On Kelvin Transformation

Bogdan, Krzysztof; Żak, Tomasz

doi:10.1007/s10959-006-0003-8

Cited by 56 publications

(86 citation statements)

References 24 publications

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“…(d) Similar results are easily found for half-space and for the complement of the unit ball by means of Kelvin transformation, see [5,17,20] (e) The results for functions supported in the full space R d are more rare: a formula for (−∆) α/2 f (x) is known when f (x) = e iy·x (Fourier transform), f (x) = |x| −a with a ∈ (0, d) (composition with Riesz kernels, [17,25]), f (x) = e −|x| 2 or f (x) = (1+|x| 2 ) −a with a = d+1 2 or a = d−α 2 + n, n = 0, 1, . .…”

Section: Introductionsupporting

confidence: 76%

Fractional Laplace Operator and Meijer G-function

2016

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We significantly expand the number of functions whose image under the fractional Laplace operator can be computed explicitly. In particular, we show that the fractional Laplace operator maps Meijer G-functions of |x| 2 , or generalized hypergeometric functions of −|x| 2 , multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the operator (1 − |x| 2 ) α/2 + (−∆) α/2 with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace operator in the unit ball in a companion paper [9].

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Section: Introductionsupporting

confidence: 76%

Fractional Laplace Operator and Meijer G-function

2016

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“…[3] and [4]. One of the other reasons why we worked on this topic is a strong need of such analytical tools to develop the potential theory of various important processes, e.g.…”

Section: Motivations and Main Resultsmentioning

confidence: 99%

“…Taking into account the results of [4] for stable processes, it is natural to ask whether such analytical constructions of conditioned processes should also be available for one-dimensional self-similar processes. In a work in progress, this question and some related topics are studied in collaboration with L. Chaumont.…”

Section: Motivations and Main Resultsmentioning

confidence: 99%

“…While the latter path transform involves time reversal from cooptional times, such as last passage time, the construction we present here involves only deterministic inversions and time changes with random clocks of the form (5). Although we only consider one dimensional diffusions in this paper, inversions of stable processes and Brownian motion in higher dimensions are studied in [4] and [27], respectively.…”

Section: Preliminaries On Dual Processes and Inversionsmentioning

confidence: 99%

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On Inversions and Doob h-Transforms of Linear Diffusions

Alili

Graczyk

Żak

2015

Lecture Notes in Mathematics

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Let X be a regular linear diffusion whose state space is an open interval E ⊆ R. We consider the dual diffusion X * whose probability law is obtained as a Doob h-transform of the law of X, where h is a positive harmonic function for the infinitesimal generator of X on E. We provide a construction of X * as a deterministic inversion I(X) of X, time changed with some random clock. Such inversions generalize the Euclidean inversions that intervene when X is a Brownian motion. The important case where X * is X conditioned to stay above some fixed level is included. The families of deterministic inversions are given explicitly for the Brownian motion with drift, Bessel processes and the 3-dimensional hyperbolic Bessel process.

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“…[4,7,11,14,18,19,24] and references given there. The idea of employing the method to prove generation theorems for semigroups of operators goes apparently back to W. Feller who constructed the semigroup of the minimal (and the reflected) Brownian motion on R + by noting that the space of odd (even) functions is left invariant by the semigroup of the unrestricted Brownian motion on R, see [12], pp.…”

Section: The General Ideamentioning

confidence: 99%

Lord Kelvin’s method of images in semigroup theory

Bobrowski

2010

Semigroup Forum

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We show that Lord Kelvin's method of images is a way to prove generation theorems for semigroups of operators. To this end we exhibit three examples: a more direct semigroup-theoretic treatment of abstract delay differential equations, a new derivation of the form of the McKendrick semigroup, and a generation theorem The general ideaLord Kelvin's method of images is an ingenious way of solving problems involving boundary conditions, see e.g. [4,7,11,14,18,19,24] and references given there. The idea of employing the method to prove generation theorems for semigroups of operators goes apparently back to W. Feller who constructed the semigroup of the minimal (and the reflected) Brownian motion on R + by noting that the space of odd (even) functions is left invariant by the semigroup of the unrestricted Brownian motion on R, see [12], pp. 341-343, comp. [3]. These semigroups are generated by the one-dimensional Laplacian with Dirichlet and Neumann boundary conditions at the origin, respectively. A question whether a similar construction can be carried out also in the case of more general boundary conditions, seems to have been opened for Communicated

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On Kelvin Transformation

Cited by 56 publications

References 24 publications

Fractional Laplace Operator and Meijer G-function

Fractional Laplace Operator and Meijer G-function

On Inversions and Doob h-Transforms of Linear Diffusions

Lord Kelvin’s method of images in semigroup theory

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