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

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

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

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