Rate of expansion of an inhomogeneous branching process of brownian particles |
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Authors: | K Bruce Erickson |
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Institution: | (1) Department of Mathematics, University of Washington, 98195 Seattle, Washington, USA |
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Abstract: | Summary Let X be the (B
0, {q
n
(x)})-branching diffusion where B
0is the exp
-subprocess of BM(R1) and q
n
(x) is the probability that a particle dying at x produces n offspring, q
0 q
1 0. Put m(x) = nq
n
(x). We assume q
n
, n 2, m and k are all continuous (but m is not necessarily bounded). If k(x)m(x) 0 as ¦x¦![rarr](/content/g810k12xx7462455/xxlarge8594.gif) , then we prove that R
t
/t (
2/2)1/2, as t![rarr](/content/g810k12xx7462455/xxlarge8594.gif) , a.s. and in mean (of any order) where R
t
is the position of the rightmost particle at time t and
0 is the largest eigenvalue of (1/2)d
2/dx
2 + Q, Q(x) = k(x)(m(x)–1).This work was supported in part by a grant from the National Science Foundation # MCS-8201470. |
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Keywords: | |
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