no. 1
Harnack inequality for symmetric stable processes on fractals
[L'inégalité de Harnack pour les processus symétriques stables sur les fractals]
Bogdan, Krzysztof1 ; Stós, Andrzej1 ; Sztonyk, Paweł1
1 Institute of Mathematics, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
Comptes Rendus. Mathématique, Tome 335 (2002) no. 1, pp. 59-63.

Nous présentons l'inégalité de Harnack pour les fonctions α-harmoniques sur d-ensembles. En particulier cas de cellule naturelle du triangle de Sierpiński nous obtenons le principe de Harnack à la frontiére. Nous donnons aussi une estimation de la vitesse de decroissance des fonctions α-harmoniques près de la frontière ainsi que l'estimation de Carleson.

We study nonnegative harmonic functions of symmetric α-stable processes on d-sets F. We prove the Harnack inequality for such functions when α∈(0,2/dw)∪(ds,2). Furthermore, we investigate the decay rate of harmonic functions and the Carleson estimate near the boundary of a region in F. In the particular case of natural cells in the Sierpiński gasket we also prove the boundary Harnack principle.

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Révisé le :
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DOI : 10.1016/S1631-073X(02)02425-1
Bogdan, Krzysztof 1 ; Stós, Andrzej 1 ; Sztonyk, Paweł 1

1 Institute of Mathematics, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland 
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Bogdan, Krzysztof; Stós, Andrzej; Sztonyk, Paweł. Harnack inequality for symmetric stable processes on fractals. Comptes Rendus. Mathématique, Tome 335 (2002) no. 1, pp. 59-63. doi : 10.1016/S1631-073X(02)02425-1. http://www.numdam.org/articles/10.1016/S1631-073X(02)02425-1/

[1] Barlow, M.T. Diffusion on fractals, Lectures on Probability Theory and Statistics, École d'Ete de Probabilites de Saint-Flour XXV, 1995, Lecture Notes in Math., 1690, Springer-Verlag, New York, 1999, pp. 1-121

[2] Barlow, M.T.; Bass, R.F. The construction of Brownian motion on the Sierpinski carpet, Ann. Inst. H. Poincaré, Volume 25 (1989), pp. 225-257

[3] Barlow, M.T.; Perkins, E.A. Brownian motion on the Sierpinski gasket, Probab. Theory Related Fields, Volume 79 (1988), pp. 543-623

[4] R.F. Bass, D.A. Levin, Harnack inequalities for jump processes, Preprint

[5] Bertoin, J. Lévy Processes, Cambridge University Press, Cambridge, 1996

[6] Blumenthal, R.M.; Getoor, R.K. Markov Processes and Potential Theory, Pure Appl. Math., Academic Press, New York, 1968

[7] Bogdan, K. The boundary Harnack principle for the fractional Laplacian, Studia Math., Volume 123 (1997), pp. 43-80

[8] Bogdan, K.; Byczkowski, T. Probabilistic proof of boundary Harnack principle for α-harmonic functions, Potential Anal., Volume 11 (1999), pp. 135-156

[9] K. Bogdan, A. Stós, P. Sztonyk, Potential theory for Lévy stable processes, Bull. Polish Acad. Sci. Math. 50 (3) (2002), to appear

[10] K. Bogdan, A. Stós, P. Sztonyk, Harnack inequality for symmetric stable processes on d-sets, Preprint

[11] Falconer, K. Fractal Geometry, Mathematical Foundations and Applications, Willey, Chichester, 1990

[12] Farkas, W.; Jacob, N. Sobolev spaces on non-smooth domains and Dirichlet forms related to subordinate reflecting diffusions, Math. Nachr., Volume 224 (2001), pp. 75-104

[13] M. Fukushima, T. Uemura, On Sobolev and capacitary inequalities for contractive Besov spaces over d-sets, Preprint, 2001

[14] Ikeda, N.; Watanabe, S. On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes, J. Math. Kyoto Univ., Volume 2 (1962), pp. 79-95

[15] Jonsson, A. Brownian motion on fractals and function spaces, Math. Z., Volume 222 (1996), pp. 495-504

[16] T. Kumagai, Some remarks for stable-like jump processes on fractals, Preprint, 2001

[17] Pietruska-Pałuba, K. On function spaces related to the fractional diffusions on d-sets, Stochastics Stochastics Rep., Volume 70 (2000), pp. 153-164

[18] Stós, A. Symmetric stable processes on d-sets, Bull. Polish. Acad. Sci. Math., Volume 48 (2000), pp. 237-245

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