Summary
Let X be the (B 0, {q n (x)})-branching diffusion where B 0is the exp \(\left( { - \int\limits_o^t {k(B_S )ds} } \right)\)-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¦→∞, then we prove that R t /t→(λ 2/2)1/2, as t→∞, a.s. and in mean (of any order) where R t is the position of the rightmost particle at time t and λ 0 is the largest eigenvalue of (1/2)d 2/dx 2 + Q, Q(x) = k(x)(m(x)−1).
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This work was supported in part by a grant from the National Science Foundation # MCS-8201470.
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Erickson, K.B. Rate of expansion of an inhomogeneous branching process of brownian particles. Z. Wahrscheinlichkeitstheorie verw Gebiete 66, 129–140 (1984). https://doi.org/10.1007/BF00532800
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DOI: https://doi.org/10.1007/BF00532800