Large-time behavior in non-symmetric Fokker-Planck equations

Abstract

We consider three classes of linear non-symmetric Fokker-Planck equations having a unique steady state and establish exponential convergence of solutions towards the steady state with explicit (estimates of) decay rates. First, "hypocoercive" Fokker-Planck equations are degenerate parabolic equations such that the entropy method to study large-time behavior of solutions has to be modified. We review a recent modified entropy method (for nonsymmetric Fokker-Planck equations with drift terms that are linear in the position variable). Second, kinetic Fokker-Planck equations with non-quadratic potentials are another example of non-symmetric Fokker-Planck equations. Their drift term is nonlinear in the position variable. In case of potentials with bounded second-order derivatives, the modified entropy method allows to prove exponential convergence of solutions to the steady state. In this application of the modified entropy method symmetric positive definite matrices solving a matrix inequality are needed. We determine all such matrices achieving the optimal decay rate in the modified entropy method. In this way we prove the optimality of previous results. Third, we discuss the spectral properties of Fokker-Planck operators perturbed with convolution operators. For the corresponding Fokker-Planck equation we show existence and uniqueness of a stationary solution. Then, exponential convergence of all solutions towards the stationary solution is proven with an uniform rate.This research was partially supported by the FWF-doctoral school "Dissipation and dispersion in nonlinear partial differential equations" and INDAM -GNFM from Italy. One author (AA) is grateful to J. Schöberl for very helpful discussions.

Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration

“…This collocation method relies on a choice of basis functions (usually polynomials up to a certain degree) to represent the exact solution, and uses a nonlinear solver per uHMC transition step. General complexity guarantees are given for their ODE solvers, and as an application of their ideas to Hamiltonian MCMC, they consider the special case of a basis of piecewise quadratic polynomials defined on a time grid of step size h. In this case, uHMC with collocation can in principle produce an ε-accurate approximation of the target µ in W 2 distance using (1) when initialized at the minimum of U and run with duration T ∝ K 1 4 L 3 4 . In each transition step of uHMC, h is chosen to satisfy h −1 ∝ LT 3 ( v + ∇U (x) T )K 1 2 (L K) 3 2 ε −1 , where x, v ∈ R d are respectively the initial position and velocity in the current transition step.…”

“…However, since in uHMC we are almost exclusively interested in a stochastic notion of accuracy of the numerical time integration [8, Theorem 3.6], it is quite natural to instead use a randomized time integrator for the Hamiltonian dynamics. 1 The aim of this paper is to suggest one such randomized integration strategy, which involves a very minor modification of the Verlet time integrator, and hence, is easy to implement. The basic idea in this new integrator is to replace the trapezoidal quadrature rule used by Verlet in each stratum with a Monte Carlo quadrature rule.…”

“…This type of quadratic "distorted" metric naturally arises in the study of the large-time behavior of dynamical systems with a Hamiltonian part; e.g., see (2.51) of[1]. Here we use the distorted metric differently, namely to quantify L 2 -accuracy of the SMC time integrator.…”