Thursday, December 5, 2024 - 17:15 in T2-234
Coexisting of branching populations
A talk in the Oberseminar Probability Theory and Mathematical Statistics series by
Nikita Elizarov from Universität Bielefeld
| Abstract: |
Consider two one-dimensional branching populations $(Z^1_n, Z_n^2)$ in a joint random environment. Quenched distributions of $Z^1_n$ and $Z^2_n$ are assumed independent. Thus, the dependence between populations is caused by the environment only. We are interested in the asymptotic behaviour of coexisting probability $\mathbb{P}(Z_n^2 > 0, Z_n^2 > 0).$ We are going to show that this problem is deeply connected to a two-dimensional random walk $\hat{S}_n$ conditioned to stay in a cone. $\hat{S}_n$ is the Doob $h$-transform of a random walk $S_n$ having i.i.d. increments with zero mean and finite variance and killed at leaving the cone. For the process $\hat{S}_n$ we estimate the probability of coming close to the boundary of the cone. This will give us upper and lower bounds for the coexistence probability.
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