本学位论文主要研究两速度分枝布朗运动粒子最大值的偏差概率的收敛速度. 在经典的分枝布朗运动中, 粒子的底运动服从一个方差恒定的布朗运动 ($s$ 时刻的方差 $\sigma^2(s)=\sigma^2 s$). 而在两速度分枝布朗运动中, 对于给定的时间 $t$, 布朗运动的方差函数是一个分段线性函数, 即 $\sigma^2(s)=\sigma_1^2 s, \,s\in [0, bt)$, $\sigma^2(s)=\sigma_1^2 bt+(s-bt)\sigma_2^2,\, s\in [bt,t]$, 其中 $b\in(0,1), \sigma_1^2>0, \sigma_2^2>0$. 关于两速度分枝布朗运动, Bovier 和 Hartung 已经证明了 $t$ 时刻所有存活粒子的最大值 $M_t$, 适当地以某个确定性函数 $m_t$ 中心化后, 是弱收敛的. 这里 $m_t$ 的形式取决于 $\sigma_1$ 和 $\sigma_2$ 的大小关系. 我们主要研究 $M_t-m_t$ 的偏差概率的收敛速度. 本论文分为三个部分. 第一部分, 研究两速度分枝布朗运动最大值的上偏差概率, 即当 $t\to\infty$ 时, 上偏差概率 $\P(M_t\ge\alpha m_t)$ 收敛于零的速度及其收敛极限, 其中 $\alpha > 1.$ 我们给出了一个完整的结果. 当 $\sigma_1 < \sigma_2$ 时, 上偏差概率的渐近行为与经典的分枝布朗运动相似. 但对于 $\sigma_1 > \sigma_2$, 上偏差概率的渐近行为与经典的分枝布朗运动存在着明显差异, 根据 $\alpha$ 与 $\sigma_2$ 之间的关系, 它们呈现出丰富的相变现象, 且不同情形下的证明方法也不同, 主要依赖截断二阶矩方法和 F-KPP 方程解的渐近行为估计. 第二部分, 研究两速度分枝布朗运动最大值的中偏差概率. 本部分是在最大值上偏差概率的基础上, 将 $\alpha m_t$ 换成 $m_t+\ell_t$, 研究中偏差概率 $\P(M_t\ge m_t+\ell_t)$ 收敛于零的速度, 其中 $\ell_t$ 满足 ${\lim_{t\to\infty}\ell_t/t=0}$ 且 ${\lim_{t\to\infty}\log t/\ell_t=0}.$ 显然, 当 $t\rightarrow\infty$ 时, 最大值的中偏差概率 $\P(M_t\geq m_t+\ell_t)$ 趋于零的速度会变慢. 不同于上偏差概率, 一阶矩估计对中偏差情形失效, 而且也不能通过 F-KPP 方程解的渐近行为估计中偏差概率. 中偏差概率需要更加精细的计算. 我们给出了中偏差概率的速率函数, 也就是指数阶衰退速率. 第三部分, 研究两速度分枝布朗运动最大值的下偏差. 此部分旨在研究当 $t\to\infty$ 时, 下偏差概率 $\P(M_t\le\alpha m_t)$ 的指数阶衰减速率, 其中$\alpha <1.$ 证明的方法主要基于首次分枝时间和首次分枝位置对于两速度分枝布朗运动最大值的影响.

In this thesis, we investigate the convergence rate of the deviation probabilities for the maximum of two speed branching Brownian motions.  In the classical branching Brownian motion, the motion of the particle obeys the Brownian motion with constant variance, i.e. $\sigma^2(s)=\sigma^2 s$. In the two speed branching Brownian motion, for a given time $t$, the variance function of the Brownian motion is a piecewise linear function satisfying $\sigma^2(s)=\sigma_1^2 s, \,s\in [0, bt)$, $\sigma^2(s)=\sigma_1^2 bt+(s-bt)\sigma_2^2,\, s\in [bt, t]$, where $b\in(0,1), \sigma_1^2>0, \sigma_2^2>0$. Bovier and Hartung have shown that the maximum $M_t$ of all surviving particles at time t, suitably centred by a deterministic function $m_t$,  converges weakly. The form of $m_t$ depends on the size of $\sigma_1$ and $\sigma_2$. We mainly study the convergence rate of the deviation probabilities of $M_t-m_t$.

This thesis consists of three parts.

In the first part, we consider the upper deviation probabilities for the maximum of two speed branching Brownian motions, i.e.,  the decay rate and convergence limits of upper deviation probabilities $\P(M_t\ge\alpha m_t)$ as $t\to\infty$, where $\alpha > 1.$

We obtain a  complete result. When $\sigma_1<\sigma_2$, the asymptotic behavior of the upper deviation probability is similar to that of the classical branching Brownian motion. But for $\sigma_1 > \sigma_2$, the asymptotic behavior of the upper deviation probability is obviously different from that of classical branching Brownian motion, showing rich phase transition phenomena.  The  proof methods vary in different situations. This mainly depends on the relationship between $\alpha$ and $\sigma_2$. We mainly rely on the second moment method and the asymptotic behavior of the solutions to F-KPP equation.

In the second part, we study the moderate deviation probabilities for the maximum of two speed branching Brownian motions. Based on the large deviation probabilities for the maximum in the first part, we replace $\alpha m_t$ with $m_t+\ell_t$,  and study the limit behaviors of the moderate deviation probabilities $\P(M_t\ge m_t+\ell_t)$, where $\ell_t$ satisfies ${\lim_{t\to\infty}\ell_t/t=0}$ and ${\lim_{t\to\infty}\log t/\ell_t=0}.$  Obviously, the moderate deviation probability $\P(M_t\geq m_t+\ell_t)$ tends to zero as $t\rightarrow\infty$  at a  slower rate. Different from the upper deviation probability, the first moment method is invalid for the moderate deviation, and the moderate deviation probability cannot be estimated from the asymptotic behavior of the solutions to F-KPP equations, which requires more elaborate calculation. We give the exponential decay rate of the moderate deviation probability.

In the third part, we consider the lower deviation probabilities for the maximum of two speed branching Brownian motions. We obtain the first order of the decay rate of the lower deviation probabilities $\P(M_t\le\alpha m_t)$ with $\alpha <1$ as $t\to\infty$ by studying the influence of the first branching time and the first branching position.