Qualitative properties of certain piecewise deterministic Markov processes

By M. Benaim, St'ephane Le Borgne, Florent Malrieu, Pierre-André Zitt. 2012

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We study a class of Piecewise Deterministic Markov Processes with state space Rd x E where E is a finite set. The continuous component evolves according to a smooth vector field that is switched at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Working under the general assumption that the process stays in a compact set, we detail a possible construction of the process and characterize its support, in terms of the solutions set of a differential inclusion. We establish results on the long time behaviour of the process, in relation to a certain set of accessible points, which is shown to be strongly linked to the support of invariant measures. Under Hormander-type bracket conditions, we prove that there exists a unique invariant measure and that the processes converges to equilibrium in total variation. Finally we give examples where the bracket condition does not hold, and where there may be one or many invariant measures, depending on the jump rates between the flows.

Subfields: Finance, Molecular Biology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/22bda9bcf3b86510c1ecf8baad8c5bd7d5da738b

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Citations in the corpus

Approximation of Invariant Measures for Regime-switching Diffusions (2014, background): “switchings. So far, the works on RSDPs have included ergodicity [3, 9, 22, 21, 26], stability [15, 21, 27, 31], recurrence and transience [18, 19, 20, 31], invariant densities [1, 2], hypoellipticity [1, 4], and so forth. Since solving RSDPs is still a challenging task, numerical schemes and/or appr”
Posterior Computation with the Gibbs Zig-Zag Sampler (2020, unknown): “By viewing (cid:101) ξ and ˜ t as parameters of the map Ψ( (cid:101) ξ, ˜ t, · ) , it follows from Lemma 6.3 of Bena¨ım et al. (2015) that there exists a neighbourhood (cid:98) C ξ of (cid:98) ξ and ε > 0 such that for C ξ = (cid:98) C ξ , equation (22) holds for all (cid:101) ξ ∈ A ξ and ˜ t ∈ ( t…”
Lotka–Volterra with randomly fluctuating environments or “how switching between beneficial environments can make survival harder” (2014, methodology): “11 in [7] (or more precisely the proof of this proposition), either has empty interior or it equates the closure of its interior.”
Piecewise deterministic Markov processes and their invariant measures (2018, background): “Note that fantom jumps and the link between the process and the embedded chain are also investigated for a different type of PDMPs in [2].”
Ergodicity of the zigzag process (2017, background): “We also note that taking canonical switching rates leads to a degenerate situation where the local Hörmander type criteria of [3, 4] do not apply.”
Sensitivity to switching rates in stochastically switched odes (2013, background): “[6], [3], [4], and [1] all study invariant measures for such processes.”