ABSTRACT
We consider Markov chains with state space 0,1,2,... and transition probabilities
Pij(t) = i(i-1)...(i-k+1)pj-i+kt + o(t), j ³ i-k, j \neq i;
Pij(t) = 1+i(i-1)...(i-k+1)pkt + o(t), j = i;
Pij(t) = o(t), j < i-k,
where t ® 0, pi > 0 (i \neq k), pk < 0, åi=0¥pi = 0, the number k is fixed. Such chains may be considered as branching processes with interaction of particles. The probability of extinction of such chain is investigates.
Refs. 10
ABSTRACT
A general multitype Markov branching process where particles interact to produce new particles is formulated: There is a set of particle producing configurations (generalized couples), each such interacting configuration giving birth independently as in branching processes. The configurations are interacting at each instant with a probability proportional to the number of such configurations present in the population. For the generating functions, differential equations are deduced, corresponding to the forward and backward Kolmogorov equations. In the case of one-type processes where k particles are needed to form an interacting configuration, the extinction probability is characterized. [MR 83i:60102 Jagers, P. (Goteborg)]
Refs. 6
ABSTRACT
A stochastic system of particles of n different kinds T1, ..., Tn, interacting as complexes, is considered. The state of the system is specified by an n-dimensional vector a=(a1, ..., an) in Nn with nonnegative integer components, and this means that there is a group Sa consisting of a1 particles of the kind T1, ..., and an particles of the kind Tn. We obtain the expression for the stationary distribution of the process and consider some particular cases.
Refs. 4
ABSTRACT
The author considers the stochastic process x(t)=(m0(t), m1(t)), t ³ 0, where m0(t) is the number of particles of type T0 and m1(t) the number of particles of type T at time t. Particles of type T0 do not interact with the others and their number does not decrease in time. The author generalizes a model considered by B. A. Sevast'yanov [Branching process, "Nauka", Moscow, 1971; MR 49#9968] and gives without proof some conditional central limit theorems. [MR 84i:60111 Cohn, Harry I. (Melbourne)]
Refs. 3
ABSTRACT
The definitions of special classes of Markov's processes M1, B1, B2, B3 are given. The generating functions method is used for investigation of processes of these classes.
Refs. 9
CONTENTS
S e m i n a r 5. Foundation concepts
- 5.1. Markov’s processes with the denumerable states. Kolmogorov’s first and second
- systems of differential equations ........................................................................................ 3
- 5.2. Birth and death processes ............................................................................................ 5
S e m i n a r 6. Special classes of the Markov’s processes
- 6.1. Multivariate generating functions ................................................................................ 7
- 6.2. Markov’s processes with interaction of particles ....................................................... 9
- 6.3. Branching processes with interaction of particles ...................................................... 11
- 6.4. Branching processes ................................................................................................... 17
- 6.5. Structure of the set of Markov’s processes ............................................................... 18
S e m i n a r 7. Applications in physics
- 7.1. Branching process with one type of particles ........................................................... 18
- 7.2. Nonlinear equation of the branching processes theory .............................................. 19
- 7.3. Models of the nuclear chain reactions ....................................................................... 21
S e m i n a r 8. Applications in chemistry and biology
- 8.1. Bimolecular reaction. The law of mass action ........................................................... 23
- 8.2. Epidemic process ...................................................................................................... 24
- 8.3. "Beast of prey and prey" process ............................................................................. 27
S e m i n a r 9. Nonequilibrium statistical physics and the random processes
- 9.1. Systems of the interacting particles in the statistical physics. Bogolubov’s chain
- equations .......................................................................................................................... 29
- 9.2. Principle of identity of particles. Finetti-Khintchine’s symmetry theorem ............. 29
- 9.3. The problem of exact solutions of Kolmogorov’s equations. Kolmogorov’s third
- equation ........................................................................................................................... 32
Refs. 34
ABSTRACT
The Markovian destruction process of quadratic type particles is considered. Explicit expression are found for transient probabilities using spesial functions. Ramification feature of transient probabilities is stated.
Refs. 15
ABSTRACT
For a special class of Markovian processes with a discrete phase spase, namely branching random processes, the first-order non-linear ordinary differential equation for generating function of the transition probability, is known. The analogous non-linear partial differential first-order equations for two processes from another spesial class of Markovian processes for a discrete phase spase, that is for the branching random processes with interaction of particles, are derived.
Refs. 13
ABSTRACT
The conditions are analysed whose realisation allows reducing the description of nonequilibrium states of physical systems to solve the kinetic equation for a single-particle distribution function. An example is presented of applying the principle of identity of particles and Finetti-Khinchin symmetry theorem to derive a kinetic equation. The model of bimolecular reaction T + T ® 3T in the form of random Markovian process at discrete phase space 0,1,2,... is taken as a system of interacting particles.
Refs. 16. Figs. 2
ABSTRACT
The author considers a two-type Markov branching process m(t) with interaction of particles, where new particles are produced by each particle of type T1 or each pair of particles of type T1 and T2. Consequently, (0,g2), g2 = 0,1,... is the set of absorbing states of the process. Let P(a1, a2) (b1, b2)(t) = P{m(t) = (b1, b2) | m(0) = (a1, a2)}; then the final probabilities are q(a1, a2)(0,g2) = limt ® ¥ P(a1,a2)(0,g2)(t). The author defines the compound generating function F(z1, z2; s) corresponding to the family of generating functions f(a1,a2)(s), a1, a2 = 0,1,... of the final probabilities q(a1,a2) (0,g2), g2 = 0,1,..., derives homogeneous second-order partial differential equations for F(z1, z2; s) in the Weiss model in epidemics and in a generalization of this model, and obtains their explicit solutions, using the Riemann method. Subsequently, integral expressions for f(a1,a2)(s) are given explicitly. Furthermore, for the Weiss model the limiting distribution of the number h(a1,a2) of final particles is given as a2 ® ¥, normalizing it by a2. [MR 2000i:60095 Nakagawa, Tetsuo. (Mibu)]
Refs. 11
ABSTRACT
Stochastic models of systems of interacting elements specified by kinetic schemes are considered. The Monte-Carlo method is presented for numerical study of such system. For an example the models the bimolecular reaction, epidemic propagation and brussellator are given in a form of Markovian random processes in the discrete phase space.
Refs. 20. Figs. 12
ABSTRACT
A discrete Markovian model of a system with external source of particles and their pairwise interaction is considered. Solutions to the stationary second Kolmogorov's equation using special functions are obtained as well as asymptotic mean and variance for the stationary distribution whose asymptotic normally is shown with large intensity of the new particles arrival.
Refs. 18. Figs. 5
ABSTRACT
For the three-dimensional Markovian process of a special type the stationary first Kolmogorov's equation is solved relative to transitional probabilities. Representation for a generating function of final probabilities is derived. Asymptotic values for both a mathematical mean and final distribution variance are found and limiting theorem are stated.
Refs. 13