TITLE: Existence of a probability space
QUESTION [3 upvotes]: Let us assume that we are given a family of Markov chains $(X^\alpha_t)_{t\geq0}$ in continuous time. Kolmogorov's result ensures that for each $\alpha \in I$ there exists a probability space $(\Omega_\alpha,\mathcal{F}_\alpha,\mathbb{P}_\alpha)$ on which $(X^\alpha_t)_{t\geq0}$ can be defined. However, I was asking my self whether there exists a probability space $(\Omega_I,\mathcal{F}_I,\mathbb{P}_I)$ on which the whole family $\{(X^\alpha_t)_{t\geq0} \mid \alpha \in I\}$ can be defined at once. In particular, does this hold if $I$ is uncountable, e.g. $I = [0;1]$? Any hint or reference would be highly appreciated.

REPLY [0 votes]: Each Markov Chain $X^\alpha$ you can associate with a probability measure $p^\alpha$ on some path space $(\Omega^\alpha,\mathcal F^\alpha)$. The existence of the suitable product space for their joint distribution depends on the dependence structure between the Markov Chains, and on the probabilities you are interested in.
For example, in case you would like to talk about independent Markov Chains and you would like to ask probabilities of the kind $p(X^{\alpha_1}\in A_1,\dots,X^{\alpha_n}\in A_n)$ then such a product measure does exist - see e.g. here (I was not able to find a formal proof though). Note that since $A_i$ here is a subset of the path space, $X^{\alpha_i}\in A_i$ is actually any measurable even associated to a single Markov Chain.