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1

A Note on Existence of Global Solutions and Invariant Measures for Jump Sdes with Locally One-Sided Lipschitz Drift

Majka Mateusz B.

Probability and Mathematical Statistics

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2020

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Vol. 40, Fasc. 1

37--55

EN

We extend some methods developed by Albeverio, Brzeźniak and Wu and we show how to apply them in order to prove existence of global strong solutions of stochastic differential equations with jumps, under a local one-sided Lipschitz condition on the drift (also known as a monotonicity condition) and a local Lipschitz condition on the diffusion and jump coefficients, while an additional global one-sided linear growth assumption is satisfied. Then we use these methods to prove existence of invariant measures for a broad class of such equations.

2

On lifting invariant probability measures

Cieśla Tomasz

Bulletin of the Polish Academy of Sciences. Mathematics

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2020

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Vol. 68, no. 1

69--74

EN

We study when an invariant probability measure lifts to an invariant measure. Consider a standard Borel space X, a Borel probability measure μ on X, a Borel map T : X → X preserving μ, a Polish space Y , a continuous map S : Y → Y , and a Borel surjection p: Y → X with p ◦ S = T ◦ p. We prove that if the fibers of p are compact then μ lifts to an S-invariant measure on Y.

3

On a limit structure of the Galton-Watson branching processes with regularly varying generating functions

Imomov Azam A.

Probability and Mathematical Statistics

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2019

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Vol. 39, Fasc. 1

61--73

EN

We investigate limit properties of discrete time branching processes with application of the theory of regularly varying functions in the sense of Karamata. In the critical situation we suppose that the offspring probability generating function has an infinite second moment but its tail regularly varies. In the noncritical case, the finite moment of type E [x ln x] is required. The lemma on the asymptotic representation of the generating function of the process and its differential analogue will underlie our conclusions.

4

Lower bound technique and its applications to function systems and stochastic partial differential equations

Szarek T.

Mathematica Applicanda

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2013

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Vol. 41, No. 2

185--198

EN

We formulate some criteria for the existence of an invariant measure for Markov chains and Markov processes. We also show their application in the theory of function systems and stochastic differential equations

PL

W pracy formułujemy kryteria dla istnienia miary niezmienniczej dla łańcuchów i procesów Markowa. Następnie pokazujemy ich użyteczność w teorii iterowanych układów funkcyjnych i stochastycznych równań rózniczkowych.

5

Ergodic theory approach to chaos: remarks and computational aspects

Mitkowski P. J., Mitkowski W.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 2

259-267

EN

We discuss basic notions of the ergodic theory approach to chaos. Based on simple examples we show some characteristic features of ergodic and mixing behaviour. Then we investigate an infinite dimensional model (delay differential equation) of erythropoiesis (red blood cell production process) formulated by Lasota. We show its computational analysis on the previously presented theory and examples. Our calculations suggest that the infinite dimensional model considered possesses an attractor of a nonsimple structure, supporting an invariant mixing measure. This observation verifies Lasota's conjecture concerning nontrivial ergodic properties of the model.

6

A note on invariant measures

Niemiec P.

Opuscula Mathematica

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2011

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Vol. 31, no. 3

425-431

EN

The aim of the paper is to show that if F is a family of continuous transformations of a nonempty compact Hausdorff space Ω, then there is no F-invariant probabilistic regular Borel measures on Ω iff there are φ1..., φp ∈ F (for some p ≥ 2) and a continuous function u: Ω, p ? R such that Σ σ ∈Spu(x&sigma(1),...,x&sigmap = 0 and lim infn?∞1/n Σ n-1/k=0 (u o Φk)9x1,...xp) ≥ 1 for each x1,...xp ∈ Ω, where Φ: Ωp ∋ (x1,...xp) ? (φ1(x1,..., φp(xp)∈ Ωp and Φk is the k-th iterate of Φ. A modified version of this result in case the family F generates an equicontinuous semigroup is proved.

7

Topological pressure for one-dimensional holomorphic dynamical systems

Gelfert K., Wolf Ch.

Bulletin of the Polish Academy of Sciences. Mathematics

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2007

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Vol. 55, no 1

53-62

EN

For a class of one-dimensional holomorphic maps ƒ of the Riemann sphere we prove that for a wide class of potentials φ the topological pressure is entirely determined by the values of φ on the repelling periodic points of ƒ. This is a version of a classical result of Bowen for hyperbolic diffeomorphisms in the holomorphic non-uniformly hyperbolic setting.

8

On a dimension of measures

Lasota A., Myjak J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2002

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Vol. 50, no 2

221-235

EN

Using the notion of the Levy concentration function we discuss a definition of the dimension for probability measures. This dimension is strongly connected with the correlation dimension of measures and with the Hausdorff dimension of sets. Moreover, we calculate some bounds of this dimension for measures generated by Iterated Function Systems and by a partial differential equation.

9

Attractors of multifunetions

Lasota A., Myjak J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2000

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Vol. 48, no 3

319-334

EN

We introduce the notion of a semiattractor and attractor for multifunctions and we show that they have properties similar to semifractals and fractals. Further we show a relationship between the multifunctions and transition functions appearing in the theory of Markov operators.

10

Semifractals on Polish spaces

Lasota A., Myjak J.

Bulletin of the Polish Academy of Sciences. Mathematics

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1998

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Vol. 46, no 2

179-196

EN

We extend the theory of semifractals to arbitrary metric spaces. We also show kow to construct semifractals on Polish spaces by a use of Markov operators and Markov chain.