We study the asymptotic behavior of mixed functionals of the formHere ξT (t) is a strong solution of the stochastic differential equation dξT (t) = aT (ξT (t)) dt + dWT (t), T > 0 is a parameter, aT = aT (x) are measurable functions such that |aT (x)| ≤ CT for all x ∈ R, WT (t) are standard Wiener processes, FT = FT (x), x ∈ R, are continuous functions, gT = gT (x), x ∈ R, are locally bounded functions, and everything is real-valued. The explicit form of the limiting processes for IT (t) is established under very nonregular dependence of gT and aT on the parameter T .Keywords Diffusion-type processes, asymptotic behavior of additive functionals, nonregular dependence on the parameter 2010 MSC 60H10, 60J60

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We introduce the notion of invariant surfaces for inhomogeneous stochastic differential equations with jumps. The results obtained enable one to determine invariant surfaces for stochastic differential equations of the type indicated.

We consider functionals of the type t 0 g(ξ(s)) dW (s), t ≥ 0. Here g is a real valued and locally square integrable function, ξ is a unique strong solution of the Itô stochastic differential equation dξ(t) = a(ξ(t)) dt + dW (t), a is a measurable real valued bounded function such that |xa(x)| ≤ C. The behavior of these functionals is studied as t → ∞. The appropriate normalizing factor and the explicit form of the limit random variable are established.

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