Convex Figures and Polyhedra by L. A. Lyusternik, T. jefferson Smith

By L. A. Lyusternik, T. jefferson Smith

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2). This treatment will recapture the above stability definitions when this condition is assumed to hold. 12) decay to zero, and say in this case that the solutions or systems are decayable. 12) for any value of p > 0 implies its (asymptotic) moment stability for every smaller value than p and stability in probability. On the other hand, one can easily show that a null solution could be (asymptotically) p-th moment stable for some p > 0 and not (asymptotically) q-th moment stable for q > p. The case most often discussed in the literature is (asymptotic) p-th moment stability with p = 2.

For arbitrary 0 ≤ s ≤ T , let Ca ([s, T ]; Lp (Ω, F, P ; H)) be the subspace of C([s, T ]; Lp (Ω, F, P ; H)) which consists of {Ft }-adapted processes. 12), however, with initial datum xs ∈ Lps (Ω; H), s ≤ t ≤ T , t Xt = T (t − s)xs + t T (t − u)F (u, Xu )du + s Xs = xs ∈ Lps (Ω; H). 4 in the following form. 5 For any 0 ≤ s ≤ t ≤ T , there exists a unique map U (t, s) : Lps (Ω; H) → Lpt (Ω; H) with properties: Stochastic Differential Equations in Infinite Dimensions 29 (i). For any s ≤ t ≤ T , xs ∈ Lps (Ω; H), U (t, s)xs is B([s, T ])×F measurable; (ii).

Kozin [1], Khas’minskii [1] and Arnold [1] (among others) clarified some of the confusion and provided a good foundation for further work. In what follows, we have no intention of listing all the possible definitions, but prefer to confine ourselves to those which are in our view of the greatest practical interest. 12). To this end, we assume that A(t, 0) = B(t, 0) = 0 and F (t, 0) = G(t, 0) = 0 for any t ≥ 0. 12) is said to be stable in probability if for arbitrarily given ε, ε > 0, there exists δ = δ(ε, ε ) > 0 such that if x0 H < δ, then P Xt (x0 ) H >ε <ε for all t ≥ 0.

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