Continuity properties of sample functions of Markov processes

We shall be concerned with the continuity properties of sample functions of Markov processes. We let ü=) \ be a space in which a completely additive probability measure, P (A), is defined over a Borel field Ja= A of subsets of Í2. A random variable is any function measurable with respect to Jw. A stochastic process is any family of random variables x ( (co), tÇ. T°\, where T" is a linear set. A sample function is the function x ( (w), considered as a function of t, for fixed a. We shall sometimes say "almost all sample functions, ""almost all to, " "with probability one, " or "almost everywhere" (written a. e. ), meaning for all a except an co-set of measure zero. A regularity condition known as separability (applied to processes) will often be used. A separable process [xt (u), tÇT0 is a process for which there is a sequence ti of parameter values, dense in T°, and a set A, P (A) =0, such that, for co (¡. A, inf xt (a¡) = inf xti (u), sup x¡ (a>) = sup x (i (co) tc. IT" l, g/r° t£IT° UQIT"for every interval I. We shall also use a somewhat stronger condition which we shall denote as property S*. A stochastic process x), t (E: T0 will be said to have property S* if there is a sequence R =/, • of parameter values dense in T°a nd a set A, P (A) =0, such that for <oQ. A, for B any closed set, I any open t-interval, ifxt (co) £B, t£IR, then xt (co) £B, tIT0. We shall suppose given a function P (X), a completely additive probability measure defined for XÇE. J, the field of linear Borel sets, and a function P (t, x; T, X), defined for OKTT', for all real x, XÇzJ, such that: (a) P (t, x; T, X) for fixed t, T, X is a Baire function of x, (b) P (t, x; T, X) for fixed t, x, T is a completely additive probability measure defined over J. (c) For 0ᵗ<s<TT', all x, XE. J, p oc (1) P (t, x; T, X) = j P (t, x; s, dy) P (s, y; T, X).