26 research outputs found
The e-property of asymptotically stable Markov semigroups
The relations between asymptotic stability and the e-property of Markov
semigroups acting on measures defined on general (Polish) metric spaces are
studied. While usually much attention is paid to asymptotic stability (and the
e-property has been for years verified only to establish it), it should be
noted that the e-property itself is also important as it, e.g., ensures that
numerical errors in simulations are negligible.
Here, it is shown that any asymptotically stable Markov-Feller semigroup with
an invariant measure such that the interior of its support is non-empty
satisfies the eventual e-property. Moreover, we prove that any Markov-Feller
semigroup, which is strongly stochastically continuous, and which possesses the
eventual e-property, also has the e-property. We also present an example
highlighting that strong stochastic continuity cannot be replaced by its weak
counterpart, unless a state space of a process corresponding to a Markov
semigroup is a compact metric space.Comment: 19 page
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
The e-property of asymptotically stable Markov-Feller operators
In this work, we prove that any asymptotically stable Markov-Feller operator
possesses the e-property everywhere outside at most a meagre set. We also
provide an example showing that this result is tight. Moreover, an equivalent
criterion for the e-property is proposed.Comment: 17 page
On absolute continuity of invariant measures associated with a piecewise-deterministic Markov processes with random switching between flows
We are concerned with the absolute continuity of stationary distributions
corresponding to some piecewise deterministic Markov process, being typically
encountered in biological models. The process under investigation involves a
deterministic motion punctuated by random jumps, occurring at the jump times of
a Poisson process. The post-jump locations are obtained via random
transformations of the pre-jump states. Between the jumps, the motion is
governed by continuous semiflows , which are switched directly after the jumps.
The main goal of this paper is to provide a set of verifiable conditions
implying that any invariant distribution of the process under consideration
that corresponds to an ergodic invariant measure of the Markov chain given by
its post-jump locations has a density with respect to the Lebesgue measure
The central limit theorem for Markov processes that are exponentially ergodic in the bounded-Lipschitz norm
In this paper, we establish a version of the central limit theorem for
Markov-Feller continuous time processes (with a Polish state space) that are
exponentially ergodic in the bounded-Lipschitz distance and enjoy a continuous
form of the Foster-Lyapunov condition. As an example, we verify the assumptions
of our main result for a specific piecewise-deterministic Markov process, whose
deterministic component evolves according to continuous semiflows, switched
randomly at the jump times of a Poisson process.Comment: 41 page
Resource engines
In this paper we aim to push the analogy between thermodynamics and quantum
resource theories one step further. Previous inspirations were based on
thermodynamic considerations concerning scenarios with a single heat bath,
neglecting an important part of thermodynamics that studies heat engines
operating between two baths at different temperatures. Here, we investigate the
performance of resource engines, which replace the access to two heat baths at
different temperatures with two arbitrary constraints on state transformations.
The idea is to imitate the action of a two--stroke heat engine, where the
system is sent to two agents (Alice and Bob) in turns, and they can transform
it using their constrained sets of free operations. We raise and address
several questions, including whether or not a resource engine can generate a
full set of quantum operations or all possible state transformations, and how
many strokes are needed for that. We also explain how the resource engine
picture provides a natural way to fuse two or more resource theories, and we
discuss in detail the fusion of two resource theories of thermodynamics with
two different temperatures, and two resource theories of coherence with respect
to two different bases.Comment: 25 pages, 4 figures, comments welcom