Intrinsic scaling properties for nonlocal operators. (English) Zbl 1371.35316
Summary: We study integrodifferential operators and regularity estimates for solutions to integrodifferential equations. Our emphasis is on kernels with a critically low singularity which do not allow for standard scaling. For example, we treat operators that have a logarithmic order of differentiability. For corresponding equations we prove a growth lemma and derive a priori estimates. We derive these estimates by classical methods developed for partial differential operators. Since the integrodifferential operators under consideration generate Markov jump processes, we are able to offer an alternative approach using probabilistic techniques.
MSC:
| 35R09 | Integro-partial differential equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 47G20 | Integro-differential operators |
| 60J75 | Jump processes (MSC2010) |
| 31B05 | Harmonic, subharmonic, superharmonic functions in higher dimensions |
References:
| [1] | Abels, H., Kassmann, M.: The Cauchy problem and the martingale problem for integrodifferential operators with non-smooth kernels. Osaka J. Math. 46, 661-683 (2009) Zbl 1196.47037 MR 2583323 · Zbl 1196.47037 |
| [2] | Bae, J.: Regularity for fully nonlinear equations driven by spatial-inhomogeneous nonlocal operators. Potential Anal. 43, 611-624 (2015)Zbl 06535137 MR 3432451 · Zbl 1385.47018 |
| [3] | Bass, R. F., Levin, D. A.: Harnack inequalities for jump processes. Potential Anal. 17, 375-388 (2002)Zbl 0997.60089 MR 1918242 · Zbl 0997.60089 |
| [4] | Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups. Ergeb. Math. Grenzgeb. 87, Springer, New York (1975)Zbl 0308.31001 MR 0481057 · Zbl 0308.31001 |
| [5] | Bertoin, J.: L´evy Processes. Cambridge Tracts in Math. 121, Cambridge Univ. Press, Cambridge (1996)Zbl 0938.60005 MR 1465812 |
| [6] | Bingham, N. H., Goldie, C. M., Teugels, J. L.: Regular Variation. Encyclopedia Math. Appl. 27, Cambridge Univ. Press, Cambridge (1987)Zbl 0667.26003 MR 1015093 · Zbl 0617.26001 |
| [7] | Caffarelli, L. A., Cabr´e, X.: Fully Nonlinear Elliptic Equations. Amer. Math. Soc. Colloq. Publ. 43, Amer. Math. Soc., Providence, RI (1995)Zbl 0834.35002 MR 1351007 · Zbl 0834.35002 |
| [8] | Caffarelli, L., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62, 597-638 (2009)Zbl 1170.45006 MR 2494809 · Zbl 1170.45006 |
| [9] | Chen, Z.-Q., Zhang, X.: H¨older estimates for nonlocal-diffusion equations with drifts. Comm. Math. Statist. 2, 331-348 (2015)Zbl 1310.60103 MR 3326235 · Zbl 1310.60103 |
| [10] | DiBenedetto, E., Gianazza, U., Vespri, V.: Harnack’s Inequality for Degenerate and Singular Parabolic Equations. Springer Monogr. Math., Springer, New York (2012) Zbl 1237.35004 MR 2865434 · Zbl 1237.35004 |
| [11] | Grzywny, T.: On Harnack inequality and H¨older regularity for isotropic unimodal L´evy processes. Potential Anal. 41, 1-29 (2014)Zbl 1302.60109 MR 3225805 · Zbl 1302.60109 |
| [12] | Jarohs, S., Weth, T.: Symmetry via antisymmetric maximum principles in nonlocal problems of variable order. Ann. Mat. Pura Appl. (4) 195, 273-291 (2016)Zbl 06548059 MR 3453602 · Zbl 1346.35010 |
| [13] | Kassmann, M., Mimica, A.: Intrinsic scaling properties for nonlocal operators. arXiv:1310.5371v2(2013) Intrinsic scaling properties for nonlocal operators1011 · Zbl 1371.35316 |
| [14] | Kassmann, M., Mimica, A.: Intrinsic scaling properties for nonlocal operators II. arXiv:1412.7566v2(2014) · Zbl 1371.35316 |
| [15] | Kassmann, M., Schwab, R. W.: Regularity results for nonlocal parabolic equations. Riv. Mat. Univ. Parma (N.S.) 5, 183-212 (2014)Zbl 1329.35095 MR 3289601 · Zbl 1329.35095 |
| [16] | Kim, S., Kim, Y.-C., Lee, K.-A.: Regularity for fully nonlinear integro-differential operators with regularly varying kernels. Potential Anal. 44, 673-705 (2016)Zbl 06580626 MR 3490545 · Zbl 1338.35086 |
| [17] | Krylov, N. V., Safonov, M. V.: An estimate for the probability of a diffusion process hitting a set of positive measure. Dokl. Akad. Nauk SSSR 245, 18-20 (1979) (in Russian) Zbl 0459.60067 MR 0525227 |
| [18] | Landis, E. M.: Second Order Equations of Elliptic and Parabolic Type. Izdat. “Nauka”, Moscow (1971) (in Russian)Zbl 0226.35001 MR 0320507 · Zbl 0226.35001 |
| [19] | Mimica, A.: Harnack inequality and H¨older regularity estimates for a L´evy process with small jumps of high intensity. J. Theor. Probab. 26, 329-348 (2013)Zbl 1279.60100 MR 3055806 · Zbl 1279.60100 |
| [20] | Mimica, A.: On harmonic functions of symmetric L´evy processes. Ann. Inst. H. Poincar´e Probab. Statist. 50, 214-235 (2014)Zbl 1298.60054 MR 3161529 · Zbl 1298.60054 |
| [21] | Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. 3rd ed., Grundlehren Math. Wiss. 293, Springer, Berlin (1999)Zbl 1087.60040 MR 1725357 · Zbl 0917.60006 |
| [22] | Sato, K.-I.: L´evy Processes and Infinitely Divisible Distributions. Cambridge Stud. Adv. Math. 68, Cambridge Univ. Press, Cambridge (2013) [ ˇSSV06]ˇSiki´c, H., Song, R., Vondraˇcek, Z.: Potential theory of geometric stable processes. Probab. Theory Related Fields 135, 547-575 (2006)Zbl 1099.60051 MR 2240700 |
| [23] | Silvestre, L.: H¨older estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 55, 1155-1174 (2006)Zbl 1101.45004 MR 2244602 · Zbl 1101.45004 |
| [24] | Silvestre, L., Schwab, R.: Regularity for parabolic integro-differential equations with very irregular kernels. · Zbl 1349.47079 |
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