On the Existence of the Dual Right Markov Process and Applications

Invariant, Super and Quasi-martingale Functions of a Markov Process

“…Assume that U = (U α ) α>0 is the resolvent of a right process X with state space E and let T 0 be the Lusin topology of E having B as Borel σ -algebra, and let m be a fixed U -excessive measure. Then by Corollary 2.4 from [8], and using also the result from [4], the following assertions hold: There exist a larger Lusin measurable space (E, B), with E ⊂ E, E ∈ B, B = B| E , and two processes X and X with common state space E, such that X is a right process on E endowed with a convenient Lusin topology having B as Borel σ -algebra (resp. X is a right process w.r.t.…”

Irreducible recurrence, ergodicity, and extremality of invariant measures for resolvents

“…A second approach to strongly continuity is given by the next result, for which we refer to [6] We would also like to stress out that if we deal with a strongly continuous sub-Markovian resolvent of contractions U on L p (E, m) then one can always find a larger Lusin topological space E ⊂ E 1 , E ∈ B(E 1 ), B = B(E 1 )| E , and an E 1 -valued right Markov process such that its resolvent U 1 regarded on L p (E, m), coincides with U and U 1 α (1 E1\E ) = 0, where m is the measure on (E 1 , B(E 1 )) extending m by zero on E 1 \ E; see [6], Theorem 2.2, and also [9] for the extension to E 1 of the adjoint resolvent. Taking into account that the properties of m-transience, m-recurrence, and mirreducibility are preserved by modifying the initial space with some zero measure set, the results presented in this paper have a probabilistic counterpart.…”