The Proof of Strong Markov Property Based on one Definition

In this paper, for countable homogeneous Markov process, we prove strong Markov property defining by [2] are valid. So for an arbitrary countable homogeneous Markov process is a strong Markov process.2000 Mathematics Subject Classification. Primary 60J25, 60J27.

Parametrix construction for certain Lévy-type processes

In this paper, we show that a non-local operator of certain type extends to the generator of a strong Markov process, admitting the transition probability density. For this transition probability density we construct the intrinsic upper and lower bounds, and prove some smoothness properties. Some examples are provided.

Quasi-Regular Dirichlet Forms: Examples and Counterexamples

We prove some new results on quasi-regular Dirichlet forms. These include results on perturbations of Dirichlet forms, change of speed measure, and tightness. The tightness implies the existence of an associated right continuous strong Markov process. We also discuss applications to a number of examples including cases with possibly degenerate (sub)-elliptic part, diffusions on loop spaces, and certain Fleming- Viot processes.

Two extreme value processes arising in hydrology

Let Tn be the time of occurrence of the nth flood peak in a hydrological system and Xn the amount by which the peak exceeds a base level. We assume that ((Tn, Xn)) is a Poisson random measure with mean measure μ(dx) K(x, dy). In this note we characterize two extreme value processes which are functionals of ((Tn, Xn)). The set-parameterized process {MA} defined by MA = sup {Xn:Tn ∈ A} is additive and we compute its one-dimensional distributions explicitly. The process (Mt), where Mt = sup{Xn: Tn ≦ t}, is a non-homogeneous strong Markov process. Our results extend but computationally simplify those of previous models.

Two extreme value processes arising in hydrology

Let Tn be the time of occurrence of the nth flood peak in a hydrological system and Xn the amount by which the peak exceeds a base level. We assume that ((Tn , Xn )) is a Poisson random measure with mean measure μ(dx) K(x, dy). In this note we characterize two extreme value processes which are functionals of ((Tn , Xn )). The set-parameterized process {MA } defined by MA = sup {Xn :Tn ∈ A} is additive and we compute its one-dimensional distributions explicitly. The process (Mt ), where Mt = sup{Xn : Tn ≦ t}, is a non-homogeneous strong Markov process. Our results extend but computationally simplify those of previous models.