Abstract
Suppose d ≥ 2 and α ∈ (1, 2). Let D be a (not necessarily bounded) C 1,1 open set in ℝd and μ = (μ 1, . . . , μ d) where each μ j is a signed measure on ℝd belonging to a certain Kato class of the rotationally symmetric α-stable process X. Let X μ be an α-stable process with drift μ in ℝd and let X μ,D be the subprocess of X μ in D. In this paper, we derive sharp two-sided estimates for the transition density of X μ,D.
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Blumenthal, R.M., Getoor, R.K.: Some theorems on stable processes. Trans. Am. Math. Soc. 95, 263–273 (1960)
Bogdan, K.: The boundary Harnack principle for the fractional Laplacian. Stud. Math. 123, 43–80 (1997)
Bogdan, K., Grzywny, T., Ryznar, M.: Heat kernel estimates for the fractional Laplacian with Dirichlet conditions. Ann. Probab. 38, 1901–1923 (2010)
Bogdan, K., Jakubowski, T.: Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Commun. Math. Phys. 271, 179–198 (2007)
Bogdan, K., Jakubowski, T.: Estimates of the Green function for the fractional Laplacian perturbed by gradient. Potent. Anal. 36, 455–481 (2012)
Bogdan, K., Kulczycki, T., Nowak, A.: Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes. Ill. J. Math. 46, 541–556 (2002)
Chen, Z.-Q., Kim, P., Song, R.: Heat kernel estimates for Dirichlet fractional Laplacian. J. Eur. Math. Soc. 12, 1307–1329 (2010)
Chen, Z.-Q., Kim, P., Song, R.: Two-sided heat kernel estimates for censored stable-like processes. Probab. Theor. Relat. Fields 146, 361–399 (2010)
Chen, Z.-Q., Kim, P., Song, R.: Dirichlet heat kernel estimates for Δα/2 + Δβ/2. Ill. J. Math. 54, 1357–1392 (2010). Special issue in honor of D. Burkholder
Chen, Z.-Q., Kim, P., Song, R.: Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation. Ann. Probab. 40(6), 2483–2538 (2012)
Chen, Z.-Q., Kim, P., Song, R.: Sharp heat kernel estimates for relativistic stable processes in open sets. Ann. Probab. 40, 213–244 (2012)
Chen, Z.-Q., Kim, P., Song, R., Vondraček, Z.: Boundary Harnack principle for Δ + Δα/2. Trans. Amer. Math. Soc. 364, 4169–4205 (2012)
Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on d-sets. Stoch. Proc. Appl. 108, 27–62 (2003)
Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory. Relat. Fields 140, 277–317 (2008)
Chen, Z.-Q., Song, R.: Estimates on Green functions and Poisson kernels of symmetric stable processes. Math. Ann. 312, 465–601 (1998)
Chung, K.L., Walsh, J.B.: Markov Processes, Brownian Motion, and Time Symmetry. Springer, New York (2005)
Getoor, R.K.: Duality of Lévy systems. Z. Wahrsch. Verw. Gebiete 19, 257–270 (1971)
Kim, P., Song, R.: Two-sided estimates on the density of Brownian motion with singular drift. Ill. J. Math 50, 635–688 (2006)
Kim, P., Song, R.: Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains. Math. Ann. 339, 135–174 (2007)
Kim, P., Song, R.: Potential theory of truncated stable processes. Math. Z. 256(1), 139–173 (2007)
Kim, P., Song, R.: Stable process with singular drift. Proc. Stoch. Appl. (to appear) (2013)
Liao, M.: Riesz Representation and Duality of Markov Processes. Ph.D. Dissertation, Department of Mathematics, Stanford University (1984)
Liao, M.: Riesz representation and duality of Markov processes. Lect. Notes. Math. 1123, 366–396 (1985). Springer, Berlin
Shur, M.G.: On dual Markov processes. Teor. Verojatnost. i Primenen. 22, 264–278 (1977), English translation: Theor. Probab. Appl. 22(1977), 257–270 (1978)
Song, R.: Estimates on the Dirichlet heat kernel of domains above the graphs of bounded C 1,1 functions. Glas. Mat. 39, 273–286 (2004)
Song, R., Wu, J.: Boundary Harnack principle for symmetric stable processes. J. Funct. Anal. 168, 403–427 (1999)
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Research of P. Kim was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (2013004822)
Research of R. Song was supported by part by a grant from the Simons Foundation (208236)
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Kim, P., Song, R. Dirichlet Heat Kernel Estimates for Stable Processes with Singular Drift In Unbounded C 1,1 Open Sets. Potential Anal 41, 555–581 (2014). https://doi.org/10.1007/s11118-013-9383-4
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DOI: https://doi.org/10.1007/s11118-013-9383-4
Keywords
- Symmetric α-stable process
- Gradient operator
- Heat kernel
- Transition density
- Green function
- Exit time
- Lévy system
- Boundary Harnack inequality
- Kato class