Download Acta Numerica 1997: Volume 6 by Arieh Iserles PDF

By Arieh Iserles

Acta Numerica surveys every year crucial advancements in numerical research. the themes and authors, selected by means of a exotic overseas panel, supply a survey of articles striking of their caliber and breadth. This quantity contains articles on multivariate integration; numerical research of semiconductor units; quickly transforms in utilized arithmetic; complexity concerns in numerical research.

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Extra resources for Acta Numerica 1997: Volume 6

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Regularity in Time This section is devoted to the analysis of the H¨ older continuity in time of the t stochastic integral process {vG,Z (x), t ∈ [0, T ]}, when x is fixed. Throughout this section, O denotes a bounded domain in R3 and we shall make the following assumption on the integrand process Z. 7. For some fixed q ∈ ]3, ∞[ and γ ∈ ]0, 1[, sup E Z(t) t∈[0,T ] q W γ,q (O T −t ) < ∞. 3). 6, it makes sense to fix the argument x ∈ O in the stochastic integral process. In addition, by the above mentioned Sobolev embedding, there is C < ∞ such that the H¨ older norm · C ρ (O) is bounded by a constant times the Sobolev norm · W γ,q (O) , provided ρ ∈ ]0, γ − 3q [.

To this t end, we first prove that for any fixed m ∈ N and t ∈ [0, T ], (vG , n ≥ mT ) is n ,Z a sequence of bounded and equicontinuous functions defined on Om with values in Lq¯(Ω), for any q¯ ∈ [1, q]. 2. REGULARITY IN TIME for any ρ ∈ ]0, ρ0 [, with ρ0 = γ ∧ τ (β, δ) − yields sup sup 3 q . Therefore, the Sobolev embedding t vG n ,Z sup E m∈N n≥mT t∈[0,T ] 31 q C ρ˜(Om ) < ∞, for any ρ˜ ∈ ]0, ρ0 − 3q [. 31) t t ˜q sup E |vG (x) − vG (y)|q¯ ≤ C|x − y|ρ¯ , n ,Z n ,Z n≥mT for every q¯ ∈ [1, q], where C does not depend on m.

32) 1 |t − t¯| I1 (t, t¯, x)| ≤ G(t, du) v0 C 1 (DT ) |u| t R3 t ¯ ≤ C v0 C 1 (DT ) |t − t|, since G(t, ·) is concentrated on Bt (0) and has total mass t. 33) yield the result stated in (a). Let us now prove (b). Clearly, sup t∈[0,T ] G(t) ∗ v˜0 ∞,D ≤ T v˜0 ∞,D T . 3. 3). Hence I3 (t, t¯, x) ≤ I31 (t, t¯, x) + I32 (t, t¯, x), where I31 (t, t¯, x) = I32 (t, t¯, x) = G(t, du) R3 R3 t¯ t¯ v˜0 (x − u) − v˜0 x − u t t G(t, du) |˜ v0 (x − u)| 1 − , t¯ . t Using the H¨ older-continuity property of v˜0 , we obtain sup I31 (t, t¯, x) ≤ C v˜0 x∈D C γ2 (D T ) |t − t¯|γ2 .

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