Unbounded functional calculus for bounded groups with applications

Abstract

Abstract. In this paper, we develop the unbounded extension of the Hille-Phillips functional calculus for generators of bounded groups. Mathematical applications include the generalised Lévy-Khintchine formula for subordinate semigroups, the analyticity of semigroups generated by fractional powers of group generators, where the power is not an odd integer, and a shifted abstract Grünwald formula. We also give an application of the theory to subsurface hydrology, modeling solute transport on a regional scale using fractional dispersion along flow lines.

Higher order Grünwald approximations of fractional derivatives and fractional powers of operators

“…In Section 3 we develop higher order Grünwald-type approximations A α h . In Corollary 3.5 we can then give the consistency error estimate (2) Ã α h f − f (α)…”

Reflected spectrally negative stable processes and their governing equations

“…g α (−x) = x −1−α g β (x −α ) for all x > 0.Then for x < 0, the first term in(5) 1+α g 1/α t (−u) α dt g(x − u) du = ) α g 1/α t (−u) α dt g(x − u) du. ) α g 1/α t (−u) α dt = exp(−[λ(−u) α ] β ) = exp(uλ 1/α ),and then the first term in (5.1) equalsx 1/α g(x − u) du = 1 (x−y)λ 1/α g(y) dy = 1 α λ 1/α−1 e xλ 1/α Lg(λ 1/α ) = Lg(λ 1/α ) αλ 1…”

“…The latter follows from (5.5) which also holds for s = ik, as g α is absolutely integrable. Then, T α t is a subordinate semigroup (where the right-translation group on BUC(R), which is strongly continuous [19,Chapter I,Section 4.15], is subordinated againstT α,t ) which is strongly continuous by [5,Theorem 4.1].…”

Mixing-driven equilibrium reactions in multidimensional fractional advection–dispersion systems

“…It can also potentially vary in space [e.g. 55, 54, 2], although in this study we will focus on cases where it does not. The fractional Laplace operator ${\mathbb{D}}_M^{\alpha }f(\mathbf{x},t)$ may be defined by its Fourier transform, which is…”