Differential Galois theory has played important roles in the the- ory of integrability of linear differential equation. In this paper we will extend the theory to nonlinear case and study the integrability of the first order non- linear differential equation. We will define for the differential equation the differential Galois group, will study the structure of the group, and will prove the equivalent between the existence of the Liouvillian first integral and the solvability of the corresponding differential Galois group.

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We prove a structure theorem for any n-rectifiable set E⊂Rn+1,n≥1, satisfying a weak version of the lower ADR condition, and having locally finite Hn (n-dimensional Hausdorff) measure. Namely, that Hn-almost all of E can be covered by a countable union of boundaries of bounded Lipschitz domains contained in Rn+1∖E. As a consequence, for harmonic measure in the complement of such a set E, we establish a non-degeneracy condition which amounts to saying that Hn|E is “absolutely continuous” with respect to harmonic measure in the sense that any Borel subset of E with strictly positive Hn measure has strictly positive harmonic measure in some connected component of Rn+1∖E. We also provide some counterexamples showing that our result for harmonic measure is optimal. Moreover, we show that if, in addition, a set E as above is the boundary of a connected domain Ω⊂Rn+1 which satisfies an infinitesimal interior thickness condition, then Hn|∂Ω is absolutely continuous (in the usual sense) with respect to harmonic measure for Ω. Local versions of these results are also proved: if just some piece of the boundary is n-rectifiable then we get the corresponding absolute continuity on that piece. As a consequence of this and recent results in “Rectifiability of harmonic measure” [Geom. Funct. Anal. 26 (2016), 703–728], we can decompose the boundary of any open connected set satisfying the previous conditions in two disjoint pieces: one that is n-rectifiable where Hausdorff measure is absolutely continuous with respect to harmonic measure and another purely n-unrectifiable piece having vanishing harmonic measure.

This paper studies the moment boundedness of solutions of linear stochastic delay differential equations with distributed delay. For a linear stochastic delay differential equation, the rst moment stability is known to be identical to that of the corresponding deterministic delay differential equation. However, boundedness of the second moment is complicated and depends on the stochastic terms. In this paper, the characteristic function of the equation is obtained through techniques of the Laplace transform. From the characteristic equation, suf cient conditions for the second moment to be bounded or unbounded are proposed.