Transition Functions and Resolvents

A stochastic process $$\left\{ {\left. {X\left( t \right),t \in \left[ {0, + \infty } \right.} \right\}} \right.$$, defined on a probability space (Ω, F, Pr), with values in a countable set E (to be called the state space of the process), is called a continuous-time parameter Markov chain if for any finite set $$ \leqslant {t_1} < {t_2} < ... < {t_n} < {t_{n + 1}}$$ of “times,” and corresponding set $${i_1},{i_2}...,{i_{n - 1}},i,j$$ of states in E such that $$\Pr \left\{ {X\left( {{t_n}} \right) = i,X\left( {{t_{n - 1}}} \right) = {i_{n - 1}},...X\left( {{t_1}} \right) = {i_1}} \right\} > 0$$, we have (1.1)$$\Pr \left\{ {X\left( {{t_{n + 1}}} \right) = j\left| {X\left( {{t_n}} \right)} \right. = i,X\left( {{t_{n - 1}}} \right) = {i_{n - 1}},...X\left( {{t_1}} \right) = {i_1}} \right\} = \Pr \left\{ {X\left( {{t_{n + 1}}} \right) = j\left| {X\left( {{t_n}} \right) = i} \right.} \right\}.$$ Equation (1.1) is called the Markov property. If for all s, t such that $$ \leqslant s \leqslant t$$ and all i,j ε E the conditional probability $$\Pr \left\{ {X\left( t \right) = j\left| {X\left( s \right)} \right. = i} \right\}$$ appearing on the right-hand side of (1.1) depends only on t − s, and not on s and t individually, we say that the process $$\left\{ {X\left( t \right),t \in \left[ {0, + \left. \infty \right)} \right.} \right\}$$ is homogeneous, or has stationary transition probabilities. In this case, then, $$\Pr \left\{ {X\left( t \right) = j\left| {X\left( s \right) = i} \right.} \right\} = \Pr \left\{ {X\left( {x - s} \right) = j\left| {X\left( 0 \right) = i} \right.} \right\}$$, and the function $${P_{ij}}\left( t \right)\mathop = \limits^{def} \Pr \left\{ {X\left( t \right) = j\left| {X\left( 0 \right)} \right. = i} \right\},ij \in E,t \geqslant 0,$$ is called the transition function of the process.