View source for Branching process with a random medium

''branching process in a random medium''

A time-inhomogeneous branching process in which the inhomogeneity is random. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175601.png" /> be a stationary sequence of random variables (the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175602.png" /> is interpreted as the state of the  "medium"  at the moment of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175603.png" />), and let to each possible state of the medium <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175604.png" /> correspond a probability distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175605.png" /> of the number of descendants of a single particle:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175606.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175607.png" /></td> </tr></table>

In order to construct a trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175608.png" /> of a branching process in a random medium the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175609.png" /> and the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756010.png" /> of the states of the medium are fixed, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756011.png" /> is determined for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756012.png" /> as a sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756013.png" /> independent random variables with distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756014.png" />. Such a complication of the branching [[Galton–Watson process|Galton–Watson process]] is natural if, for example, the branching process in a random medium is regarded as a model of a biological population.

The properties of branching processes in a random medium are analogous to those of ordinary branching processes. For instance, the generating distribution function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756015.png" />, under the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756016.png" />, has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

(for a branching Galton–Watson process, i.e. for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756018.png" />, the right-hand side of (*) equals the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756019.png" />-fold iteration of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756020.png" />). Branching processes in a random medium may be subcritical, critical and supercritical: the  "criticality parameter"  here (see [[#References|[1]]]) is the variable

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756021.png" /></td> </tr></table>

(for ordinary branching processes, the  "criticality parameter"  is the mathematical expectation of the number of  "descendants"  of a single particle). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756022.png" />, the branching process in a random medium is said to be subcritical and, for the random variable

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which is the probability of extinction of the branching process in a random medium for a given trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756024.png" />, the relation

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is valid. There is also the analogue of the limit theorem of the subcritical Galton–Watson branching process: For almost all realizations of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756026.png" /> the limits

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exist and satisfy

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756028.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756029.png" />, the branching process in a random medium is said to be critical, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756030.png" /></td> </tr></table>

and, for almost all realizations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756031.png" />,

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756033.png" />, the branching process in a random medium is said to be supercritical; in such a case

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756034.png" /></td> </tr></table>

and, if certain additional conditions are met, there exists for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756035.png" /> a non-negative random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756036.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756037.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K.B. Athreya,   P.E. Ney,   "Branching processes" , Springer  (1972)</TD></TR></table>



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