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By Masaaki Yoshida

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Denote Ai = {ω ∈ Ω : (ti , x(ti , t0 , x0 )) ∈ S} . 3) yields P(Ai ) = 1, hence P(Ai ) = 0. This means that the probability of their countable union, i Ai = {ω : ∃i, (ti , x(ti , t0 , x0 )) ∈ S} , Invariant Sets for Systems with Random Perturbations 55 equals zero. So, the probability of the set i Ai = {ω : (ti , x(ti , t0 , x0 )) ∈ S, ∀ti ≥ 0} is 1. This shows that the trajectory of the process (t, x(t, t0 , x0 , ω0 )), for ω0 ∈ ∩i Ai , belongs to S for all t ≥ 0, because otherwise there would exist a rational point ti , due to continuity of the trajectories and the set S, such that (ti , x(ti , t0 , x0 )) ∈ S.

61) Proof. 62) for any t ≥ τ ≥ 0, where the constants γ and K are positive are independent of t, τ , and x0 . 63) K exp{−γA} < . 51). Let us find an estimate for the mean of the distance between this solution and the set S over [0, A]. 10, we have t E|x(t, x0 ) − y(t, x0 )| ≤ 0 LE|x(s, x0 ) − y(s, x0 )| ds A +ε E|g(s, x(s, x0 ), ξ(s))| ds + ε 0 E|Ii (x(ti , x0 ), ηi )| 0

F. Tsarkov and his collaborators [190, 188, 189, 191, 179]. Related results are obtained by V. V. Anisimov in [4, 5, 3], and V. S. Korolyuk in [78, 79]. Limit behavior of semi-Markov processes with switchings were studied by A. V. Svishchuk [180]. However, in the mentioned works, the right-hand sides of the systems, the values of the impulses and their times were assumed to satisfy the conditions that implied that the solutions of the impulsive systems would be Markov or semi-Markov processes, which permitted to study them with analytical methods used in the theory of Markov processes, and these methods are not very “sensible” to the fact that the trajectories of the solutions are discontinuous.

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Fuchsian Differential Equations: With Special Emphasis on the Gauss-Schwarz Theory by Masaaki Yoshida

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Categories: Differential Equations