Approximation and Entropy Numbers of Volterra Operators with Application to Brownian Motion M.A. Lifshits, W.Linde We consider the Volterra integral operator from Lp (0, ∞) → Lq (0, ∞), defined by Z s (Tρ,ψ f )(s) = ρ(s) ψ(t)f (t)dt 0
and investigate its degree of compactness (entropy numbers and approximation numbers). The entropy estimates are applied to investigate the small ball behaviour of weighted Wiener processes ρW in the Lq -norm. For example, if ρ satisfies some weak monotonicity conditions at zero and infinity, then lim ¯2 log P (kρW kq ≤ ¯) = −kq kρk22q .
(TÏ,Ïf)(s) = Ï(s). â« s. 0 Ï(t)f(t)dt and investigate its degree of compactness (entropy numbers and approximation numbers). The entropy estimates are applied to ...
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