Ergodicity of a certain Markov process of Poisson type

One proves the ergodicity of a Markov process of Poisson type with reflection at zero.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 832–835, June, 1990.     

On the structure of Markov flows

A new infinitesimal characterization of completely positive but not necessarily homomorphic Markov flows from a C^*-algebra to bounded operators on the boson Fock space over L²(R) is given. Contrarily to previous characterizations, based on stochastic differential equations, this characterization is universal, i.e., valid for arbitrary Markov flows. With this result the study of Markov flows is reduced to the study of four C₀-semigroups. This includes the classical case and even in this case it seems to be new. The result is applied to deduce a new existence theorem for Markov flows.     

On the structure of certain po-semigroups

Communicated by M. W. Mislove     

On a Markov process generated by non-decreasing concave functions

A discrete-time Markov process on [0, ∞) is considered. The process is generated by selecting at each time, in an independent and stationary way, a concave non-decreasing function. Sufficient conditions for the existence of a unique stationary limiting distribution are given.     

On the transition from a Markov chain to a continuous time process

Starting from a real-valued Markov chain X₀,X₁,…,X_n with stationary transition probabilities, a random element {Y(t);t[0, 1]} of the function space D[0, 1] is constructed by letting Y(k/n)=X_k, k= 0,1,…,n, and assuming Y (t) constant in between. Sample tightness criteria for sequences {Y(t);t[0,1]};_n of such random elements in D[0, 1] are then given in terms of the one-step transition probabilities of the underlying Markov chains. Applications are made to Galton-Watson branching processes.     

On the structure of certain valued fields

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the n-th residue rings are isomorphic for each $n\ge 1$, then $K_1$ and $K_2$ are isometric and isomorphic. More generally, for $n_1\ge 1$, there is $n_2$ depending only on the ramification indices of $K_1$ and $K_2$ such that any homomorphism from the $n_1$-th residue ring of $K_1$ to the $n_2$-th residue ring of $K_2$ can be lifted to a homomorphism between the valuation rings. Moreover, we get a functor from the category of certain principal Artinian local rings of length n to the category of certain complete discrete valuation rings of mixed characteristic with perfect residue fields, which naturally generalizes the functorial property of unramified complete discrete valuation rings. Our lifting result improves Basarab's relative completeness theorem for finitely ramified henselian valued fields, which solves a question posed by Basarab, in the case of perfect residue fields.     

On the structure of 1-dependent Markov chains

Any stationary 1-dependent Markov chain with up to four states is a 2-block factor of independent, identically distributed random variables. There is a stationary 1-dependent Markov chain with five states which is not, even though every 1-dependent renewal process is a 2-block factor.     

Zero-one laws for the excursions and range of a Lévy process

Summary A 01 law for the excursions of a Lévy process X from 0 is established, and is used to prove that the closed range of X is either a nowhere dense set, or else contains an interval including X ₀.     

On certain exact relations for sojourn probabilities of a wiener process

New exact relations are proved for the sojourn probability of a Wiener process between two time-de-pendent boundaries. The proof is based on the investigation of the heat-conduction equation in the domain determined by these functions-boundaries. The relations are given in the form of series. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 842–846, June, 1999