Absorption in Invariant Domains for Semigroups of Quantum Channels

We introduce a notion of absorption operators in the context of quantum Markov processes. The absorption problem in invariant domains (enclosures) is treated for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous times: We define a well-behaving set of positive operators which can correspond to classical absorption probabilities, and we study their basic properties, in general, and with respect to accessibility structure of channels, transience and recurrence. In particular, we can prove that no accessibility is allowed between the null and positive recurrent subspaces. In the case, when the positive recurrent subspace is attractive, ergodic theory will allow us to get additional results, in particular about the description of fixed points.

The connection between evolution algebras, random walks and graphs

Evolution algebras are a new type of non-associative algebras which are inspired from biological phenomena. A special class of such algebras, called Markov evolution algebras, is strongly related to the theory of discrete time Markov chains. The winning of this relation is that many results coming from Probability Theory may be stated in the context of Abstract Algebra. In this paper, we explore the connection between evolution algebras, random walks and graphs. More precisely, we study the relationships between the evolution algebra induced by a random walk on a graph and the evolution algebra determined by the same graph. Given that any Markov chain may be seen as a random walk on a graph, we believe that our results may add a new landscape in the study of Markov evolution algebras.

Evolution of states in a continuum migration model

The Markov evolution of states of a continuum migration model is studied. The model describes an infinite system of entities placed in $${\mathbbm {R}}^d$$ R d in which the constituents appear (immigrate) with rate b(x) and disappear, also due to competition. For this model, we prove the existence of the evolution of states $$\mu _0 \mapsto \mu _t$$ μ 0 ↦ μ t such that the moments $$\mu _t(N_\Lambda ^n)$$ μ t ( N Λ n ) , $$n\in {\mathbbm {N}}$$ n ∈ N , of the number of entities in compact $$\Lambda \subset {\mathbbm {R}}^d$$ Λ ⊂ R d remain bounded for all $$t>0$$ t > 0 . Under an additional condition, we prove that the density of entities and the second correlation function remain point-wise bounded globally in time.

A continuum individual based model of fragmentation: dynamics of correlation functions

An individual-based model of an infinite system of point particles in R^d is proposed and studied. In this model, each particle at random produces a finite number of new particles and disappears afterwards. The phase space for this model is the set Γ of all locally finite subsets of R^d. The system's states are probability measures on Γ the Markov evolution of which is described in terms of their correlation functions in a scale of Banach spaces. The existence and uniqueness of solutions of the corresponding evolution equation are proved.

On the Risk of Extinction for a Population Subject to a Random Markov Evolution with Jumps

We consider a one-dimensional population whose evolution is described by a jump-diffusion equation and we study the effects of changing the coefficients of the equation on the extinction time, that is the instant at which the population becomes arbitrarily small. It is shown that, under the same diffusion coefficient, if one reduces the drift and the size of jumps, the speed of extinction increases; moreover, the probability of reaching a higher population state than the present one before reaching a lower population size decreases. If the diffusion coefficient is state-independent, the speed of extinction increases with it. Furthermore, if no jumps are allowed (i.e. for a simple-diffusion equation), then under certain conditions on the coefficients of the equation both large and small values of the diffusion coefficient result in a higher extinction risk.