View source for Branching process with diffusion

A model of a branching process in which the reproducing particles diffuse in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175701.png" />. Let the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175702.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175703.png" />-dimensional, with an absorbing boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175704.png" />, and let the particles in the domain itself execute mutually independent Brownian motions. Each particle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175705.png" /> is independently converted, within a time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175706.png" />, into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175707.png" /> particles with a probability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757010.png" />. Let the daughter particles begin their independent evolution from the point of their genesis. Let

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be the generating function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757012.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757013.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757014.png" /> be the number of particles in a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757015.png" /> at the moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757016.png" /> if there initially was one particle at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757017.png" />. The generating functional

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satisfies the quasi-linear parabolic equation

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with the initial condition

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and the boundary condition

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Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757022.png" /> the eigen values, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757023.png" /> be the eigen function of the problem

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corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757025.png" />. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757026.png" /> the asymptotic relation

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holds. For this reason the problem is said to be subcritical if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757028.png" />, critical if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757029.png" /> and supercritical if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757030.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757031.png" />, a branching process with diffusion dies out with probability one, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757032.png" />, both the probability of dying out and the probability of the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757033.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757034.png" /> will in general be positive. Depending on their criticality, branching processes with diffusion obey limit theorems analogous to those valid for branching processes without diffusion.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.A. [B.A. Sevast'yanov] Sewastjanow,   "Verzweigungsprozesse" , Akad. Wissenschaft. DDR  (1974)  (Translated from Russian)</TD></TR></table>



====Comments====
Additional references can be found in the article [[Branching process|Branching process]].