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Limiting distribution of the rightmost particle in catalytic branching Brownian motion. (English) Zbl 1346.60128
Summary: We study the model of binary branching Brownian motion with spatially-inhomogeneous branching rate $$\beta \delta _0(\cdot )$$, where $$\delta _0(\cdot )$$ is the Dirac delta function and $$\beta$$ is some positive constant. We show that the distribution of the rightmost particle centred about $$\frac{\beta } {2}t$$ converges to a mixture of Gumbel distributions according to a martingale limit. Our results form a natural extension to [S. Lalley and T. Sellke, Ann. Probab. 16, No. 3, 1051–1062 (1988; Zbl 0658.60113)] for the degenerate case of catalytic branching.

##### MSC:
 60J55 Local time and additive functionals 60J65 Brownian motion 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
##### Keywords:
Brownian motion; local time; catalytic branching
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