A spatial individual-based contact model with age structure

Dominika Jasińska

Displaying similar documents to “A spatial individual-based contact model with age structure”

Chaotic behaviour of continuous dynamical system generated by Euler equation branching and its application in macroeconomic equilibrium model

Barbora Volná (2015)

Mathematica Bohemica

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We focus on the special type of the continuous dynamical system which is generated by Euler equation branching. Euler equation branching is a type of differential inclusion $\dot{x} \in {f (x), g (x)}$ , where $f, g : X \subset ℝ^{n} \to ℝ^{n}$ are continuous and $f (x) \neq g (x)$ at every point $x \in X$ . It seems this chaotic behaviour is typical for such dynamical system. In the second part we show an application in a new formulated overall macroeconomic equilibrium model. This new model is based on the fundamental macroeconomic aggregate equilibrium model called...

Coexistence probability in the last passage percolation model is $6 - 8 log 2$

David Coupier, Philippe Heinrich (2012)

Annales de l'I.H.P. Probabilités et statistiques

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A competition model on $ℕ^{2}$ between three clusters and governed by directed last passage percolation is considered. We prove that coexistence, i.e. the three clusters are simultaneously unbounded, occurs with probability $6 - 8 log 2$ . When this happens, we also prove that the central cluster almost surely has a positive density on $ℕ^{2}$ . Our results rely on three couplings, allowing to link the competition interfaces (which represent the borderlines between the clusters) to some particles in the multi-TASEP,...

Existentially closed II₁ factors

Ilijas Farah, Isaac Goldbring, Bradd Hart, David Sherman (2016)

Fundamenta Mathematicae

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We examine the properties of existentially closed ( $^{ω}$ -embeddable) II₁ factors. In particular, we use the fact that every automorphism of an existentially closed ( $^{ω}$ -embeddable) II₁ factor is approximately inner to prove that Th() is not model-complete. We also show that Th() is complete for both finite and infinite forcing and use the latter result to prove that there exist continuum many nonisomorphic existentially closed models of Th().

Probabilistic properties of a Markov-switching periodic $G A R C H$ process

Billel Aliat, Fayçal Hamdi (2019)

Kybernetika

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In this paper, we propose an extension of a periodic $G A R C H$ ( $P G A R C H$ ) model to a Markov-switching periodic $G A R C H$ ( $M S$ - $P G A$ $R C H$ ), and provide some probabilistic properties of this class of models. In particular, we address the question of strictly periodically and of weakly periodically stationary solutions. We establish necessary and sufficient conditions ensuring the existence of higher order moments. We further provide closed-form expressions for calculating the even-order moments as well...

Odd cutsets and the hard-core model on $ℤ^{d}$

Ron Peled, Wojciech Samotij (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the hard-core lattice gas model on $ℤ^{d}$ and investigate its phase structure in high dimensions. We prove that when the intensity parameter exceeds $C d^{- 1 / 3} {(log d)}^{2}$ , the model exhibits multiple hard-core measures, thus improving the previous bound of $C d^{- 1 / 4} {(log d)}^{3 / 4}$ given by Galvin and Kahn. At the heart of our approach lies the study of a certain class of edge cutsets in $ℤ^{d}$ , the so-called odd cutsets, that appear naturally as the boundary between different phases in the hard-core model. We provide a refined...

The scaling limits of a heavy tailed Markov renewal process

Julien Sohier (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In this paper we consider heavy tailed Markov renewal processes and we prove that, suitably renormalised, they converge in law towards the $α$ -stable regenerative set. We then apply these results to the strip wetting model which is a random walk $S$ constrained above a wall and rewarded or penalized when it hits the strip $[0, \infty) \times [0, a]$ where $a$ is a given positive number. The convergence result that we establish allows to characterize the scaling limit of this process at criticality.

On the central limit theorem for some birth and death processes

Tymoteusz Chojecki (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Suppose that ${X n : n \geq 0}$ is a stationary Markov chain and $V$ is a certain function on a phase space of the chain, called an observable. We say that the observable satisfies the central limit theorem (CLT) if $Y_{n} : = N^{- 1 / 2} \sum_{n = 0}^{N} V (X_{n})$ converge in law to a normal random variable, as $N \to + \infty$ . For a stationary Markov chain with the $L^{2}$ spectral gap the theorem holds for all $V$ such that $V (X_{0})$ is centered and square integrable, see Gordin [7]. The purpose of this article is to characterize a family of observables $V$ for which the CLT holds...

The Dynamics of an Interactional Model of Rabies Transmitted between Human and Dogs

Wei Yang, Jie Lou (2009)

Bollettino dell'Unione Matematica Italiana

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Assuming that the population of dogs is constant and the population of human satisfies the Logistical model, an interactional model of rabies transmitted between human and dogs is formulated. Two thresholds $R_{0}$ and $R_{1}$ which determine the outcome of the disease are identified. Utilizing the method of Lyapunov function and the property of the cooperative systems, we get the global asymptotic stability for both the disease-free equilibrium and the endemic equilibrium. A critical vaccination...

Covariance structure of wide-sense Markov processes of order k ≥ 1

Arkadiusz Kasprzyk, Władysław Szczotka (2006)

Applicationes Mathematicae

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A notion of a wide-sense Markov process $X_{t}$ of order k ≥ 1, $X_{t} \sim W M (k)$ , is introduced as a direct generalization of Doob’s notion of wide-sense Markov process (of order k=1 in our terminology). A base for investigation of the covariance structure of $X_{t}$ is the k-dimensional process $x_{t} = (X_{t - k + 1}, . . ., X_{t})$ . The covariance structure of $X_{t} \sim W M (k)$ is considered in the general case and in the periodic case. In the general case it is shown that $X_{t} \sim W M (k)$ iff $x_{t}$ is a k-dimensional WM(1) process and iff the covariance function of $x_{t}$ has the triangular...

On the structure of the set of higher order spreading models

Bünyamin Sarı, Konstantinos Tyros (2014)

Studia Mathematica

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We generalize some results concerning the classical notion of a spreading model to spreading models of order ξ. Among other results, we prove that the set $S M_{ξ}^{w} (X)$ of ξ-order spreading models of a Banach space X generated by subordinated weakly null ℱ-sequences endowed with the pre-partial order of domination is a semilattice. Moreover, if $S M_{ξ}^{w} (X)$ contains an increasing sequence of length ω then it contains an increasing sequence of length ω₁. Finally, if $S M_{ξ}^{w} (X)$ is uncountable, then it contains an antichain...

Evaluating default priors with a generalization of Eaton’s Markov chain

Brian P. Shea, Galin L. Jones (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let $𝛷$ be a class of functions on the parameter space and consider estimating elements of $𝛷$ under quadratic loss. If the formal Bayes estimator of every function in $𝛷$ is admissible, then the prior is strongly admissible with respect to $𝛷$ . Eaton’s method for establishing strong admissibility is based on studying the stability properties of a particular Markov chain associated with...

Time-varying Markov decision processes with state-action-dependent discount factors and unbounded costs

Beatris A. Escobedo-Trujillo, Carmen G. Higuera-Chan (2019)

Kybernetika

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In this paper we are concerned with a class of time-varying discounted Markov decision models $ℳ_{n}$ with unbounded costs $c_{n}$ and state-action dependent discount factors. Specifically we study controlled systems whose state process evolves according to the equation $x_{n + 1} = G_{n} (x_{n}, a_{n}, ξ_{n}), n = 0, 1, ...$ , with state-action dependent discount factors of the form $α_{n} (x_{n}, a_{n})$ , where $a_{n}$ and $ξ_{n}$ are the control and the random disturbance at time $n$ , respectively. Assuming that the sequences of functions ${α_{n}}$ , ${c_{n}}$ and ${G_{n}}$ converge, in certain sense, to $α_{\infty}$ ,...

A Deformed Quon Algebra

Hery Randriamaro (2019)

Communications in Mathematics

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The quon algebra is an approach to particle statistics in order to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. The quons are particles whose annihilation and creation operators obey the quon algebra which interpolates between fermions and bosons. In this paper we generalize these models by introducing a deformation of the quon algebra generated by a collection of operators $a_{i, k}$ , $(i, k) \in ℕ^{*} \times [m]$ , on an infinite dimensional vector space satisfying...

Stein’s method in high dimensions with applications

Adrian Röllin (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Let $h$ be a three times partially differentiable function on $ℝ^{n}$ , let $X = (X_{1}, ..., X_{n})$ be a collection of real-valued random variables and let $Z = (Z_{1}, ..., Z_{n})$ be a multivariate Gaussian vector. In this article, we develop Stein’s method to give error bounds on the difference $𝔼 h (X) - 𝔼 h (Z)$ in cases where the coordinates of $X$ are not necessarily independent, focusing on the high dimensional case $n \to \infty$ . In order to express the dependency structure we use Stein couplings, which allows for a broad range of applications, such as classic...

Ground states of supersymmetric matrix models

Gian Michele Graf (1998-1999)

Séminaire Équations aux dérivées partielles

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We consider supersymmetric matrix Hamiltonians. The existence of a zero-energy bound state, in particular for the $d = 9$ model, is of interest in M-theory. While we do not quite prove its existence, we show that the decay at infinity such a state would have is compatible with normalizability (and hence existence) in $d = 9$ . Moreover, it would be unique. Other values of $d$ , where the situation is somewhat different, shall also be addressed. The analysis is based on a Born-Oppenheimer approximation....

Quasi-polynomial mixing of the 2D stochastic Ising model with “plus” boundary up to criticality

Eyal Lubetzky, Fabio Martinelli, Allan Sly, Fabio Lucio Toninelli (2013)

Journal of the European Mathematical Society

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We considerably improve upon the recent result of [37] on the mixing time of Glauber dynamics for the 2D Ising model in a box of side $L$ at low temperature and with random boundary conditions whose distribution $P$ stochastically dominates the extremal plus phase. An important special case is when $P$ is concentrated on the homogeneous all-plus configuration, where the mixing time $T_{M I X}$ is conjectured to be polynomial in $L$ . In [37] it was shown that for a large enough inverse-temperature $β$ and...

Uniform regularity for an isentropic compressible MHD- $P 1$ approximate model arising in radiation hydrodynamics

Tong Tang, Jianzhu Sun (2021)

Czechoslovak Mathematical Journal

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It is well known that people can derive the radiation MHD model from an MHD- $P 1$ approximate model. As pointed out by F. Xie and C. Klingenberg (2018), the uniform regularity estimates play an important role in the convergence from an MHD- $P 1$ approximate model to the radiation MHD model. The aim of this paper is to prove the uniform regularity of strong solutions to an isentropic compressible MHD- $P 1$ approximate model arising in radiation hydrodynamics. Here we use the bilinear commutator and...

Distortion and spreading models in modified mixed Tsirelson spaces

S. A. Argyros, I. Deliyanni, A. Manoussakis (2003)

Studia Mathematica

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The results of the first part concern the existence of higher order ℓ₁ spreading models in asymptotic ℓ₁ Banach spaces. We sketch the proof of the fact that the mixed Tsirelson space T[(ₙ,θₙ)ₙ], $θ_{n + m} \geq θ ₙ θ ₘ$ and $l i m_{n} θ ₙ^{1 / n} = 1$ , admits an $ℓ ₁^{ω}$ spreading model in every block subspace. We also prove that if X is a Banach space with a basis, with the property that there exists a sequence (θₙ)ₙ ⊂ (0,1) with $l i m_{n} θ ₙ^{1 / n} = 1$ , such that, for every n ∈ ℕ, $| | \sum_{k = 1}^{m} x_{k} | | \geq θ ₙ \sum_{k = 1}^{m} | | x_{k} | |$ for every ₙ-admissible block sequence ${(x_{k})}_{k = 1}^{m}$ of vectors in X, then there exists c...

Distortion inequality for the Frobenius-Perron operator and some of its consequences in ergodic theory of Markov maps in $ℝ^{d}$

Piotr Bugiel (1998)

Annales Polonici Mathematici

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Asymptotic properties of the sequences (a) ${P_{φ}^{j} g}_{j = 1}^{\infty}$ and (b) ${j^{- 1} \sum_{i = 0}^{j - 1} P ⁱ_{φ} g}_{j = 1}^{\infty}$ , where $P_{φ} : L ¹ \to L ¹$ is the Frobenius-Perron operator associated with a nonsingular Markov map defined on a σ-finite measure space, are studied for g ∈ G = f ∈ L¹: f ≥ 0 and ⃦f ⃦ = 1. An operator-theoretic analogue of Rényi’s Condition is introduced. It is proved that under some additional assumptions this condition implies the L¹-convergence of the sequences (a) and (b) to a unique g₀ ∈ G. The general result is applied to some smooth Markov...