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Let

  • $I\subseteq[0,\infty)$ be closed under addition and $0\in I$
  • $(\Omega,\mathcal A)$ be a measurable space
  • $E$ be a Polish space and $\mathcal E$ be the Borel $\sigma$-algebra on $E$
  • $X=(X_t)_{t\in I}$ be a stochastic process on $(\Omega,\mathcal A)$ with values in $(E,\mathcal E)$
  • $(\mathcal F_t)_{t\in I}$ be the filtration generated by $X$

$X$ is called a Markov process with distributions $(\operatorname P_x)_{x\in E}$ $:\Leftrightarrow$

  1. $\operatorname P_x$ is a probability measure on $(\Omega,\mathcal A)$ with $$\operatorname P_x\left[X_0=x\right]=1\;,$$ For all $x\in E$
  2. $\kappa : E\times\mathcal E^{\otimes I}\to[0,1]$ with $$\kappa(x,A):=\operatorname P_x\left[X\in A\right]$$ is a stochastic kernel
  3. $X$ has the weak Markov property, i.e. $$\operatorname P_x\left[X_{s+t}\in B\mid\mathcal F_s\right]=\kappa_t(X_s,B)\;\;\;\text{for all }x\in E\;s,t\in I\;\text{and }B\in\mathcal E\;\tag 1$$ where $$\kappa_t:E\times\mathcal E\to[0,1]\;,\;\;\;(x,B)\mapsto\kappa\left(x,\left\{y\in E^I:y(t)\in B\right\}\right)=\operatorname P_x\left[X_t\in B\right]$$

Question:$\;\;\;$Let $x\in E$ and $B\in\mathcal E$. Under which conditions does $$q(x,B):=\frac 1t\lim_{t\downarrow 0}\kappa_t(x,B)$$ exist?

Probably, we need at least right-continuity of $X$, but I'm unsure which further assumptions we need in detail.

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  • $\begingroup$ @saz I'm sorry, I've made a mistake in phrasing the question. I'm not interested in the existence of $$\lim_{t\downarrow 0}\operatorname P_x\left[X_t\in B\right]\;,$$ but $$\lim_{t\downarrow 0}\frac 1t\operatorname P_x\left[X_t\in B\right]\;.$$ I assume that's an important difference, since (for $x\in E$) $$\mu_t:=\frac 1t\operatorname P_x\left[X_t\in\;\cdot\;\right]\;\;\;\text{for }t\in I\setminus\left\{0\right\}$$ is not a family of (sub-)probability measures anymore. $\endgroup$ – 0xbadf00d Oct 5 '15 at 15:53
  • $\begingroup$ If you take $B=E$ then $q(x,E)=\infty$. Is that a problem? $\endgroup$ – Nate Eldredge Oct 5 '15 at 16:04
  • $\begingroup$ @NateEldredge Is your comment related to my previous one? In general: No, that's not a problem. As you may guess, I'm especially interested in the question: "Under which conditions does the so called $Q$-matrix exist?" In that special scenario, one assumes $E$ to be countable and $I=[0,\infty)$. And the sets $B$ of interest are $B=\left\{y\right\}$ for $y\in E$. However, I've asked myself how restrictive we really need to be in general. $\endgroup$ – 0xbadf00d Oct 5 '15 at 16:21
  • $\begingroup$ @0xbadf00d I see. So far, I've seen these kind of results only for (real-valued) Lévy processes and, more generally, Feller processes. $\endgroup$ – saz Oct 5 '15 at 16:39
  • $\begingroup$ @saz Do you know, that $$q(x,y):=\lim_{t\downarrow 0}\frac 1t\operatorname P_x\left[X_t=y\right]\;\;\;\text{for }x,y\in E$$ ($E$ assumed to be at most countable) is the so-called transition rate matrix (or $Q$-matrix) and is considered in "continuous-time Markov chains"? I'm working with the textbook Probability Theory: A Comprehensive Course (see link below for the relevant chapter) and the author introduces this matrix without stating under which conditions $q$ exists. $\endgroup$ – 0xbadf00d Oct 5 '15 at 16:47

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