Densities for piecewise deterministic Markov processes with boundary

Abstract

We study the existence of densities for distributions of piecewise deterministic Markov processes. We also obtain relationships between invariant densities of the continuous time process and that of the process observed at jump times. In our approach we use functional-analytic methods and the theory of linear operator semigroups. By imposing general conditions on the characteristics of a given Markov process, we show the existence of a substochastic semigroup describing the evolution of densities for the process and we identify its generator. Our main tool is a new perturbation theorem for substochastic semigroups, where we perturb both the action of the generator and of its domain, allowing to treat general transport-type equations with non-local boundary conditions. A couple of particular examples illustrate our general results.

Dynamics of antibody levels: Asymptotic properties

“…The proof of this result is rather standard, so we only sketch it omitting the computational part. Some new and general results concerning piecewise deterministic Markov processes with boundary can be found in Gwiżdż and Tyran‐Kamińska 15 …”

Transport equations and perturbations of boundary conditions

“…Let the state space be equipped with a σ ‐finite measure m . By imposing general conditions on the characteristics (see Theorem 4.7), we showed in Gwiżdż and Tyran‐Kamińska 4 the existence of a substochastic semigroup (a positive contraction C 0 ‐semigroup of linear operators) on L 1 ( E , m ) describing the evolution of densities for the process. However, in general, it might happen that τ ∞ is finite with positive probability, so that the minimal process is explosive, leading to a loss of mass.…”

“…In Gwiżdż and Tyran‐Kamińska, 4 our main tool was a perturbation result for substochastic semigroups on L 1 spaces. We considered initial‐boundary value problems given in the general abstract form where Ψ 0 and Ψ are positive and possibly unbounded operators defined on a linear subspace with values in a boundary space , the operator is positive, and is such that the operator A 0 , defined as the restriction of A to the nullspace ker(Ψ 0 ), that is, is the generator of a substochastic semigroup on L 1 .…”

“…However, if Ψ and B are both unbounded, the well‐possedness of the problem () might not hold in general. In Gwiżdż and Tyran‐Kamińska, 4 we provided sufficient conditions for the operator A + B with domain ker(Ψ−Ψ 0 ) to have an extension G generating a substochastic semigroup on L 1 . In the particular case of Ψ 0 =Ψ=0, we recovered Kato's perturbation theorem 13,16,17 on L 1 .…”

“…In Section 3, we show how the results from Section 2 can be applied to problems as in () with both B =0 and B ≠0. We also complete the characterization of the generator G from Gwiżdż and Tyran‐Kamińska 4 …”