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Teoriya Veroyatnostei i ee Primeneniya, 1958, Volume 3, Issue 2, Pages 121–136 (Mi tvp4924)  

This article is cited in 30 scientific papers (total in 30 papers)

Branching Stochastic Processes for Particles Diffusing in a Bounded Domain with Absorbing Boundaries

B. A. Sevast'yanov

Moscow
Abstract: Particles diffusing in a multidimensional restricted domain with absorbing boundaries produce independent new particles according to a scheme for branching processes with the probability-generating function $F(z)=\sum_{k=0}^\infty P_{k^{z^k}}$, where $P_k$ is the conditional probability that a particle turns into $k$ particles, if at all. Let $P_n(x,t)$ be the probability that a particle existing at point $x$ after $t$ generations turns into $n$ particles. The probability-generating function
$$F(x,t,z)=\sum_n P_n(x,t)z^n$$
satisfies (11). Let $P_0(x)=\lim _{t\to 0}P_0(x,t)$ be the probability of extinction if the initial particle was at point $x$. $P_0(x)$ satisfies a non-linear integral equation of Hammerstein’s type (21). There is a critical value $A_0>1$ for $A= F'$ (1). If $A\leq A_0$, then $P_0(x)\equiv1$; if $A>A_0$, then $P_0(x)<1$. The critical value is
$$A_0=\frac{D_{\lambda 1}+c}{c}$$
where $D$ is the coefficient of diffusion, $1/c$ is the mean value for the lifetime of a particle, $\lambda_1$ is the least eigenvalue of the boundary value problem in §1.
In §7 the probability distribution for a number of final particles is investigated.
Received: 24.01.1958
English version:
Theory of Probability and its Applications, 1958, Volume 3, Issue 2, Pages 111–126
DOI: https://doi.org/10.1137/1103011
Document Type: Article
Language: Russian
Citation: B. A. Sevast'yanov, “Branching Stochastic Processes for Particles Diffusing in a Bounded Domain with Absorbing Boundaries”, Teor. Veroyatnost. i Primenen., 3:2 (1958), 121–136; Theory Probab. Appl., 3:2 (1958), 111–126
Citation in format AMSBIB
\Bibitem{Sev58}
\by B.~A.~Sevast'yanov
\paper Branching Stochastic Processes for Particles Diffusing in a Bounded Domain with Absorbing Boundaries
\jour Teor. Veroyatnost. i Primenen.
\yr 1958
\vol 3
\issue 2
\pages 121--136
\mathnet{http://mi.mathnet.ru/tvp4924}
\transl
\jour Theory Probab. Appl.
\yr 1958
\vol 3
\issue 2
\pages 111--126
\crossref{https://doi.org/10.1137/1103011}
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  • https://www.mathnet.ru/eng/tvp/v3/i2/p121
  • This publication is cited in the following 30 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
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