报告人：侯浩杰 博士后（北京理工大学）

时间：2025年08月28日 15:00-

地点：理科楼LA103

摘要：Consider a subcritical branching random walk $\{Z_k\}_{k\geq 0}$ with offspring distribution $\{p_k\}_{k\geq 0}$ and step size $X$.  Let $M_n$ denote the rightmost position reached by $\{Z_k\}_{k\geq 0}$ up to generation $n$  and define $M := \sup_{n\geq 0} M_n$. In this talk, we  give asymptotics of tail probability of  $M$ under optimal assumptions $\sum^{\infty}_{k=1}(k\log k)p_k<\infty$ and $\mathbb{E}[Xe^{\gamma X}]<\infty$, where $\gamma >0$ is a constant such that $\mathbb{E}[e^{\gamma X}]=\frac{1}{m}$ and  $m=\sum_{k=0}^\infty kp_k\in (0,1)$.  Moreover, we confirm the conjecture of Neuman and Zheng [Probab. Theory Related Fields, 2017] by establishing the existence of a critical value $m\mathbb{E}[X e^{\gamma X}]$ such that

   \begin{align}

      \lim_{n\to\infty} \mathbb{P}(M_n\geq cn\big| M\geq cn)=

      \begin{cases}

         &1,~c\in\big(0,m\mathbb{E}[Xe^{\gamma X}]\big); \\

         &0,~c\in\big(m\mathbb{E}[Xe^{\gamma X}], \infty\big).

      \end{cases}

   \end{align}

Finally, we extend these results to the maximal displacement of branching random walks with killing. Based on an ongoing joint work with Shuxiong Zhang (Anhui Normal University).

邀请人：数学研究中心

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