Difference between revisions of "Branching processes, regularity of"
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A property of branching processes ensuring that the number of particles at any moment of time is finite. The problem of the regularity of a branching process is usually reduced to the problem of uniqueness of the solution of some differential or integral equation. For instance, in a continuous-time branching process the differential equation | A property of branching processes ensuring that the number of particles at any moment of time is finite. The problem of the regularity of a branching process is usually reduced to the problem of uniqueness of the solution of some differential or integral equation. For instance, in a continuous-time branching process the differential equation | ||
| − | + | $$ | |
| + | |||
| + | \frac{\partial F (t; s) }{\partial t } | ||
| + | = \ | ||
| + | f (F (t; s)) | ||
| + | $$ | ||
| + | |||
| + | with the initial condition $ F(0; s) = s $ | ||
| + | has a unique solution if and only if, for any $ \epsilon > 0 $, | ||
| + | the integral | ||
| − | + | $$ | |
| + | \int\limits _ {1 - \epsilon } ^ { 1 } | ||
| + | { | ||
| + | \frac{dx}{f (x) } | ||
| + | } | ||
| + | $$ | ||
| − | + | is divergent. In the branching [[Bellman–Harris process|Bellman–Harris process]] the generating function $ F(t; s) $ | |
| + | of the number of particles is the solution of the non-linear integral equation | ||
| − | + | $$ \tag{* } | |
| + | F (t; s) = \ | ||
| + | \int\limits _ { 0 } ^ { t } | ||
| + | h (F (t - u; s)) | ||
| + | dG (u) + s | ||
| + | (1 - G (t)), | ||
| + | $$ | ||
| − | + | where $ G(t) $ | |
| + | is the distribution function of the lifetimes of particles and $ h(t) $ | ||
| + | is the generating function of the number of daughter particles ( "direct descendants" ) of a single particle. If, for given $ t _ {0} , c _ {1} , c _ {2} > 0 $ | ||
| + | and an integer $ n \geq 1 $, | ||
| + | the inequalities | ||
| − | + | $$ | |
| + | c _ {1} t ^ {n} \leq G (t) | ||
| + | \leq c _ {2} t ^ {n} | ||
| + | $$ | ||
| − | + | are valid for all $ 0 \leq t \leq t _ {0} $, | |
| + | the solution of equation (*) is unique if and only if the equation | ||
| − | + | $$ | |
| − | + | \frac{d ^ {n} \phi }{dt ^ {n} } | |
| + | = \ | ||
| + | h ( \phi ) - 1 | ||
| + | $$ | ||
with initial conditions | with initial conditions | ||
| − | + | $$ | |
| + | \phi (0) = 1,\ \ | ||
| + | \phi ^ {(r)} (0) = 0,\ r = 1 \dots n - 1, | ||
| + | $$ | ||
has a unique solution | has a unique solution | ||
| − | + | $$ | |
| + | 0 \leq \phi (t) \leq 1. | ||
| + | $$ | ||
For a branching process described by equation (*) to be regular, it is necessary and sufficient for the integral | For a branching process described by equation (*) to be regular, it is necessary and sufficient for the integral | ||
| − | + | $$ | |
| + | \int\limits _ { 0 } ^ \epsilon | ||
| + | { | ||
| + | \frac{dx}{x ^ {1-1/n } (1-h(1-x)) ^ {1/n} } | ||
| + | } | ||
| + | $$ | ||
| − | to diverge for any | + | to diverge for any $ \epsilon > 0 $. |
====References==== | ====References==== | ||
Latest revision as of 06:29, 30 May 2020
2020 Mathematics Subject Classification: Primary: 60J80 [MSN][ZBL]
A property of branching processes ensuring that the number of particles at any moment of time is finite. The problem of the regularity of a branching process is usually reduced to the problem of uniqueness of the solution of some differential or integral equation. For instance, in a continuous-time branching process the differential equation
$$ \frac{\partial F (t; s) }{\partial t } = \ f (F (t; s)) $$
with the initial condition $ F(0; s) = s $ has a unique solution if and only if, for any $ \epsilon > 0 $, the integral
$$ \int\limits _ {1 - \epsilon } ^ { 1 } { \frac{dx}{f (x) } } $$
is divergent. In the branching Bellman–Harris process the generating function $ F(t; s) $ of the number of particles is the solution of the non-linear integral equation
$$ \tag{* } F (t; s) = \ \int\limits _ { 0 } ^ { t } h (F (t - u; s)) dG (u) + s (1 - G (t)), $$
where $ G(t) $ is the distribution function of the lifetimes of particles and $ h(t) $ is the generating function of the number of daughter particles ( "direct descendants" ) of a single particle. If, for given $ t _ {0} , c _ {1} , c _ {2} > 0 $ and an integer $ n \geq 1 $, the inequalities
$$ c _ {1} t ^ {n} \leq G (t) \leq c _ {2} t ^ {n} $$
are valid for all $ 0 \leq t \leq t _ {0} $, the solution of equation (*) is unique if and only if the equation
$$ \frac{d ^ {n} \phi }{dt ^ {n} } = \ h ( \phi ) - 1 $$
with initial conditions
$$ \phi (0) = 1,\ \ \phi ^ {(r)} (0) = 0,\ r = 1 \dots n - 1, $$
has a unique solution
$$ 0 \leq \phi (t) \leq 1. $$
For a branching process described by equation (*) to be regular, it is necessary and sufficient for the integral
$$ \int\limits _ { 0 } ^ \epsilon { \frac{dx}{x ^ {1-1/n } (1-h(1-x)) ^ {1/n} } } $$
to diverge for any $ \epsilon > 0 $.
References
| [S] | B.A. Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) MR0408018 Zbl 0291.60039 |
Comments
Additional references can be found in the article Branching process.
Branching processes, regularity of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branching_processes,_regularity_of&oldid=26373