We prove that in the Euclidean space of arbitrary dimension the inversion of the isotropic stable Lévy process killed at the origin is, after an appropriate change of time, the same stable process conditioned in the sense of Doob by the Riesz kernel. Using this identification we derive and explain transformation rules for the Kelvin transform acting on the Green function and the Poisson kernel of the stable process and on solutions of Schrödinger equation based on the fractional Laplacian. The Brownian motion and the classical Laplacian are included as a special case.
Similar content being viewed by others
References
Axler S., Bourdon P., Ramey W. (1992). Harmonic Function Theory. Springer-Verlag.
Bañuelos R., Bogdan K. (2004). Symmetric stable processes in cones. Potential Anal. 21(3): 263–288
Berg, C., and Forst, G. (1975). Potential Theory on Locally Compact Abelian Groups. Springer-Verlag.
Bertoin, J. (1996). Lévy processes. Cambridge University Press.
Bliedtner, J., and Hansen, W. (1986). Potential Theory. An Analytic and Probabilistic Approach to Balayage. Springer-Verlag
Blumenthal, R. M., and Getoor, R. K. (1968). Markov Processes and Potential Theory. Academic Press.
Blumenthal R.M., Getoor R.K., Ray D.B. (1961). On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99, 540–554
Bochner, S. (1995). Harmonic Analysis and the Theory of Probability. University of California Press.
Bogdan K. (1999). Representation of α-harmonic functions in Lipschitz domains. Hiroshima Math. J. 29(2): 227–243
Bogdan, K. (2000). Kelvin transform and Riesz potentials. unpublished notes.
Bogdan K., Burdzy K., Chen Z.-Q. (2003). Censored stable processes. Probab. Theory Related Fields 127, 89–152
Bogdan K., Byczkowski T. (1999). Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133(1): 53–92
Bogdan K., Byczkowski T. (2000). Potential theory of Schrödinger operator based on fractional Laplacian. Probab. Math. Statist. 20(2): 293–335
Bogdan K., Stós A., Sztonyk P. (2003). Harnack inequality for stable processes on d-sets. Studia Math. 158(2): 163–198
Bogdan K., Sztonyk P. (2005). Harnack’s inequality for stable Lévy processes. Potential Anal. 22(2): 133–150
Brelot-Collin B., Brelot M. (1973). Allure á la frontiere des solutions positives de l’équation de Weinstein \(L_k(u) = \Delta u+\frac{k}{x_n}\frac{\partial u}{\partial x_n}=0\) dans le demi-espace E (x n > 0) de R n (n ≥ 2). Bull. Acad. Royale de Belg. 59, 1100–1117
Chen Z.Q., Song R. (1998). Martin boundary and integral representation for harmonic functions of symmetric stable processes. J. Funct. Anal. 159, 267–294
Chung, K. L., and Zhao, Z. (1995). From Brownian motion to Schrödinger’s equation. Springer-Verlag.
Clerc J.-L. (2000). Kelvin transform and multi-harmonic polynomials. Acta. Math. 185, 81–99
Courant R., Hilbert D. (1954). Methoden der Mathematischen Physik, I. Springer-Verlag.
Doetsch, G. (1974). Introduction to the Theory and Application of the Laplace Transformation. Springer.
Doob J.L. (1957). Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France 85, 431–458
Durret R. (1984). Brownian Motion and Martingales in Analysis. Wadsworth, Belmont
Dziubański J. (1991). Asymptotic behaviour of densities of stable semigroups of measures. Probab. Theory Related Fields 87, 459–467
Folland, G. (1976). Introduction to Partial Differential Equations. Princeton University Press.
Hunt G.A. (1956). Some theorems concerning Brownian motion. Trans. Amer. Math. Soc. 81, 294–319
Hunt G. A. (1957). Markov Processes and Potentials: I, II, III, Illinois J. Math. 1 44–93; 316–369, 2 (1958) 151–213.
Jacob N. (2002). Pseudo-Differential Operators and Markov Processes, Generators and Their Potential Theory. Vol. II, Imperial College Press, London
Jakubowski T. (2003). The estimates for the Green function in Lipschitz domains for the symmetric stable processes. Probab. Math. Statist. 22(2): 419–441
Kac M. (1957). Some remarks on stable processes. Publ. Inst. Statist. Univ. Paris 6, 303–306
Landkof N.S. (1972). Foundations of Modern Potential Theory. Springer-Verlag.
Ma, Z. -M., and Röckner, M. (1992). Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer-Verlag.
Michalik K., Ryznar M. (2004). Relative Fatou theorem for α-harmonic functions in Lipschitz domains. Illinois J. Math. 48(3): 977–998
Michalik K., Samotij K. (2000). Martin representation for α-harmonic functions. Probab. Math. Statist. 20(1): 75–91
Port S.C. (1967). Hitting times and potentials for recurrent stable processes. J. Analyse Math. 20, 371–395
Ransford, T. (1995). Potential Theory in the Complex Plane. Cambridge University Press.
Revuz D., Yor M. (1999). Continuous martingales and Brownian motion, 3rd ed., Springer, Berlin
Riesz M. (1938). Intégrales de Riemann-Liouville et potentiels. Acta Sci. Math. Szeged 9, 1–42
Riesz M. (1938). Rectification au travail “Intégrales de Riemann-Liouville et potentiels”. Acta Sci. Math. Szeged 9, 116–118
Rogers L., Williams, D. (1987). Diffusions, Markov Processes and Martingales Vol. 2: Itô calculus. John Wiley & Sons.
Sato, K. (1999). Lévy processes and infinitely divisible distributions. Cambridge University Press.
Stein, E. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press.
Williams, D. (1979). Diffusions, Markov Processes, and Martingales Vol. 1: Foundations. John Wiley & Sons.
Yor M. (1985). À propos de l’inverse du mouvement brownien dans R n(n ≥ 3). Ann. Inst. H. Poincaré Probab. Statist. 21(1): 27–38
Yosida, K. (1980). Functional Analysis. Springer-Verlag.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bogdan, K., Żak, T. On Kelvin Transformation. J Theor Probab 19, 89–120 (2006). https://doi.org/10.1007/s10959-006-0003-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-006-0003-8
Keywords
- Inversion
- Kelvin transform
- isotropic stable Lévy process
- Brownian motion
- Doob conditional process
- Riesz kernel
- Green function
- Schrödinger equation
- Laplace transform
- resolvent