On the distribution of first hits for the symmetric stable processes.

Blumenthal, Robert; Getoor, R. K.; Ray, Daniel

doi:10.1090/s0002-9947-1961-0126885-4

Cited by 74 publications

(73 citation statements)

References 12 publications

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“…The Green function G(x, y) for (−∆) α/2 in the unit ball was found in [20]. Traditionally, it is written in the form that was first stated in [3],…”

Section: Unit Ball and Its Complementmentioning

confidence: 99%

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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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“…The Green function G(x, y) for (−∆) α/2 in the unit ball was found in [20]. Traditionally, it is written in the form that was first stated in [3],…”

Section: Unit Ball and Its Complementmentioning

confidence: 99%

“…For a formal derivation of the expression for the harmonic measure, see Section IV.5 in [17]. The expression for the Green function in its most common form was obtained in [3].…”

Section: Introductionmentioning

confidence: 99%

Fractional Laplace Operator and Meijer G-function

2016

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show abstract

“…Although a lot is known about symmetric stable processes and their potential theory (see [1,2,3,4,9,10,12,14,16,17,20,21] and the references therein), little is known about the counterparts to some of the deep results for Brownian motion, such as sharp estimates on Green functions and Poisson kernels of bounded domains. In the special case of balls, the explicit formulae for the Poisson kernels and Green functions for symmetric stable processes are known.…”

Section: −T|ξ|mentioning

confidence: 99%

“…In the special case of balls, the explicit formulae for the Poisson kernels and Green functions for symmetric stable processes are known. The formula for the Poisson kernel of balls were obtained by M. Riesz and the formula for Green function of balls were obtained by Blumenthal, Getoor and Ray in [3]. Prior to that, Kac, Polland, Spitzer, Widom, Kesten and Kinney had obtained some results for 1-dimensional case (cf.…”

Section: −T|ξ|mentioning

confidence: 99%