By Gabriella Caristi, Enzo Mitidieri (auth.), Erik Koelink, Jan van Neerven, Ben de Pagter, Guido Sweers, Annemarie Luger, Harald Woracek (eds.)
Capturing the state-of-the-art of the interaction among partial differential equations, sensible research, maximal regularity, and chance idea, this quantity was once initiated on the Delft convention at the party of the retirement of Philippe Clément. will probably be of curiosity to researchers in PDEs and practical analysis.
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Additional info for Partial Differential Equations and Functional Analysis: The Philippe Clément Festschrift
Some a posteriori error estimators for elliptic partial diﬀerential equations. Math. , 44, 170, 283–301.  Becker, R. and Rannacher, R. (1996). A feed-back approach to error control in ﬁnite element methods: basic analysis and examples. East-West J. Numer. Math, 4, 4, 237–264.  Becker, R. and Rannacher, R. (2001). An optimal control approach to a posteriori error estimation in ﬁnite element methods. Acta Numerica, Cambridge University Press, 2001, 1–102. , Dahmen, W. and DeVore, R. (2004).
3. 5) P L1−2γ Λ ≤ U (t) < +∞, t ≥ 0 = 1, where Λ is the random variable deﬁned by ∞ µ2n IAn , Λ := n=1 and An is deﬁned for each n ∈ N by sup |ux (t)|C[0,L] ∨ sup |uy (t)|C[0,L] ∈ [n − 1, n) An := t≥0 . 6) t≥0 Proof. We have t M t L 2 [g(ξ, ux (s, ξ)) − g(ξ, uy (s, ξ))] dξ ds = 0 0 and hence L d M t = [g(ξ, ux (t, ξ)) − g(ξ, uy (t, ξ))]2 dξ. 6) d M dt ∞ t ≥ L1−2γ µ2n IAn Z 2γ (t) = L1−2γ Λ Z 2γ (t) t ≥ 0. 2) in Hypothesis 2 we have d M dt L t ≤ L2 ρ2 (t, ξ) dξ = L2 |ρ(t)|2L2 (0,L) . 7) 54 S. s.
48 S. 7) we obtain t X(t) ≤ x − y + t ≥ 0. X(s)γ dB(s), 0 Thus, in the case γ > 1 we cannot use comparison with the geometric Brownian motion, but with the process I(t), t ≥ 0, obtained from a Bessel process Y (t) of parameter δ := 1 + γ/(γ − 1) by the formula Y (t) = 1 I 1−γ (t), γ−1 t ≥ 0. This forces us to use results on the asymptotic behavior and on the set of polar points of Bessel processes. 2. Preliminaries and hypotheses Throughout the paper we shall assume that the reaction term f satisﬁes the following conditions.
Partial Differential Equations and Functional Analysis: The Philippe Clément Festschrift by Gabriella Caristi, Enzo Mitidieri (auth.), Erik Koelink, Jan van Neerven, Ben de Pagter, Guido Sweers, Annemarie Luger, Harald Woracek (eds.)