Low-Lying Eigenvalues and Convergence to the Equilibrium of Some Piecewise Deterministic Markov Processes Generators in the Small Temperature Regime

Abstract

In this work we study the number of small eigenvalues and the convergence to the equilibrium of the Bouncy Particle Sampler process and the ZigZag process generators in the small temperature regime. Such processes, which fall in the class of Piecewise Deterministic Markov Processes, are non diffusive and non reversible.

On explicit $L^2$-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms

“…Other theoretical studies of the PDMPs include scaling limits and spectral analysis: The work [18] established the scaling limit of first coordinate for BPS, and [9] proved scaling limits of ZZ and BPS for several statistical observables. Spectral analysis of PDMP were considered in [10,30] in one-dimension and [25] for the metastable regime.…”

Complexity of zigzag sampling algorithm for strongly log-concave distributions

“…A central limit theorem of the zigzag process is established in [2], and a large deviation principle is established for the empirical measure in [6]. The spectrum of the zigzag process has been studied in [5,32]. A dimension independent exponential convergence rate for the zigzag process is established in [1], using the hypocoercivity framework developed in [24].…”

Eyring-Kramers type formulas for some piecewise deterministic Markov processes

“…In particular, U has a finite number of critical points on T. It is proved in [16] that P h with domain [16], the authors prove the following spectral result on P h in the limit h → 0.…”

“…In many applications in statistical physics where one needs to sample from π, the constant h is very small compared to the energetic barriers of U. Recently, the long time behavior of the semigroups generated by the generators L h of these processes on L 2 have been investigated in [16] when h ≪ 1, where it has also been proved that in the set {Re(z) ≥ −ǫ 0 h}, L h has exactly n 0 eigenvalues (n 0 being the number of local minima of U), which are non positive, real, and exponentially small as h → 0. Such results exhibit a metastable behavior of the PDMP when h ≪ 1, as it is the case for diffusion processes [11].…”

“…Let us mention that dim Ker(P h ) = 1 holds for all h > 0 (see [16]). In [16], it is assumed that min T α > 0, but all the results of [16] still hold if the less stringent assumption (1.4) is satisfied. Indeed, if (1.4)…”