Equicontinuous Families of Markov Operators in View of Asymptotic Stability

Relation between equicontinuity, the so called e property and stability of Markov operators is studied. In particular, it is shown that any asymptotically stable Markov operator with an invariant measure such that the interior of its support is nonempty satisfies the e property

Law of the Iterated Logarithm for some Markov operators

The Law of the Iterated Logarithm for some Markov operators, which converge exponentially to the invariant measure, is established. The operators correspond to iterated function systems which, for example, may be used to generalize the cell cycle model examined by A. Lasota and M.C. Mackey, J. Math. Biol. (1999).Comment: 23 page

Modelling with measures: Approximation of a mass-emitting object by a point source

We consider a linear diffusion equation on $\Omega:=\mathbb{R}^2\setminus\bar{\Omega_\mathcal{O}}$, where $\Omega_\mathcal{O}$ is a bounded domain. The time-dependent flux on the boundary $\Gamma:=\partial\Omega_\mathcal{O}$ is prescribed. The aim of the paper is to approximate the dynamics by the solution of the diffusion equation on the whole of $\mathbb{R}^2$ with a measure-valued point source in the origin and provide estimates for the quality of approximation. For all time $t$, we derive an $L^2([0,t];L^2(\Gamma))$-bound on the difference in flux on the boundary. Moreover, we derive for all $t>0$ an $L^2(\Omega)$-bound and an $L^2([0,t];H^1(\Omega))$-bound for the difference of the solutions to the two models

Differentiability in perturbation parameter of measure solutions to perturbed transport equation

We consider a linear perturbation in the velocity field of the transport equation. We investigate solutions in the space of bounded Radon measures and show that they are differentiable with respect to the perturbation parameter in a proper Banach space, which is predual to the H\"older space $\mathcal{C}^{1+\alpha}(\mathbb{R}^d)$. This result on differentiability is necessary for application in optimal control theory, which we also discuss

Continuous dependence of an invariant measure on the jump rate of a piecewisedeterministic Markov process

We investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right after the jump is attained by a randomly selected continuous transformation. It is assumed that the jumps appear at random moments, which coincide with the jump times of a Poisson process with intensity . The model of this type, although in a more general version, was examined in our previous papers, where we have shown, among others, that the Markov process under consideration possesses a unique invariant probability measure, say . The aim of this paper is to prove that the map 7! is continuous (in the topology of weak convergence of probability measures). The studied dynamical system is inspired by certain stochastic models for cell division and gene expression

On the continuous dependence of the stationary distribution of a piecewise deterministic Markov process on its jump intensity

We examine a piecewise deterministic Markov process, whose whole randomness stems from the jumps, which occur at the random time points according to a Poisson process, and whose post-jump locations are attained by randomly selected transformations of the pre-jumps states. Between the jumps, the process is deterministically driven by a continuous semiflow. The aim of the paper is to establish the continuous dependence of the invariant measure of this process on the jump intensity.Narodowe Centrum Nauk (grant nr 2018/02/X/ST1/01518