General birth-and-death as well as hopping stochastic dynamics of infinite multicomponent particle systems in the continuum are considered. We derive the corresponding evolution equations for ...quasi-observables and correlation functions. We also present sufficient conditions that allow us to consider these equations on suitable Banach spaces.
We study the tail asymptotic of subexponential probability densities on the real line. Namely, we show that the n-fold convolution of a subexponential probability density on the real line is ...asymptotically equivalent to this density multiplied by n. We prove Kesten's bound, which gives a uniform in n estimate of the n-fold convolution by the tail of the density. We also introduce a class of regular subexponential functions and use it to find an analogue of Kesten's bound for functions on ℝd. The results are applied to the study of the fundamental solution to a nonlocal heat equation.
We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol (m)n can be extended from a natural number m∈N to the falling ...factorials (z)n=z(z−1)⋯(z−n+1) of an argument z from F=R or C, and Stirling numbers of the first and second kinds are the coefficients of the expansions of (z)n through zk, k≤n and vice versa. When taking into account spatial positions of elements in a locally compact Polish space X, we replace N by the space of configurations—discrete Radon measures γ=∑iδxi on X, where δxi is the Dirac measure with mass at xi. The spatial falling factorials (γ)n:=∑i1∑i2≠i1⋯∑in≠i1,…,in≠in−1δ(xi1,xi2,…,xin) can be naturally extended to mappings M(1)(X)∋ω↦(ω)n∈M(n)(X), where M(n)(X) denotes the space of F-valued, symmetric (for n≥2) Radon measures on Xn. There is a natural duality between M(n)(X) and the space CF(n)(X) of F-valued, symmetric continuous functions on Xn with compact support. The Stirling operators of the first and second kind, s(n,k) and S(n,k), are linear operators, acting between spaces CF(n)(X) and CF(k)(X) such that their dual operators, acting from M(k)(X) into M(n)(X), satisfy (ω)n=∑k=1ns(n,k)⁎ω⊗k and ω⊗n=∑k=1nS(n,k)⁎(ω)k, respectively. In the case where X has only a single point, the Stirling operators can be identified with Stirling numbers. We derive combinatorial properties of the Stirling operators, present their connections with a generalization of the Poisson point process and with the Wick ordering under the canonical commutation relations.
In ecology, one of the most fundamental questions relates to the persistence of populations, or conversely to the probability of their extinction. Deriving extinction thresholds and characterizing ...other critical phenomena in spatial and stochastic models is highly challenging, with few mathematically rigorous results being available for discrete‐space models such as the contact process. For continuous‐space models of interacting agents, to our knowledge no analytical results are available concerning critical phenomena, even if continuous‐space models can arguably be considered to be more natural descriptions of many ecological systems than lattice‐based models.
Here we present both mathematical and simulation‐based methods for deriving extinction thresholds and other critical phenomena in a broad class of agent‐based models called spatiotemporal point processes. The mathematical methods are based on a perturbation expansion around the so‐called mean‐field model, which is obtained at the limit of large‐scale interactions. The simulation methods are based on examining how the mean time to extinction scales with the domain size used in the simulation. By utilizing a constrained Gaussian process fitted to the simulated data, the critical parameter value can be identified by asking when the scaling between logarithms of the time to extinction and the domain size switches from sublinear to superlinear.
As a case study, we derive the extinction threshold for the spatial and stochastic logistic model. The mathematical technique yields rigorous approximation of the extinction threshold at the limit of long‐ranged interactions. The asymptotic validity of the approximation is illustrated by comparing it to simulation experiments. In particular, we show that species persistence is facilitated by either short or long spatial scale of the competition kernel, whereas an intermediate scale makes the species vulnerable to extinction.
Both the mathematical and simulation methods developed here are of very general nature, and thus we expect them to be valuable for predicting many kinds of critical phenomena in continuous‐space stochastic models of interacting agents, and thus to be of broad interest for research in theoretical ecology and evolutionary biology.
We study propagation over
of the solution to a doubly nonlocal reaction-diffusion equation of the Fisher-KPP-type with anisotropic kernels. We present both necessary and sufficient conditions which ...ensure linear in time propagation of the solution in a direction. For kernels with a finite exponential moment over
we prove front propagation in all directions for a general class of initial conditions decaying in all directions faster than any exponential function (that includes, for the first time in the literature about the considered type of equations, compactly supported initial conditions).
The aim of this paper is to develop foundations of umbral calculus on the space D′ of distributions on Rd, which leads to a general theory of Sheffer polynomial sequences on D′. We define a sequence ...of monic polynomials on D′, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on D′ to be of binomial type or a Sheffer sequence, respectively. We also construct a lifting of a sequence of monic polynomials on R of binomial type to a polynomial sequence of binomial type on D′, and a lifting of a Sheffer sequence on R to a Sheffer sequence on D′. Examples of lifted polynomial sequences include the falling and rising factorials on D′, Abel, Hermite, Charlier, and Laguerre polynomials on D′. Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role.
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