What's the meaning of the state space with locally compact topological space?

Question

I have encountered a statement in one paper describing the continuous-time controlled Markov chain with space state which is locally compact topological space. What does this mean? In my previous experience, the state space of the continuous-time controlled Markov chain is either finite or countable infinite. I hope somebody could explain the relationship between the state space of finite or countable infinite and the state space which is the locally comopact topological space.

Besides, what's the meaning of the trajectories of the stochastic processes?

Thanks in advance!

Answer 1

Locally Compact spaces are not necessarily metric, and not necessarily separable then not a Polish space which is the natural setting for probability and the state space of Markov process. The sate space can be a functionnal space see here.

Definition of trajectories need some elementary notation.

$(\Omega, \mathscr{F}, \mathscr{F}_t, \mathbb P)$ is a filtred probability space, where $\mathscr{F}_t, \, t \in \mathbb R_{\geq 0}$ is a filtration.

1. A Stochastic Process is a family of Random Variables indexed on $t$.
2. For a fixed $\omega$ we call the map $t \to X_t(\omega)$ a trajectory or a sample path.

There are three different notions of equality for a stochastic process.

• Two processes are indistinguishable if $\mathbb{P}(\{X_t = Y_t, \, \, \forall t\}) = 1$
• One process $X_t$ is a modification of $Y_t$ if $\forall t, \, \, \mathbb{P}(X_t = Y_t) = 1$
• Two processes are Equally Distributed if for any choice of $A_k \in \mathscr{B}(\mathbb R^d)$ and $t_k$ we have that $\mathbb{P}(X_{t_k} \in A_k) = \mathbb{P}(Y_{t_k} \in A_k)$.