Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions

Murali Rao1, Renming Song2, Zoran Vondraček3
1University of Florida
2Department of Mathematics, University of Illinois, Urbana, U.S.A.
3Department of Mathematics, University of Zagreb, Zagreb, Croatia

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Barlow, M.T.: Diffusions on fractals, in Lecture Notes on probability theory and Statistics, École d’ Été de Probabilités de Saint-Flour XXV – 1995, Lect. Notes Math. 1690, Springer, 1998, pp. 1–121.

Bass, R.F. and Kassmann, M.: ‘Harnack inequalities for non-local operators of variable order’, Trans. Amer. Math. Soc. 357 (2005), 837–850.

Bass, R.F. and Levin, D.A.: ‘Harnack inequalities for jump processes’, Potential Anal. 17 (2002), 375–388.

Bass, R.F. and Levin, D.A.: ‘Transition probabilities for symmetric jump processes’, Trans. Amer. Math. Soc. 354 (2002), 2933–2953.

Berg C. and Forst, G.: Potential Theory on Locally Compact Abelian Groups, Springer, Berlin, 1975.

Bertoin, J.: Lévy Processes, Cambridge University Press, Cambridge, 1996.

Bingham, N.H., Goldie, C.M. and Teugels, J.L.: Regular Variation, Cambridge University Press, Cambridge, 1987.

Bogdan, K., Stós, A. and Sztonyk, P.: ‘Potential theory for Lévy stable processes’, Bull. Pol. Acad. Sci., Math. 50 (2002), 361–372.

Chen, Z.-Q. and Kumagai, T.: ‘Heat kernel estimates for stable-like processes on d-sets’, Stoch. Process. Appl. 108 (2003), 27–62.

Chen, Z.-Q. and Song, R.: ‘Drift transform and Green function estimates for discontinuous processes’, J. Funct. Anal. 201 (2003), 262–281.

Davies, E.B.: Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.

Geman, H., Madan, D.B. and Yor, M.: ‘Time changes for Lévy processes’, Math. Finance 11 (2001), 79–96.

Jacob, N.: Pseudo Differential Operators and Markov Processes, Vol. 1, Imperial College Press, London, 2001.

Jacob, N. and Schilling, R.L.: ‘Some Dirichlet spaces obtained by subordinate reflected diffusions’, Rev. Mat. Iberoamericana 15 (1999), 59–91.

Krylov, N.V. and Safonov, M.V.: ‘An estimate of the probability that a diffusion process hits a set of positive measure’, Sov. Math. Dokl. 20 (1979), 253–255.

Kumagai, T.: Some remarks for stable-like jump processes on fractals, in P. Grabner and W. Woess (eds), Trends in Math., Fractals in Graz 2001, Birkhäuser, Basel, 2002, pp. 185–196.

Riesz, M.: ‘Integrals de Riemann–Liouville et potentiels’, Acta Szeged 9 (1938), 1–42.

Ryznar, M.: ‘Estimates of Green functions for relativistic α-stable process’, Potential Anal. 17 (2002), 1–23.

Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999.

Schilling, R.L.: Zum Pfadenverhalten vom Markovschen Prozessen, die mit Lévy–Prozessen vergleichbar sind, Dissertation Universität Erlangen–Nürnberg, Erlangen, 1994.

Selmi, M.: ‘Comparison des semi-groupes et des résolvantes d’ordre α associés à des opérateurs différentiels de type divergence’, Potential Anal. 3 (1994), 15–45.

Song, R.: ‘Sharp bounds on the density, Green function and jumping function of subordinate killed BM’, Probab. Theory Related Fields 128 (2004), 606–628.

Song, R. and Vondraček, Z.: ‘Potential theory of subordinate killed Brownian motion in a domain’, Probab. Theory Related Fields 125 (2003), 578–592.

Song, R. and Vondraček, Z.: ‘Harnack inequalities for some classes of Markov processes’, Math. Z. 246 (2004), 177–202.

Song, R. and Vondraček, Z.: ‘Harnack inequality for some discontinuous Markov processes with a diffusion part, Glas. Mat. 40 (2005), 177–187.

Stós, A.: ‘Symmetric α-stable processes on d-sets’, Bull. Pol. Acad. Sci. Math. 48 (2000), 237–245.

Vondraček, Z.: ‘Basic potential theory of certain nonsymmetric strictly α-stable processes’, Glas. Mat. 37 (2002), 193–215.