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Gregory Derfel

Publications and source records attributed to Gregory Derfel.

8 recordsLinked to original sources
arXiv2018

Probabilistic approach to a cell growth model

We consider the time evolution of the supercritical Galton-Watson model of branching particles with extra parameter (mass). In the moment of the division the mass of the particle (which is growing linearly after the birth) is divided in random proportion between two offsprings (mitosis). Using the technique of moment equations we study asymptotic of the mass distribution of the particles. Mass distribution of the particles is the solution of the equation with linearly transformed argument: functional, functional-differential or integral. We derive several limit theorems describing the fluctuations of the density of the particles, first two moments of the total masses etc.

math.PR↗
arXiv2015

On bounded continuous solutions of the archetypal equation with rescaling

The `archetypal' equation with rescaling is given by $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,μ(\mathrm{d}a,\mathrm{d}b)$ ($x\in\mathbb{R}$), where $μ$ is a probability measure; equivalently, $y(x)=\mathbb{E}\{y(α(x-β))\}$, with random $α,β$ 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 $\mathbb{E}\{\ln|α|\}=0$ such a theorem holds subject to uniform continuity of $y(x)$; the latter is guaranteed under mild regularity assumptions on $β$, 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 $y(x)=\mathbb{E}\{y(X_τ)\,|\,X_0=x\}$ (with a suitable stopping time $τ$) due to Doob's optional stopping theorem applied to the martingale $y(X_n)$.

math.PR↗
arXiv2015

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

We study the archetypal functional equation of the form $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,μ(\mathrm{d}a,\mathrm{d}b)$ ($x\in\mathbb{R}$), where $μ$ is a probability measure on $\mathbb{R}^2$; equivalently, $y(x)=\mathbb{E}\{y(α(x-β))\}$, where $\mathbb{E}$ is expectation with respect to the distribution $μ$ of random coefficients $(α,β)$. Existence of non-trivial (i.e., non-constant) bounded continuous solutions is governed by the value $K:=\iint_{\mathbb{R}^2}\ln|a|\,μ(\mathrm{d}a,\mathrm{d}b)=\mathbb{E}\{\ln|α|\}$; namely, under mild technical conditions no such solutions exist whenever $K<0$, whereas if $K>0$ (and $α>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(α,β)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $\mathbb{P}(α<0)>0$ is drastically different from that with $α>0$; 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 $x\rightsquigarrowα(x-β)$.

math.PR↗
arXiv2012

Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincaré 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 Sierpiński gasket.

math.SP↗
arXiv2007

Complex asymptotics of Poincaré functions and properties of Julia sets

The asymptotic behaviour of the solutions of Poincaré's functional equation $f(λz)=p(f(z))$ ($λ>1$) for $p$ a real polynomial of degree $\geq2$ is studied in angular regions of the complex plain. The constancy of an occurring periodic function is characterised in terms of geometric properties of the Julia set of $p$. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of $f$ is related to the harmonic measure on the Julia set of $p$.

math.CV↗
arXiv2007

On bounded solutions of the balanced generalized pantograph equation

The question about the existence and characterization of bounded solutions to linear functional-differential equations with both advanced and delayed arguments was posed in early 1970s by T. Kato in connection with the analysis of the pantograph equation, y'(x)=ay(qx)+by(x). In the present paper, we answer this question for the balanced generalized pantograph equation of the form -a_2 y''(x)+a_1 y'(x)+y(x)=int_0^infty y(qx) m(dq), where a_1 > or = 0, a_2 > or = 0, a_1^2+a_2^2>0, and m is a probability measure. Namely, setting K:=int_0^infty ln(q) m(dq), we prove that if K < or = 0 then the equation does not have nontrivial (i.e., nonconstant) bounded solutions, while if K>0 then such a solution exists. The result in the critical case, K=0, settles a long-standing problem. The proof exploits the link with the theory of Markov processes, in that any solution of the balanced pantograph equation is an L-harmonic function relative to the generator L of a certain diffusion process with "multiplication" jumps. The paper also includes three "elementary" proofs for the simple prototype equation y'(x)+y(x)=(1/2)y(qx)+(1/2)y(x/q), based on perturbation, analytical, and probabilistic techniques, respectively, which may appear useful in other situations as efficient exploratory tools.

math.PR↗
arXiv2005

The Zeta Function of the Laplacian on Certain Fractals

We prove that the zeta-function $ζ_Δ$ of the Laplacian $Δ$ on a self-similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane. We characterise the poles, compute their residues, and give expressions for some special values of the zeta-function. Furthermore, we discuss the presence of oscillations in the eigenvalue counting function.

math.SP↗