Tail generating functions for extendable branching processes

The Galton-Watson process with immigration which is time-homogeneous but not permitted when the process is in state 0 (so that this state is absorbing) is briefly studied in the subcritical and supercritical cases. Results analogous to those for the ordinary Galton-Watson process are found to hold. Partly-new techniques are required, although known end-results on the standard process with and without immigration are used also. In the subcritical case a new parameter is found to be relevant, replacing to some extent the criticality parameter.

The Kesten-Stigum theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an $L \\log L$ condition is decisive. In critical and subcritical cases, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least $n$ generations. We give conceptual proofs of these theorems based on comparisons of Galton-Watson measure to another measure on the space of trees. This approach also explains Yaglom's exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution.

It is well-known that branching processes can be represented as a sum of random number of independent random variables. This relation serves as a starting point for investigation of the branching process. However, characteristics of the population representing as sums of dependent terms are not usually amenable to investigation by the traditional methods in the theory of branching processes. If we use some results in the theory of summation of dependent variables: (a) it is possible to study some new characteristics of population in the branching process without immigration (the number of r-tuples at time t1 that have at time t2>t1 nonempty offspring sets differing from each other in number by at most d; the number of pairs of particles having the same number of offspring at time t; reduced branching processes and so on ). (b) it is possible to study branching processes with immigration, depending on reproduction. Key wordsDiscrete-timeimmigrant(k,i)-processreproduction dependent immigrationstopping timedependent indicators

Suppose that a road network model is given, together with some given demand for travel by (say) car and that the demand for travel varies with time of day but not from day to day. Suppose that this demand is given in the form of specified total outflow rates from each origin headed towards each destination, for each origin-destination pair and for each time of day, and that some initial time-dependent routeinflow rates, meeting the given demand, are given. Finally, suppose that within-day time is represented by a continuous variable. This paper specifies a natural smooth day-to-day route-swapping procedure wherein drivers swap toward less expensive routes as day succeeds day, and shows that under reasonable conditions there is an equilibrium state of this dynamical system. If such a collection of route-inflows has arisen today, say, then there is no incentive for any route-inflow to change tomorrow, in the sense that at each moment of today each of today's route-inflows isalready on a route which today yielded the smallest travel cost. Such a set of “no-incentive-to-change” route-inflows is called adynamic equilibrium, or adynamic user-equilibrium, and may be regarded as a solution of the dynamic equilibrium traffic assignment problem. Thus, the paper introduces a smooth day-to-day dynamic assignment model and, using this model, shows that there is a dynamic user-equilibrium in a continuous time setting. The paper briefly considers the day-to-day stability of the route-swapping process, also in a continuous setting. Finally, the paper gives a simple dynamical example illustrating the stability of the route-swapping process in a simple two-route network when there is deterministic queueing at bottlenecks.

The multitype branching random walk, II

Let μ ( t ) be the number of particles at time t of a continuous-time critical branching process. It is known that the probability of non-extinction of the process at time t Q ( t ) = P { μ ( t ) > 0 | μ (0) = 1} → 0 as t → ∞. Hence it follows that Q m 0 = P { μ ( t ) > 0 | μ (0) = m } ∼ mQ ( t ) → 0 for any m = 2,3, . . . Let for any integer m > r ≥ 1 In this paper, we prove that Q mr ( t ) ∼ ( m − r ) Q ( t ) as t → ∞ for any critical continuous-time Markov branching process. Earlier, this result was obtained for branching processes with finite variation of the number of particles.

Let Zn be the number of particles at time n in a critical branching process in random environment and Zm,n be the number of particles in this process at time m ≤ n which have non-empty offspring at time n. We prove limit theorems for the process {Z[nt],n, t ∈ (0, 1]} conditioned on the event {Zn > 0}. Quenched and annealed approaches are considered.*

We study both a continuous time non-binary Galton−Watson random tree and its explo- ration (or height) process in the subcritical, critical and supercritical cases. We then renormalize our branching process and exploration process, and take the weak limit as the size of the population tends to infinity.

Motivated by a problem arising in the mining industry, we estimate the energy ε(η) that is needed to reduce a unit mass to fragments of size at most η in a fragmentation process, when η→0. We assume that the energy used in the instantaneous dislocation of a block of size s into a set of fragments (s1,s2,…) is sβφ(s1/s,s2/s,…), where φ is some cost function and β a positive parameter. Roughly, our main result shows that if α>0 is the Malthusian parameter of an underlying Crump-Mode-Jagers branching process (with α = 1 when the fragmentation is mass-conservative), then there exists a c∈(0,∞) such that ε(η)∼cηβ-α when β<α. We also obtain a limit theorem for the empirical distribution of fragments of size less than η that result from the process. In the discrete setting, the approach relies on results of Nerman for general branching processes; the continuous approach follows by considering discrete skeletons. In the continuous setting, we also provide a direct approach that circumvents restrictions induced by the discretization.

Let $Z_n$ be the number of particles at moment n in a branching process in a random environment. Assuming that $Z_n$ is ``critical in a certain sense we prove theorems describing the asymptotic behavior as $n\to\infty$ of the distribution of the vector $(Z_{[nt_1]},\ldots,Z_{[nt_{b}]})$, of the number of particles in the process at moments $0 0$.

The paper is concerned with critical branching processes in a Markov random environment. A conditional functional limit theorem for the number of particles in a process and a conditional invariance principle are proved. The asymptotic tail behaviour for the distributions of the maximum and the total number of particles in a process is found.

This paper deals with the computation of survival probabilities and extinction times for multitype positively regular branching processes. If all of the generating functions of the offspring distributions are of the linear fractional form and have the same denominator, explicit expressions may be obtained for all of their iterates. It is then possible to obtain formulae for survival probabilities and bounds on the mean time to extinction, given extinction, of a line descended from a single individual. If there are two types and the offspring distributions are bivariate Poisson, their generating functions may be bounded by linear fractional generating functions. It is then possible to compute upper and lower bounds on mean times to extinction, given extinction, and this is done for some special cases.

In this paper we study blow up of the equation\(u_t = u_{xx} + u^\gamma \dot W_{tx}\), where\(\dot W_{tx}\) is a two-dimensional white noise field and where Dirichlet boundary conditions are enforced. It is known that if γ<3/2, then the solution exists for all time; in this paper we show that if γ is much larger than 3/2, then the solution blows up in finite time with positive probability. We prove this by considering how peaks in the solution propagate. If a peak of high mass forms, we rescale the equation and divide the mass of the peak into a collection of peaks of smaller mass, and these peaks evolve almost independently. In this way we compare the evolution ofu to a branching process. Large peaks are regarded as particles in this branching process. Offspring are peaks which are higher by some factor. We show that the expected number of offspring is greater than one when γ is much larger than 3/2, and thus the branching process survives with positive probability, corresponding to blowup in finite time.

We describe an improvement of Chergui and Moulaï’s method (2008) that generates the whole efficient set of a multiobjective integer linear fractional program based on the branch and cut concept. The general step of this method consists in optimizing (maximizing without loss of generality) one of the fractional objective functions over a subset of the original continuous feasible set; then if necessary, a branching process is carried out until obtaining an integer feasible solution. At this stage, an efficient cut is built from the criteria’s growth directions in order to discard a part of the feasible domain containing only nonefficient solutions. Our contribution concerns firstly the optimization process where a linear program that we define later will be solved at each step rather than a fractional linear program. Secondly, local ideal and nadir points will be used as bounds to prune some branches leading to nonefficient solutions. The computational experiments show that the new method outperforms the old one in all the treated instances.

We consider dynamic cooperative games in characteristic function form in the sense that the characteristic function evolves over time in accordance with a difference or differential equation that is influenced not only by the current ("instantaneous") characteristic function but also by the solution concept used to allocate the benefits of cooperation among the players. The latter solution concept can be any one of a number of now standard solution concepts of cooperative game theory but, for demonstration purposes, we focus on the core and the Shapley value. In the process, we introduce some new mechanisms by which players may regard the evolution of cooperative game over time and analyse them with respect to the goal of attaining time consistency either in discrete or in continuous time setting. In discrete time, we illustrate the phenomena that can arise when an allocation according to a given solution concept is used to adapt the values of coalitions at successive time points. In continuous time, we introduce the notion of an "instantaneous" game and its integration over time.