$\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}$ Let $Q$ be a [[Markov Kernel]] on $(E,\EEE)$, an operator on $B(E)$. The big idea behind Markov Processes is to iterate this operator. Specifically, >[!definition] Transition Semigroup >A collection $(Q_t)_{t\geq 0}$ of MKs on $E$ is a ==**transition semigroup**== if: >- For every $x\in E$, $Q_0(x, dy) = \delta_x(dy)$. >- **Chapman-Kolmogorov Relation:** For every $s,t\geq 0$, $Q_{t+s} = Q_tQ_s$. >- For every $A\in \EEE$, $(t,x)\mapsto Q_t(x,A)$ is measurable with respect to $\BBB(\RR_+)\otimes \EEE$. >[!idea] >- This $Q_0$ is the identity operator on $B(E)$. >- Explicitly, $Q_{t + s}(x,A) = \int_E Q_t(x, dy) Q_s(y,A)$ >- This is a regularity condition, which is not used until we define the **resolvent** below. Now, fix a filtered probability space. >[!definition] Markov Process >A ==**Markov Process**== is an adapted process $X$ on $E$ such that, for every $s,t\geq 0$ and $f\in B(E)$,$\EE[f(X_{s+t}) | \FFF_s] = Q_t f(X_s).$ Note that Markov Processes wrt a filtration $\FFF_t$ are automatically Markov Processes wrt $\FFF^X_t$. >[!idea] Key idea >The RHS is notably $\sigma(X_s)$-measurable! Hence we obtain that, say, for any $A\in \EEE$, >$\PP\left[X_{s+t}\in A | X_r, 0\leq r \leq s\right] = \PP\left[X_{s+t}\in A | X_s\right] = Q_t(X_s,A).$ >This should be very clearly reminiscent of our naive understanding of Markov Processes. >[!example] Writing it out >Let $\gamma(dx)$ be the law of $X_0$. Then, for $0 = t_0 < t_1 < \dots < t_p$, $f_0,\dots, f_p\in B(E)$, $\EE\left[f_0(X_0)\dots f_p(X_{t_p})\right] = \int \gamma(dx_0)f_0(x_0) \int Q_{t_1}(x_0, dx_1)f_1(x_1)\dots \int Q_{t_p - t_{p-1}}(x_{p-1}, dx_p) f_p(x_p).$ > >Obvious. By the CK relations, we can check that these marginal distributions are compatible. By the [[Kolmogorov Extension]] theorem, so long as $E$ is a separable metric space, we obtain a canonical process $X$ whose finite-dimensional distributions are given by the above, for any $\gamma$ and $(Q_t)_{t\geq 0}$. >[!idea]- The unimportant details > 1. The space of all mappings $X: \RR_+\to E$ is denoted $\Omega^* = E^{\RR_+}$, with product $\sigma$-field $\FFF^*$ generated by coordinate mappings. > 2. We already have the finite-dimensional distributions via $\gamma$ and $(Q_t)_{t\geq 0}$. > 3. By the KET, we obtain a probability distribution $\PP_\gamma$ on $\FFF^*$. This is not the pushforward of anything, so some people like to call the elements of $\Omega^*$ $\omega$; I think this is too much notation. Each element of $\FFF^*$ is literally a path, hence $X_t = X(t)$ is a canonical process on $\Omega^*$. > 4. In the specific case $\gamma = \delta_x$, call the associated distribution $\PP_x$ on $\FFF^*$. For each $A\in \FFF^*$, $x\mapsto \PP_x(A)$ is measurable. Furthermore, it is true that $\PP_\gamma(A) = \int \gamma(dx) \PP_x(A).$ >[!example] BM >On $E = \RR^d$, pre-Bownian motion is generated by $Q_t(x,dy) = \frac{1}{(2\pi t)^{d/2}} e^{-\frac{z^2}{2t}}$ # The Resolvent >[!definition] Resolvent >For $\lambda > 0$, the ==**$\lambda$-resolvent**== is the linear operator $R_\lambda: B(E)\to B(E)$ defined via$R_\lambda f(x) = \int_0^\infty e^{-\lambda t} Q_t f(x) dt.$ Notably, the regularity condition has been used to ensure $t\mapsto Q_tf(x)$ is $\BBB(\RR_+)\otimes \EEE$-measurable, and hence the integral yields a $\EEE$-measurable function (this is [[Fubini-Tonelli]]). In the future, we will write $R_\lambda = \int_0^\infty e^{-\lambda t} Q_t$ with the obvious action; $R_\lambda$ is the **Legendre Transform** of $Q$. The intuition is that $Q_t = e^{tL}$ for some $L$, such that this is formally $\frac{1}{\lambda - L}$. >[!claim] >If $0\leq f\leq 1$, then $0\leq R_\lambda f\leq \frac{1}{\lambda}$. In other words, $\norm{R_\lambda}\leq \frac{1}{\lambda}$, but it also preserves sign; if $0\leq f$, then $0\leq R_\lambda f$. >[!claim] Resolvent Equation >If $\lambda, \mu > 0$, then $R_\lambda - R_\mu + (\lambda - \mu)R_\lambda R_\mu = 0$. You can path-find the solution from the corresponding identity for real numbers (if $Q$ were finite-dimensional, you could spectral). >[!claim] Supermartingale >Let $h\in B(E)$ be non-negative and let $\lambda > 0$. Then, $e^{-\lambda t} R_\lambda h(X_t)$ is an $\FFF_t$-supermartingale. This follows from the quick computation$\EE\left[e^{-\lambda(t+s)}R_\lambda h(X_{t+s})\right] = e^{-\lambda(t+s)}Q_sR_\lambda h(X_t)\leq e^{-\lambda t}R_\lambda h(X_t).$