By Yuri Kabanov

Two-scale platforms defined by means of singularly perturbed SDEs were the topic of considerable literature. notwithstanding, this new monograph develops topics that have been hardly ever addressed and will take delivery of the collective description "Stochastic Tikhonov-Levinson conception and its applications." The booklet presents a mathematical equipment designed to research the dynamic behaviour of a randomly perturbed method with speedy and gradual variables. unlike the deterministic Tikhonov-Levinson thought, the fundamental version is defined in a extra real looking method via stochastic differential equations. This ends up in a few new theoretical questions yet concurrently permits us to regard in a unified means an incredibly broad spectrum of functions like quickly modulations, approximate filtering, and stochastic approximation.Two-scale structures defined through singularly perturbed SDEs were the topic of considerable literature. even if, this new monograph develops matters that have been hardly ever addressed and will take delivery of the collective description "Stochastic Tikhonov-Levinson concept and its applications." The booklet offers a mathematical gear designed to research the dynamic behaviour of a randomly perturbed approach with quick and sluggish variables. not like the deterministic Tikhonov-Levinson concept, the elemental version is defined in a extra practical manner through stochastic differential equations. This ends up in a couple of new theoretical questions yet at the same time permits us to regard in a unified manner an incredibly broad spectrum of functions like quick modulations, approximate filtering, and stochastic approximation.

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3 that for any γ > 0 there exists r0 = r0 (γ) > 0 such that |˜ yr0 − ϕ(0, xo )| ≤ γ/4. By virtue of continuity of ϕ for sufficiently small ε |ϕ(0, xo ) − ϕ(r0 ε, xo )| ≤ γ/4. Therefore, 48 2 The Tikhonov Theory for SDEs |Δεt0 | = |ytε0 − ϕ(t0 , xεt0 )| ≤ |˜ yrε0 − y˜r0 | + |˜ yr0 − ϕ(0, xo )| +|ϕ(0, xo ) − ϕ(r0 ε, xo )| + |ϕ(t0 , xo ) − ϕ(t0 , xεt0 )| ≤ |˜ yrε0 − y˜r0 | + |ϕ(t0 , xo ) − ϕ(t0 , xεt0 )| + γ/2. ε By the local Lipschitz condition in the set {σN ≥ t0 } (where |xεt0 | ≤ N ) we have with L = LN that |ϕ(t0 , xo ) − ϕ(t0 , xεt0 )| ≤ L|xo − xεt0 | ≤ L|xo − x ˜εr0 |.

E. 2) have an exponential decay with a rate γ. The rate of exponential decay is an essential parameter and plays an important role in our applications of the moment bounds. It is well-known that for time-dependent A, even smooth, it is not possible to guarantee the asymptotic stability only in spectral terms. 2) has a solution of the exponential growth. 7). 2; results of the latter are helpful in our stochastic version of the Tikhonov theory. 3 we show also that √ ln T is the exact growth rate for the Ornstein–Uhlenbeck process.

Thus, the representation Vθ1 = θ Φε (θ, s)αε,m s ds t0 and above estimates lead to the inequality IΓ |Vθ1 | ≤ CN,m ε + (ρN (η) + φN (δ))η + CN /m.

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