For certain Sheffer sequences \((s_n)_{n=0}^\infty\) on \(\mathbb C\), Grabiner (1988) proved that, for each \(\alpha\in0,1\), the corresponding Sheffer operator \(z^n\mapsto s_n(z)\) extends to a ...linear self-homeomorphism of \(\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)\), the Fréchet topological space of entire functions of order at most \(\alpha\) and minimal type (when the order is equal to \(\alpha>0\)). In particular, every function \(f\in \mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)\) admits a unique decomposition \(f(z)=\sum_{n=0}^\infty c_n s_n(z)\), and the series converges in the topology of \(\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)\). Within the context of a complex nuclear space \(\Phi\) and its dual space \(\Phi'\), in this work we generalize Grabiner's result to the case of Sheffer operators corresponding to Sheffer sequences on \(\Phi'\). In particular, for \(\Phi=\Phi'=\mathbb C^n\) with \(n\ge2\), we obtain the multivariate extension of Grabiner's theorem. Furthermore, for an Appell sequence on a general co-nuclear space \(\Phi'\), we find a sufficient condition for the corresponding Sheffer operator to extend to a linear self-homeomorphism of \(\mathcal E^{\alpha}_{\mathrm{min}}(\Phi')\) when \(\alpha>1\). The latter result is new even in the one-dimensional case.
We study a Markov birth-and-death process on a space of locally finite configurations, which describes an ecological model with a density dependent fecundity regulation mechanism. We establish ...existence and uniqueness of this process and analyze its properties. In particular, we show global time-space boundedness of the population density and, using a constructed Foster-Lyapunov-type function, we study return times to certain level sets of tempered configurations. We find also sufficient conditions that the degenerate invariant distribution is unique for the considered process.
We study stability of stationary solutions for a class of non-local semilinear parabolic equations. To this end, we prove the Feynman--Kac formula for a L\'{e}vy processes with time-dependent ...potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the zero solution, and use them to the study of the spatial logistic equation arising in population ecology. For this equation, we find conditions which imply that its positive stationary solution is asymptotically stable. We consider also the case when the initial condition is given by a random field.
The aim of this paper is to develop foundations of umbral calculus on the space \(\mathcal D'\) of distributions on \(\mathbb R^d\), which leads to a general theory of Sheffer polynomial sequences on ...\(\mathcal D'\). We define a sequence of monic polynomials on \(\mathcal D'\), a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on \(\mathcal D'\) to be of binomial type or a Sheffer sequence, respectively. We also construct a lifting of a sequence of monic polynomials on \(\mathbb R\) of binomial type to a polynomial sequence of binomial type on \(\mathcal D'\), and a lifting of a Sheffer sequence on \(\mathbb R\) to a Sheffer sequence on \(\mathcal D'\). Examples of lifted polynomial sequences include the falling and rising factorials on \(\mathcal D'\), Abel, Hermite, Charlier, and Laguerre polynomials on \(\mathcal D'\). Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role.
We discuss general concept of Markov statistical dynamics in the continuum. For a class of spatial birth-and-death models, we develop a perturbative technique for the construction of statistical ...dynamics. Particular examples of such systems are considered. For the case of Glauber type dynamics in the continuum we describe a Markov chain approximation approach that gives more detailed information about statistical evolution in this model.
We consider a Fisher-KPP-type equation, where both diffusion and nonlinear part are nonlocal, with anisotropic probability kernels. Under minimal conditions on the coefficients, we prove existence, ...uniqueness, and uniform space-time boundedness of the positive solution. We investigate existence, uniqueness, and asymptotic behavior of monotone traveling waves for the equation. We also describe the existence and main properties of the front of propagation.
There is studied an infinite system of point entities in \(\mathbb{R}^d\) which reproduce themselves and die, also due to competition. The system's states are probability measures on the space of ...configurations of entities. Their evolution is described by means of a BBGKY-type equation for the corresponding correlation (moment) functions. It is proved that: (a) these functions evolve on a bounded time interval and remain sub-Poissonian due to the competition; (b) in the Vlasov scaling limit they converge to the correlation functions of the time-dependent Poisson point field the density of which solves the kinetic equation obtained in the scaling limit from the equation for the correlation functions. A number of properties of the solutions of the kinetic equation are also established.
We consider spatial population dynamics given by Markov birth-and-death process with constant mortality and birth influenced by establishment or fecundity mechanisms. The independent and density ...dependent dispersion of spreading are studied. On the base of general methods of 14, we construct the state evolution of considered microscopic ecological systems. We analyze mesoscopic limit for stochastic dynamics under consideration. The corresponding Vlasov-type non-linear kinetic equations are derived and studied.
The aim of this paper is to analyze different regulation mechanisms in spatial continuous stochastic development models. We describe the density behavior for models with global mortality and local ...establishment rates. We prove that the local self-regulation via a competition mechanism (density dependent mortality) may suppress a unbounded growth of the averaged density if the competition kernel is superstable.
