explicit proof of that fact than that generally found in the…