Diffusion approximation of evolutionary systems with equilibrium in asymptotic split phase space. (English) Zbl 1075.60099

Korolyuk, Vladimir S.; Limnios, Nicolaos

Teor. Jmovirn. Mat. Stat. 70, 63-73 (2004) and Theory Probab. Math. Stat. 70, 71-82 (2005).

The authors deal with additive functionals of the form \(\xi(t)=\int_0^t\eta(ds;x(s))\), where \(x(t),t\geq0,\) is a switching Markov process and \(\eta(t;x),t\geq0,\) is a switched \(\mathbb R^d\)-valued Markov process with locally independent increments. In their previous papers they discussed average and diffusion approximations of these additive functionals for a reducible Markov process \(x(t),t\geq0,\) with the balance condition \(\int_E\pi(ds)a(u;x)=0\), where \(\pi(dx)\) is the stationary distribution of \(x(t),t\geq0,\) and \(a(u;x)\) is the drift velocity of \(\eta(t;x)\). In this paper they propose nonhomogeneous diffusion approximation results without this balance condition on the drift parameter. A diffusion approximation result for an asymptotic split phase space of the switching Markov process is presented.

MSC:

60J55	Local time and additive functionals
60J75	Jump processes (MSC2010)
60F17	Functional limit theorems; invariance principles

Keywords:

additive functional; Markov process with locally independent increments; nonhomogeneous diffusion