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), tÇ^T0} 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<(o>), 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, t^IT0.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 O^KT^T', 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 0^t<s<T^T', all x, XE.J, p oc(1) P(t, x; T,X) = j P(t, x; s, dy)P(s, y; T, X).