Stochastic Kolmogorov systems driven by wideband noises

Two-Time-Scale Regime-Switching Stochastic Kolgomorov Systems With Wideband Noises

“…where o(ε) → 0 as ε → 0. Using boundedness of x ε,N (•) owing to the truncation, the continuity of f (•, •, ι) and σ(•, •, ι), the boundedness of {ξ(u)}, ξ ε (u) = ξ(u/ε 2 ), and {ξ(t)} being stationary mixing, similar to the approach in [38], it can be shown that…”

“…More precise description and the conditions of the wideband noise will be given in the subsequent section. In [38], we showed how the idealized Brownian model can be approximated by a physically realizable model. The models that we are dealing with is non-Markovian.…”

“…Moreover, the letter N is used as an index to reflect the dependence on the truncation level N . Comparing to the work [38], the terms involve an α ε (t) dependence, which makes the analysis more difficult.…”

“…In our recent work [38], we examined a class of such non-Markovian systems, from the consideration of approximation and scaling limit. Although the mathematical idealization enables us to take advantage of the Markovian structure, in reality, very often, one does not have true "white" noise, but only has something close to "white" noise.…”

“…The limit systems are driven by a Brownian motion. In this paper, taking the result of [38] as a point of departure, we further examine Kolmogorov systems in which in addition to the wideband noise, the systems are hybrid with a switching component that are subject to both strong and weak interactions. Mathematically, the systems of interest are singularly perturbed systems; see also related work [29,30] among others.…”