A cell growth model revisited

In this paper a stochastic model for the simultaneous growth and division of a cell-population cohort structured by size is formulated. This probabilistic approach gives straightforward proof of the existence of the steady-size distribution and a simple derivation of the functional-differential equation for it. The latter one is the celebrated pantograph equation (of advanced type). This firmly establishes the existence of the steady-size distribution and gives a form for it in terms of a sequence of probability distribution functions. Also it shows that the pantograph equation is a key equation for other situations where there is a distinct stochastic framework

The Pantograph Equation in the Complex Plane

AbstractThe subject matter of this paper focuses on two functional differential equations with complex lag functions. We address ourselves to the existence and uniqueness of solutions and to their asymptotic behaviour

Functional-Differential Equations with Compressed Arguments and Polynomial Coefficients: Asymptotics of the Solutions

AbstractFunctional-differential equations with linearly compressed arguments and polynomial coefficients are considered. We prove, under some mild restrictions on the coefficients, that each solution y(t) of such an equation, satisfying estimate |y(t)| ≤ C exp{γ In2 |t|} (t → ∞), where 0 < γ < γ̃, is polynomial

On bounded continuous solutions of the archetypal equation with rescaling

The `archetypal' equation with rescaling is given by $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,\mu(\mathrm{d}a,\mathrm{d}b)$ ($x\in\mathbb{R}$), where $\mu$ is a probability measure; equivalently, $y(x)=\mathbb{E}\{y(\alpha(x-\beta))\}$, with random $\alpha,\beta$ and $\mathbb{E}$ denoting expectation. Examples include: (i) functional equation $y(x)=\sum_{i} p_{i} y(a_i(x-b_i))$; (ii) functional-differential (`pantograph') equation $y'(x)+y(x)=\sum_{i} p_{i} y(a_i(x-c_i))$ ($p_{i}>0$, $\sum_{i} p_{i}=1$). Interpreting solutions $y(x)$ as harmonic functions of the associated Markov chain $(X_n)$, we obtain Liouville-type results asserting that any bounded continuous solution is constant. In particular, in the `critical' case $\mathbb{E}\{\ln|\alpha|\}=0$ such a theorem holds subject to uniform continuity of $y(x)$; the latter is guaranteed under mild regularity assumptions on $\beta$, satisfied e.g.\ for the pantograph equation (ii). For equation (i) with $a_i=q^{m_i}$ ($m_i\in\mathbb{Z}$, $\sum_i p_i m_i=0$), the result can be proved without the uniform continuity assumption. The proofs utilize the iterated equation $y(x)=\mathbb{E}\{y(X_\tau)\,|\,X_0=x\}$ (with a suitable stopping time $\tau$) due to Doob's optional stopping theorem applied to the martingale $y(X_n)$.Comment: Substantially revised. The title is modifie

Analysis of the archetypal functional equation in the non-critical case

We study the archetypal functional equation of the form $y(x)=\iint_{R^2} y(a(x-b))\,\mu(da,db)$ ($x\in R$), where $\mu$ is a probability measure on $R^2$; equivalently, $y(x)=E\{y(\alpha (x-\beta))\}$, where $E$ is expectation with respect to the distribution $\mu$ of random coefficients $(\alpha,\beta)$. Existence of non-trivial (i.e. non-constant) bounded continuous solutions is governed by the value $K:=\iint_{R^2}\ln |a| \mu(da,db) =E \{\ln |\alpha|\}$; namely, under mild technical conditions no such solutions exist whenever $K0$ (and $\alpha>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(\alpha,\beta)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $P(\alpha0$ is drastically different from that with $\alpha>0$; in particular, we prove that a bounded solution $y(\cdot)$ possessing limits at $\pm\infty$ must be constant. The proofs employ martingale techniques applied to the martingale $y(X_n)$, where $(X_n)$ is an associated Markov chain with jumps of the form $x\rightsquigarrow\alpha (x-\beta)$

Generalized Refinement Equations and Subdivision Processes

AbstractThe concept of subdivision schemes is generalized to schemes with a continuous mask, generating compactly supported solutions of corresponding functional equations in integral form. A necessary and a sufficient condition for uniform convergence of these schemes are derived. The equivalence of weak convergence of subdivision schemes with the existence of weak compactly supported solutions to the corresponding functional equations is shown for both the discrete and integral cases. For certain non-negative masks stronger results are derived by probabilistic methods. The solution of integral functional equations whose continuous masks solve discrete functional equations, are shown to be limits of discrete nonstationary schemes with masks of increasing support. Interesting functions created by these schemes are C∞ functions of compact support including the up-function of Rvachev

Spectral analysis on infinite Sierpinski fractafolds

A fractafold, a space that is locally modeled on a specified fractal, is the fractal equivalent of a manifold. For compact fractafolds based on the Sierpinski gasket, it was shown by the first author how to compute the discrete spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian. A similar problem was solved by the second author for the case of infinite blowups of a Sierpinski gasket, where spectrum is pure point of infinite multiplicity. Both works used the method of spectral decimations to obtain explicit description of the eigenvalues and eigenfunctions. In this paper we combine the ideas from these earlier works to obtain a description of the spectral resolution of the Laplacian for noncompact fractafolds. Our main abstract results enable us to obtain a completely explicit description of the spectral resolution of the fractafold Laplacian. For some specific examples we turn the spectral resolution into a "Plancherel formula". We also present such a formula for the graph Laplacian on the 3-regular tree, which appears to be a new result of independent interest. In the end we discuss periodic fractafolds and fractal fields

Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincar\'e and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpi\'nski gasket

Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals

We investigate the existence of the meromorphic extension of the spectral zeta function of the Laplacian on self-similar fractals using the classical results of Kigami and Lapidus (based on the renewal theory) and new results of Hambly and Kajino based on the heat kernel estimates and other probabilistic techniques. We also formulate conjectures which hold true in the examples that have been analyzed in the existing literature

The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields

We consider an "elastic" version of the statistical mechanical monomer-dimer problem on the n-dimensional integer lattice. Our setting includes the classical "rigid" formulation as a special case and extends it by allowing each dimer to consist of particles at arbitrarily distant sites of the lattice, with the energy of interaction between the particles in a dimer depending on their relative position. We reduce the free energy of the elastic dimer-monomer (EDM) system per lattice site in the thermodynamic limit to the moment Lyapunov exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value and covariance function are the Boltzmann factors associated with the monomer energy and dimer potential. In particular, the classical monomer-dimer problem becomes related to the MLE of a moving average GRF. We outline an approach to recursive computation of the partition function for "Manhattan" EDM systems where the dimer potential is a weighted l1-distance and the auxiliary GRF is a Markov random field of Pickard type which behaves in space like autoregressive processes do in time. For one-dimensional Manhattan EDM systems, we compute the MLE of the resulting Gaussian Markov chain as the largest eigenvalue of a compact transfer operator on a Hilbert space which is related to the annihilation and creation operators of the quantum harmonic oscillator and also recast it as the eigenvalue problem for a pantograph functional-differential equation.Comment: 24 pages, 4 figures, submitted on 14 October 2011 to a special issue of DCDS-