On time reversal of piecewise deterministic Markov processes

Abstract

We study the time reversal of a general PDMP. The time reversed process is defined as X (T −t)− , where T is some given time and X t is a stationary PDMP. We obtain the parameters of the reversed process, like the jump intensity and the jump measure.

Numerical methods for piecewise deterministic Markov processes with boundary

“…We now state a first convergence result Theorem 2 Under hypotheses (H) and (HD), let T > δt. Let (p (K) n ) i∈E,n∈N,K∈M be the solution of Scheme (13), (14), (15), (16), (17) which satisfies (23)-(24). Then, assuming δt → 0, τ → 0, and h/τ → 0, P D (dx, dt) (resp.…”

“…n ) n∈N,K∈M be the solution of Scheme (13), (14), (15), (16), (17) which satisfies (23)-(24). Then, assuming δt → 0, τ → 0, and h/τ → 0, P D (t)dx (defined by (22)) tends to ρ t , unique solution of (5) (or (41)), for the weak topology of the measures, for almost every t > 0.…”

Densities for piecewise deterministic Markov processes with boundary

“…For general background on PDMPs we refer the reader to [17]. A variety of applications has generated a renewed interest in PDMPs, see [9,12,13,27,30,34] and the references therein.…”

“…preserving the norm of nonnegative elements from L 1 . Although stability and ergodicity of PDMPs are developed in great generality in [15], the general problem of existence of absolutely continuous invariant measures has not been treated at all except for specific examples, see [30] for a recent account of different models where the existence is known. If we know already that the process induces a substochastic semigroup then we can use the methods presented in [32,34] to get existence of invariant densities.…”

Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation

“…Recently, the asymptotic behavior of PDMPs have been investigated in [4,5,12,37], limit theorems for infinite dimensional PDMPs in [35], control problems in [10,11,22], numerical methods in [33,6], time reversal in [27] and to end up this list with no claim of completeness, estimation of the jump rates for PDMPs in [2]. Hybrid systems are the object of great attention because they offer an accurate description of a large class of phenomena arising in various domains such as physics or biology.…”