$\require{physics}\newcommand{\cbrt}[1]{\sqrt[3]{#1}}\newcommand{\sgn}{\text{sgn}}\newcommand{\ii}[1]{\textit{#1}}\newcommand{\eps}{\varepsilon}\newcommand{\EE}{\mathbb E}\newcommand{\PP}{\mathbb P}\newcommand{\Var}{\mathrm{Var}}\newcommand{\Cov}{\mathrm{Cov}}\newcommand{\pperp}{\perp\kern-6pt\perp}\newcommand{\LL}{\mathcal{L}}\newcommand{\pa}{\partial}\newcommand{\AAA}{\mathscr{A}}\newcommand{\BBB}{\mathscr{B}}\newcommand{\CCC}{\mathscr{C}}\newcommand{\DDD}{\mathscr{D}}\newcommand{\EEE}{\mathscr{E}}\newcommand{\FFF}{\mathscr{F}}\newcommand{\WFF}{\widetilde{\FFF}}\newcommand{\GGG}{\mathscr{G}}\newcommand{\HHH}{\mathscr{H}}\newcommand{\PPP}{\mathscr{P}}\newcommand{\Ff}{\mathcal{F}}\newcommand{\Gg}{\mathcal{G}}\newcommand{\Hh}{\mathbb{H}}\DeclareMathOperator{\ess}{ess}\newcommand{\CC}{\mathbb C}\newcommand{\FF}{\mathbb F}\newcommand{\NN}{\mathbb N}\newcommand{\QQ}{\mathbb Q}\newcommand{\RR}{\mathbb R}\newcommand{\ZZ}{\mathbb Z}\newcommand{\KK}{\mathbb K}\newcommand{\SSS}{\mathbb S}\newcommand{\II}{\mathbb I}\newcommand{\conj}[1]{\overline{#1}}\DeclareMathOperator{\cis}{cis}\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}\DeclareMathOperator*{\range}{range}\DeclareMathOperator*{\nul}{null}\DeclareMathOperator*{\Tr}{Tr}\DeclareMathOperator*{\tr}{Tr}\newcommand{\id}{1\!\!1}\newcommand{\Id}{1\!\!1}\newcommand{\der}{\ \mathrm {d}}\newcommand{\Zc}[1]{\ZZ / #1 \ZZ}\newcommand{\Zm}[1]{\left(\ZZ / #1 \ZZ\right)^\times}\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\End}{End}\newcommand{\GL}{\mathbb{GL}}\newcommand{\SL}{\mathbb{SL}}\newcommand{\SO}{\mathbb{SO}}\newcommand{\OO}{\mathbb{O}}\newcommand{\SU}{\mathbb{SU}}\newcommand{\U}{\mathbb{U}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\Cl}{\mathrm{Cl}}\newcommand{\gr}{\mathrm{gr}}\newcommand{\gl}{\mathfrak{gl}}\newcommand{\sl}{\mathfrak{sl}}\newcommand{\so}{\mathfrak{so}}\newcommand{\su}{\mathfrak{su}}\newcommand{\sp}{\mathfrak{sp}}\newcommand{\uu}{\mathfrak{u}}\newcommand{\fg}{\mathfrak{g}}\newcommand{\hh}{\mathfrak{h}}\DeclareMathOperator{\Ad}{Ad}\DeclareMathOperator{\ad}{ad}\DeclareMathOperator{\Rad}{Rad}\DeclareMathOperator{\im}{im}\renewcommand{\BB}{\mathcal{B}}\newcommand{\HH}{\mathcal{H}}\DeclareMathOperator{\Lie}{Lie}\DeclareMathOperator{\Mat}{Mat}\DeclareMathOperator{\span}{span}\DeclareMathOperator{\proj}{proj}$ # Jump Processes on a Finite State Space If $\abs{E} < \infty$ with discrete topology, all cadlag $f\in D(E)$ are of the following form: ![[Pasted image 20240422144120.png]] The $T_i$ form an increasing sequence of stopping times. By Markov, we really only need to discuss the distribution of $(T_1, X_{T_1}$. The main results: > [!claim] Wait along an exponential process, and then jump independently of how long you waited. > Under $P_x$, there is some real number $q(x)\geq 0$ such that $T_1$ is $q(x)$-exponential. If $q(x) > 0$, then $T_1$ and $X_{T_1}$ are independent under $P_x$. > > Recall that a $\lambda$-exponential RV has distribution $\PP(U > r) = e^{-\lambda r}$. If $r = 0$, then $U = \infty$ a.s.; such a state is absorbing. We assume $q(y) > 0$ everywhere now. >[!claim] Generator (Informal) >Let $\Pi(x,y) = \PP_x(X_{T_1} = y)$. Note that $\Pi(x,\bullet)$ is a probability measure and $\Pi(x,x) = 0$. > >The generator $L$ has domain $D(L) = C_0(E) = B(E)$. For every $\phi\in B(E)$ and $x\in E$,$L\phi(x) = \sum_{y\in E} L(x,y) \phi(y),\qquad L(x,y) = \begin{cases} >q(x)\Pi(x,y) & y\neq x\\ >-q(x) & y = x >\end{cases}$ $L(x,y)$ is the instantaneous rate of transition from $x$ to $y$. >[!claim] 18.615 (Informal) >The jump times $T_1 < T_2 < \dots$ are all finite $\PP_x$-AS, and $X_{T_i}$ form a discrete Markov chain with TK $\Pi$. Conditioned on $(X_{T_i})$, $(T_{i+1} - T_i)$ are still independent and $q(X_{T_i})$-exponentially distributed. # Levy Processes >[!definition] Levy Process >A ==**pre-Levy Process**== $Y$ is a real process such that: >1. $Y_0 = 0$ a.s. >2. For every $0\leq s \leq t$, $(Y_t - Y_s)\pperp \sigma(Y_r: 0\leq r \leq s)$, and $Y_t - Y_s$ has the same law as $Y_{t-s}$. >3. $Y_t\to 0$ in probability as $t\downarrow 0$. > > Then, for all $t\geq 0$, we can let $Q_t(0,dy)$ be the law of $Y_t$. For all $x\in \RR$, we can let $Q_t(x,dy)$ be the image of $Q_t(0,dy)$ under translation. > > This collection forms a Feller semigroup, and thus $Y$ is a Feller process. A ==**Levy Process**== is a cadlag modification. Example: Brownian motion. # Continuous-State Branching Processes Imagine cell growth. Okay, define this object. >[!definition] CSBP >A ==**Continuous-State Branching Process**== is a Markov process such that for each $x,y\in \RR_+$ and $t\geq 0$,$Q_t(x,\bullet)* Q_t(y, \bullet) = Q_t(x+y,\bullet).$ > >Here, $*$ means convolution. Notably, $Q_t(0,\bullet) = \delta_0$ for every $t\geq 0$. Also, if $X,X'$ are two independent CSBPs with the same semigroup, then $X + X'$ is also a CSBP with semigroup $Q$. These processes are also well-understood. Notably, if: 1. $Q_t(x,\{0\}) < 1$ for every $x > 0$ and $t > 0$ (i.e., populations don't just die for small enough $x$). 2. $Q_t(x,\bullet)\to \delta_x(\bullet)$ as $t\downarrow 0$ in [[Random Measure|weak convergence of probability measures]], i.e. you don't do infinitely many things in finite time. then $Q$ is Feller.