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1 2006

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Bessel potentials, Green functions and exponential functionals on half-spaces

Byczkowski T., Ryznar M., Byczkowska H.

Probability and Mathematical Statistics

2006

Vol. 26, Fasc. 1

155--173

The purpose of the paper is to provide precise estimates for the Green function corresponding to the operator (I—Δ)α/2, 0 <α< 2. The potential theory of this operator is based on Bessel potentials Jα=(I—Δ) -α/2. In probabilistic terms it corresponds to a subprobabilistic process obtained from the so-called relativistic a-stable process. We are interested in the theory of the killed process when exiting a fixed half-space. The crucial role in our research is played by (recently found) an explicit form of the Green function of a half-space. We also examine properties of some exponential functionals corresponding to the operator (I—Δ) α/2.