Asymptotic expansion for transport processes in semi-Markov media (original) (raw)

Asymptotic behaviour of a transport equation

Annales Polonici Mathematici, 1992

We study the asymptotic behaviour of the semigroup of Markov operators generated by the equation u t + bux + cu = a ax 0 u(t, ax − y)µ(dy). We prove that for a > 1 this semigroup is asymptotically stable. We show that for a ≤ 1 this semigroup, properly normalized, converges to a limit which depends only on a.

Continuous Markov Semigroups and Stability of Transport Equations

Journal of Mathematical Analysis and Applications, 2000

A new theorem for asymptotic stability of Markov semigroups is proved. This result is applied to transport equations connected with diffusion and jumping processes and randomly perturbed dynamical systems.

Two-parameter semigroups, evolutions and their applications to Markov and diffusion fields on the plane

Journal of Applied Mathematics and Stochastic Analysis, 1996

We study two-parameter coordinate-wiseC0-semigroups and their generators, as well as two-parameter evolutions and differential equations up to the second order for them. These results are applied to obtain the Hille-Yosida theorem for homogeneous Markov fields of the Feller type and to establish forward, backward, and mixed Kolmogorov equations for nonhomogeneous diffusion fields on the plane.

Asymptotic expansion of semi-Markov random evolutions

Stochastics, 2009

Regular and singular parts of asymptotic expansions of semi-Markov random evolutions are given. Regularity of boundary conditions is shown. An algorithm for calculation of initial conditions is proposed.

Randomly flashing diffusion: Asymptotic properties

J Statist Phys, 1996

The theory of abstract Markov operators and semigroups is applied for studying asymptotics of a randomly flashing diffusion process. The probability distribution of the process is determined by a set of two partial differential equations and sufficient conditions for the existence of a stationary solution of the equations are formulated, and convergence of solutions to the stationary solution is proved.

Invariant measures and the Kolmogorov equation for the stochastic fast diffusion equation

Stochastic Processes and their Applications, 2010

We prove the existence of an invariant measure µ for the transition semigroup P t associated with the fast diffusion porous media equation in a bounded domain O ⊂ R d , perturbed by a Gaussian noise. The Kolmogorov infinitesimal generator N of P t in L 2 (H −1 (O), µ) is characterized as the closure of a secondorder elliptic operator in H −1 (O). Moreover, we construct the Sobolev space W 1,2 (H −1 (O), µ) and prove that D(N ) ⊂ W 1,2 (H −1 (O), µ).

Asymptotic expansion of a semi-Markov random evolution

Ukrainian Mathematical Journal, 2006

We determine the regular and singular components of the asymptotic expansion of a semi-Markov random evolution and show the regularity of boundary conditions. In addition, we propose an algorithm for finding initial conditions for t = 0 in explicit form using the boundary conditions for the singular component of the expansion. 1 determines the transition probabilities for the imbedded Markov chain κ n = κ (τ n) , n ≥ 0, and the distribution functions