Symmetrization, symmetric stable processes, and Riesz capacities
HTML articles powered by AMS MathViewer
- by Dimitrios Betsakos PDF
- Trans. Amer. Math. Soc. 356 (2004), 735-755 Request permission
Addendum: Trans. Amer. Math. Soc. 356 (2004), 3821-3821.
Abstract:
Let $\texttt {X}_t$ be a symmetric $\alpha$-stable process killed on exiting an open subset $D$ of $\mathbb R^n$. We prove a theorem that describes the behavior of its transition probabilities under polarization. We show that this result implies that the probability of hitting a given set $B$ in the complement of $D$ in the first exit moment from $D$ increases when $D$ and $B$ are polarized. It can also lead to symmetrization theorems for hitting probabilities, Green functions, and Riesz capacities. One such theorem is the following: Among all compact sets $K$ in $\mathbb R^n$ with given volume, the balls have the least $\alpha$-capacity ($0<\alpha <2$).References
- Albert Baernstein II, A unified approach to symmetrization, Partial differential equations of elliptic type (Cortona, 1992) Sympos. Math., XXXV, Cambridge Univ. Press, Cambridge, 1994, pp. 47–91. MR 1297773
- Rodrigo Bañuelos, RafałLatała, and Pedro J. Méndez-Hernández, A Brascamp-Lieb-Luttinger-type inequality and applications to symmetric stable processes, Proc. Amer. Math. Soc. 129 (2001), no. 10, 2997–3008. MR 1840105, DOI 10.1090/S0002-9939-01-06137-8
- Dimitrios Betsakos, Polarization, conformal invariants, and Brownian motion, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 59–82. MR 1601843
- R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc. 95 (1960), 263–273. MR 119247, DOI 10.1090/S0002-9947-1960-0119247-6
- R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR 0264757
- Krzysztof Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math. 123 (1997), no. 1, 43–80. MR 1438304, DOI 10.4064/sm-123-1-43-80
- Krzysztof Bogdan and Tomasz Byczkowski, Potential theory for the $\alpha$-stable Schrödinger operator on bounded Lipschitz domains, Studia Math. 133 (1999), no. 1, 53–92. MR 1671973, DOI 10.4064/sm-133-1-53-92
- Friedemann Brock and Alexander Yu. Solynin, An approach to symmetrization via polarization, Trans. Amer. Math. Soc. 352 (2000), no. 4, 1759–1796. MR 1695019, DOI 10.1090/S0002-9947-99-02558-1
- Zhen-Qing Chen and Renming Song, Intrinsic ultracontractivity and conditional gauge for symmetric stable processes, J. Funct. Anal. 150 (1997), no. 1, 204–239. MR 1473631, DOI 10.1006/jfan.1997.3104
- Zhen-Qing Chen and Renming Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann. 312 (1998), no. 3, 465–501. MR 1654824, DOI 10.1007/s002080050232
- Zhen-Qing Chen and Renming Song, Martin boundary and integral representation for harmonic functions of symmetric stable processes, J. Funct. Anal. 159 (1998), no. 1, 267–294. MR 1654115, DOI 10.1006/jfan.1998.3304
- Kai Lai Chung, Lectures from Markov processes to Brownian motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 249, Springer-Verlag, New York-Berlin, 1982. MR 648601, DOI 10.1007/978-1-4757-1776-1
- Kai Lai Chung and Murali Rao, Equilibrium and energy, Probab. Math. Statist. 1 (1980), no. 2, 99–108 (1981). MR 626304
- V. N. Dubinin, Capacities and geometric transformations of subsets in $n$-space, Geom. Funct. Anal. 3 (1993), no. 4, 342–369. MR 1223435, DOI 10.1007/BF01896260
- V. N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable, Uspekhi Mat. Nauk 49 (1994), no. 1(295), 3–76 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 1, 1–79. MR 1307130, DOI 10.1070/RM1994v049n01ABEH002002
- E. B. Dynkin, Markov processes. Vols. I, II, Die Grundlehren der mathematischen Wissenschaften, Band 121, vol. 122, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. MR 0193671, DOI 10.1007/978-3-662-00031-1
- E. B. Dynkin, Markov processes. Vols. I, II, Die Grundlehren der mathematischen Wissenschaften, Band 121, vol. 122, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. MR 0193671, DOI 10.1007/978-3-662-00031-1
- N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027, DOI 10.1007/978-3-642-65183-0
- Pertti Mattila, Orthogonal projections, Riesz capacities, and Minkowski content, Indiana Univ. Math. J. 39 (1990), no. 1, 185–198. MR 1052016, DOI 10.1512/iumj.1990.39.39011
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- Sidney C. Port, On hitting places for stable processes, Ann. Math. Statist. 38 (1967), 1021–1026. MR 214151, DOI 10.1214/aoms/1177698769
- Sidney C. Port, A remark on hitting places for transient stable process, Ann. Math. Statist. 39 (1968), 365–371. MR 225386, DOI 10.1214/aoms/1177698397
- Sidney C. Port and Charles J. Stone, Brownian motion and classical potential theory, Probability and Mathematical Statistics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0492329
- Jukka Sarvas, Symmetrization of condensers in $n$-space, Ann. Acad. Sci. Fenn. Ser. A. I. 522 (1972), 44. MR 348108
- A. Yu. Solynin, Polarization and functional inequalities, Algebra i Analiz 8 (1996), no. 6, 148–185 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 6, 1015–1038. MR 1458141
- Renming Song and Jang-Mei Wu, Boundary Harnack principle for symmetric stable processes, J. Funct. Anal. 168 (1999), no. 2, 403–427. MR 1719233, DOI 10.1006/jfan.1999.3470
- G. Szegö, Über einige Extremalaufgaben der Potentialtheorie, Math. Zeitschrift 31 (1930), 583-593.
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Additional Information
- Dimitrios Betsakos
- Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
- MR Author ID: 618946
- Email: betsakos@auth.gr
- Received by editor(s): July 14, 2002
- Received by editor(s) in revised form: January 23, 2003
- Published electronically: September 22, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 735-755
- MSC (2000): Primary 31B15, 60J45
- DOI: https://doi.org/10.1090/S0002-9947-03-03298-7
- MathSciNet review: 2022718
Dedicated: Dedicated to Albert Baernstein on the occasion of the thirty years of his star-function