Manh Hong Duong

Stationary solutions of the Vlasov-Fokker-Planck equation: existence, characterization and phase-transition

In this paper, we study the set of stationary solutions of the Vlasov-Fokker-Planck (VFP) equation. This equation describes the time evolution of the probability distribution of a particle moving under the influence of a double-well potential, an interaction potential, a friction force and a stochastic force. We prove, under suitable assumptions, that the VFP equation does not have a unique stationary solution and that there exists a phase transition. Our study relies on the recent results by Tugaut and coauthors regarding the McKean-Vlasov equation.

Formulation of the relativistic heat equation and the relativistic kinetic Fokker–Planck equations using GENERIC

In this paper, we formulate the relativistic heat equation and the relativistic kinetic Fokker–Planck equations into the GENERIC (General Equation for Non-Equilibrium Reversible–Irreversible Coupling) framework. We also show that the relativistic Maxwellian distribution is the stationary solution of the latter. The GENERIC formulation provides an alternative justification that the two equations are meaningful relativistic generalizations of their non-relativistic counterparts.

On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type

In this paper, we construct the fundamental solution to a degenerate diffusion of Kolmogorov type and develop a time-discrete variational scheme for its adjoint equation. The so-called mean squared derivative cost function plays a crucial role occurring in both the fundamental solution and the variational scheme. The latter is implemented by minimizing a free energy functional with respect to the Kantorovich optimal transport cost functional associated with the mean squared derivative cost. We establish the convergence of the scheme to the solution of the adjoint equation generalizing previously known results for the Fokker-Planck equation and the Kramers equation.

The Vlasov-Fokker-Planck equation in non-convex landscapes: convergence to equilibrium

In this paper, we study the long-time behaviour of solutions to the Vlasov-Fokker-Planck equation where the confining potential is non-convex. This is a nonlocal nonlinear partial differential equation describing the time evolution of the probability distribution of a particle moving under the influence of a non-convex potential, an interaction potential, a friction force and a stochastic force. Using the free-energy approach, we show that under suitable assumptions solutions of the Vlasov-Fokker-Planck equation converge to an invariant probability.

Analysis of the mean squared derivative cost function

In this paper, we investigate the mean squared derivative cost functions that arise in various applications such as in motor control, biometrics and optimal transport theory. We provide qualitative properties, explicit analytical formulas and computational algorithms for the cost functions. We also perform numerical simulations to illustrate the analytical results. In addition, as a by-product of our analysis, we obtain an explicit formula for the inverse of a Wronskian matrix that is of independent interest in linear algebra and differential equations theory. Copyright