Limiting distribution of the rightmost particle in catalytic branching Brownian motion

Sergey Bocharov, Simon C. Harris
2016 Electronic Communications in Probability  
We study the model of binary branching Brownian motion with spatially-inhomogeneous branching rate βδ0(·), where δ0(·) is the Dirac delta function and β is some positive constant. We show that the distribution of the rightmost particle centred about β 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 [10] for the degenerate case of catalytic branching. Before we state the main result of this
more » ... (Theorem 1.1) let us define the notation and recall some of the existing results for this catalytic model in [1]. Let us denote by P the probability measure associated to the branching process with E the corresponding expectation. We denote the set of all the particles in the system at time t by N t . For every particle u ∈ N t we denote by X u t its spatial position at time t. Finally, we define
doi:10.1214/16-ecp22 fatcat:3g6zahwsybfd7fdlkdyvdma33y