Stability of iterates of Markov operators
Strong and weak stability of some Markov operators
An integral Markov operator appearing in biomathematics is investigated. This operator acts on the space of probabilistic Borel measures. Let and be probabilistic Borel measures. Sufficient conditions for weak and strong convergence of the sequence to are given.
Strangely sweeping one-dimensional diffusion
Let X(t) be a diffusion process satisfying the stochastic differential equation dX(t) = a(X(t))dW(t) + b(X(t))dt. We analyse the asymptotic behaviour of p(t) = ProbX(t) ≥ 0 as t → ∞ and construct an equation such that and .
Asymptotic behaviour of a transport equation
We study the asymptotic behaviour of the semigroup of Markov operators generated by the equation . We prove that for a > 1 this semigroup is asymptotically stable. We show that for a ≤ 1 this semigroup, properly normalized, converges to a limit which depends only on a.
On a one-dimensional analogue of the Smale horseshoe
We construct a transformation T:[0,1] → [0,1] having the following properties: 1) (T,|·|) is completely mixing, where |·| is Lebesgue measure, 2) for every f∈ L¹ with ∫fdx = 1 and φ ∈ C[0,1] we have , where μ is the cylinder measure on the standard Cantor set, 3) if φ ∈ C[0,1] then for Lebesgue-a.e. x.
Markov operators: applications to diffusion processes and population dynamics
This note contains a survey of recent results concerning asymptotic properties of Markov operators and semigroups. Some biological and physical applications are given.
Relative entropy and stability of stochastic semigroups
Reinforced walk on graphs and neural networks
A directed-edge-reinforced random walk on graphs is considered. Criteria for the walk to end up in a limit cycle are given. Asymptotic stability of some neural networks is shown.
On the Information Dimensions
A relationship between the information dimension and the average dimension of a measure is given. Properties of the average dimension are studied.
Fragmentation-Coagulation Models of Phytoplankton
We present two new models of the dynamics of phytoplankton aggregates. The first one is an individual-based model. Passing to infinity with the number of individuals, we obtain an Eulerian model. This model describes the evolution of the density of the spatial-mass distribution of aggregates. We show the existence and uniqueness of solutions of the evolution equation.
Professor Andrzej Lasota (1932-2006)
Dnia 28 grudnia 2006 r. odszedł od nas Profesor Andrzej Lasota, członek honorowy Polskiego Towarzystwa Matematycznego, członek rzeczywisty Polskiej Akademii Nauk, członek czynny Polskiej Akademii Umiejętności, doktor honoris causa Uniwersytetu Śląskiego. W osobie Zmarłego nauka polska straciła wybitnego uczonego, nauczyciela kilka pokoleń matematyków, wspaniałego humanistę, Człowieka wielkiej życzliwości dla uczniów i przyjaciół.
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