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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.

60J55 Local time and additive functionals
60J65 Brownian motion
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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