Asymptotic expansion for the functional of Markovian evolution in R^d in the circuit of diffusion approximation

Abstract

Is studied asymptotic expansion for solution of singularly perturbed equation for functional of Markovian evolution in R d . The view of regular and singular parts of solution is found. Mathematics Subject Classification (2000): primary 60J25,secondary 35C20.

Asymptotic expansion of semi-Markov random evolutions

“…The condition (20) may be used to construct a solution of the Markov renewal equation (18) for the singular terms. The boundary condition for singular terms W k (+∞) = 0, k ≥ 1 and the Markov renewal limit theorem [19] provide by a calculation:…”

“…We are going to write down an algorithm for the construction of initial conditions (at t = 0) for the regular terms using the boundary conditions for the singular terms as τ → ∞ (see Remark 1.1). For the first singular term W 1 (τ ) we have the equation (see (18)):…”

“…The last term of this equality may be rewritten without the use of singular terms. This may be done, like in the work [18], using the Laplace transform of the W i (τ ).…”

Asymptotic expansion for the distribution of a Markovian random motion

“…Let us put v = ε −1 and λ = ε −2 , where ε > 0 is a small parameter. It is well known [8], [10], [11] that the solution of Eq. (3) in hydrodynamical limit (as ε → 0) weakly converges to the corresponding functional of Wiener process.…”

“…+ ε 2k u k (t, x, y, z). In [10] for the solution of a singularly perturbed equation of type (3) the remainder of asymptotic expansion in the circuit of diffusion approximation was studied.…”