Low-Lying Eigenvalues and Convergence to the Equilibrium of Some Piecewise Deterministic Markov Processes Generators in
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nnales Henri Poincar´ e
Low-Lying Eigenvalues and Convergence to the Equilibrium of Some Piecewise Deterministic Markov Processes Generators in the Small Temperature Regime Arnaud Guillin and Boris Nectoux 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. They have recently been used a lot for simulation issues, falling in the domain of Markov chain Monte Carlo method, due to their numerically observed astonishing performances. Nevertheless, they are far from being theoretically understood, in particular at the spectral level, which is the scope of our study. Mathematics Subject Classification. 35P15, 35P20, 47F05, 35Q82, 35Q92.
1. Introduction and Main Results 1.1. Purpose and Setting of This Work 1.1.1. Purpose. The quite recent growing interest for Piecewise Deterministic Markov Processes [12] (PDMP hereafter) stems from their use within the Markov chain Monte Carlo methodology. It aims at simulating a target probability distribution π by choosing a good Markov chain in the sense that it is ergodic and has stationary probability measure π. Let us be a little more precise concerning this probability measure π. Let M be the position space (for instance, a compact manifold without boundary, which will be the case in this work) and π be the Gibbs measure 2
e− h U (x) π(dx) = − 2 U dx, e h M
(1)
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A. Guillin, B. Nectoux
Ann. Henri Poincar´e
associated with the potential function U : M → R and the parameter h > 0, dx being the Lebesgue measure on M. The parameter h is proportional to the Boltzmann constant kB through the relation h = kB T , T being the temperature of the underlying system. The Hastings–Metropolis algorithm [32] is surely the most well-known method to create such a Markov chain by ensuring reversibility with respect to π. However, its performance may be questioned in terms of speed of convergence, computational cost, and behavior with respect to the dimension of the problem. PDMP may be shortly described as follows: Between two jumps (possibly of only part of the coordinates) whose rates may of course depend of the position of the process, they have a deterministic behavior (see, for instance, Remarks 1 and 2 ). PDMP may show remarkable features [4,18], as they are by essence non-reversible and may thus exhibit faster rates of convergence toward equilibrium (see [3]). Of course, there is still a lot of work to do to correctly assess the rate of convergence of such PDMP, see, for example, [1,8,13,16], and the behavior with respect to the dimension still has to be precisely understood (see, however, [6]). In practice, the second main advantage of these processes is that they can be used to sample the Gibbs measure (1) without sampling Brownian motions, as, for example, in Langevin-type method such as MALA, but only
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