Abstract
We study translation-invariant integrodifferential operators that generate Lévy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided.
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Abatangelo, N., Jarohs, S., Saldaña, A.: Green function and Martin kernel for higher-order fractional Laplacians in balls. Nonlinear Analysis 175, 173–190 (2018)
Bae, J., Kassmann, M.: Schauder estimates in generalized Hölder spaces. Preprint, 2015, arXiv:1505.05498 (2015)
Bass, R.F.: Regularity results for stable-like operators. J. Funct. Anal. 257(8), 2693–2722 (2009)
Bertoin, J.: Lévy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996)
Blumenthal, R.M., Getoor, R.K., Ray, D.B.: On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99, 540–554 (1961)
Bogdan, K.: Representation of α-harmonic functions in Lipschitz domains. Hiroshima Math. J. 29(2), 227–243 (1999)
Bogdan, K., Byczkowski, T.: Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133(1), 53–92 (1999)
Bogdan, K., Grzywny, T., Pietruska-Pałuba, K., Rutkowski, A.: Extension and trace for nonlocal operators. To appear in J. Math. Pure Appl. https://doi.org/10.1016/j.matpur.2019.09.005 (2019)
Bogdan, K., Grzywny, T., Ryznar, M.: Barriers, exit time and survival probability for unimodal Lévy processes. Probab. Theory Related Fields 162(1-2), 155–198 (2015)
Bogdan, K., Żak, T.: On Kelvin transformation. J. Theoret. Probab. 19(1), 89–120 (2006)
Bucur, C.: Some observations on the Green function for the ball in the fractional Laplace framework. Commun. Pure Appl. Anal. 15(2), 657–699 (2016)
Burch, C.C.: The Dini condition and regularity of weak solutions of elliptic equations. J. Differential Equations 30(3), 308–323 (1978)
Chung, K.L., Zhao, Z.X.: From Brownian Motion to Schrödinger’s Equation, vol 312 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1995)
Duzaar, F., Gastel, A., Mingione, G.: Elliptic systems, singular sets and Dini continuity. Comm. Partial Differential Equations 29(7-8), 1215–1240 (2004)
Dyda, B., Kassmann, M.: Function spaces and extension results for nonlocal dirichlet problems. J. Funct. Anal. 277(11), 108–134 (2019)
Dynkin, E.B.: Markov processes. Vols. I, II, vol 122 of Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, Bände 121. Academic Press Inc., Publishers, New York; Springer, Berlin (1965)
Felsinger, M., Kassmann, M., Voigt, P.: The Dirichlet problem for nonlocal operators. Math. Z. 279(3-4), 779–809 (2015)
Grubb, G.: Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators. Anal. PDE 7(7), 1649–1682 (2014)
Grzywny, T.: On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes. Potential Anal. 41(1), 1–29 (2014)
Grzywny, T., Kwaśnicki, M.: Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes. Stochastic Process. Appl. 128(1), 1–38 (2018)
Grzywny, T., Ryznar, M.: Hitting times of points and intervals for symmetric Lévy processes. Potential Anal. 46(4), 739–777 (2017)
Grzywny, T., Ryznar, M, Trojan, B.: Asymptotic behaviour and estimates of slowly varying convolution semigroups. International Mathematics Research Notices 2019(23), 7193–7258 (2019)
Grzywny, T., Szczypkowski, K.: Estimates of heat kernel for non-symmetric Lévy processes. Preprint, 2017, arXiv:1710.07793 (2017)
Grzywny, T., Szczypkowski, K.: Lévy processes: concentration function and heat kernel bounds. Preprint, 2019, arXiv:1907.00778 (2019)
Grzywny, T., Szczypkowski, K.: Kato classes for Lévy processes. Potential Anal. 47(3), 245–276 (2017)
Hartman, P., Wintner, A.: On uniform Dini conditions in the theory of linear partial differential equations of elliptic type. Amer. J. Math. 77, 329–354 (1955)
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. 2, 79–95 (1962)
Kac, M.: Some remarks on stable processes. Publ. Inst. Statist. Univ. Paris 6, 303–306 (1957)
Kim, M., Kim, P., Lee, J., Lee, K.-A.: Boundary regularity for nonlocal operators with kernels of variable orders. J. Funct. Anal. 277(1), 279–332 (2019)
Kim, P., Mimica, A.: Harnack inequalities for subordinate Brownian motions. Electron. J. Probab. 17(37), 1–23 (2012)
Kovats, J.: Fully nonlinear elliptic equations and the Dini condition. Comm. Partial Differential Equations 22(11-12), 1911–1927 (1997)
Kulczycki, T., Ryznar, M.: Gradient estimates of harmonic functions and transition densities for Lévy processes. Trans. Amer. Math. Soc. 368(1), 281–318 (2016)
Matiı̆čuk, M. I., Èı̆del’man, S. D.: Boundary value problems for second order parabolic and elliptic equations in Dini spaces. Dokl. Akad. Nauk SSSR 198, 533–536 (1971)
McShane, E.J.: Extension of range of functions. Bull. Amer. Math. Soc. 40(12), 837–842 (1934)
Pruitt, W.E.: The growth of random walks and Lévy processes. Ann. Probab. 9(6), 948–956 (1981)
Ros-Oton, X.: Nonlocal elliptic equations in bounded domains: a survey. Publ. Mat. 60(1), 3–26 (2016)
Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101(3), 275–302 (2014)
Ros-Oton, X., Serra, J.: Boundary regularity for fully nonlinear integro-differential equations. Duke Math. J. 165(11), 2079–2154 (2016)
Ros-Oton, X., Serra, J.: Regularity theory for general stable operators. J. Differential Equations 260(12), 8675–8715 (2016)
Rutkowski, A.: The Dirichlet problem for nonlocal Lévy-type operators. Publ. Mat. 62(1), 213–251 (2018)
Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions, vol 68 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999). Translated from the 1990 Japanese original, Revised by the author
Schilling, R.L., Song, R., Vondraček, Z.: Bernstein functions: Theory and applications, vol 37 of de Gruyter Studies in Mathematics. 2nd edn. (2019)
Sztonyk, P.: On harmonic measure for Lévy processes. Probab. Math. Statist. 20(2), 383–390 (2000)
Watanabe, T.: The isoperimetric inequality for isotropic unimodal Lévy processes. Z. Wahrsch. Verw. Gebiete 63(4), 487–499 (1983)
Zhao, Z.: A probabilistic principle and generalized Schrödinger perturbation. J. Funct. Anal. 101(1), 162–176 (1991)
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We would like to express our deep gratitude to the anonymous reviewers for numerous helpful and valuable comments and suggestions which significantly contributed to the quality of this article.
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Appendix A: Potential Theory for Recurrent Unimodal Lévy Processes
Appendix A: Potential Theory for Recurrent Unimodal Lévy Processes
In this appendix we establish a formula for the Green function for a bounded open set D in case of recurrent isotropic unimodal Lévy process Xt. Contrary to the transient case, here the potential kernel \(U(x)={\int \limits }_{0}^{\infty }p_{t}(x) \mathrm {d} t\) is infinite (see [41, Chapter 7] for a detailed discussion on the subject), so the classical Hunt formula has no application. Instead, one can define the λ-potential kernel Uλ by setting
Similarly, we define the λ-Green function for an open set D
Note that both Uλ and \(G_{D}^{\lambda }\) exist. An analogue of the Hunt formula for \(G_{D}^{\lambda }\) holds, namely, for \(x,y \in {\mathbb {R}^{d}}\),
Lemma A.1
Let \(d \geqslant 1\). For any fixed \(x_{0} \in {\mathbb {R}^{d}} \setminus \{0\}\) we have λUλ(x0) → 0 as λ → 0+.
Proof
In the following part we introduce a mild ambiguity by denoting by 1, depending on the context, either a real number or the vector \((0,...,0,1) \in {\mathbb {R}^{d}}\). Set x0 = 1. Let \(f_{\lambda }(r)= {\int \limits }_{|x|<\sqrt {r}}\mathrm {d} x {\int \limits }_{0}^{\infty } e^{-\lambda u}p_{u}(x) \mathrm {d} u\). We have
By the proof of [19, Lemma 6],
Hence, we have for λ > 0,
By monotonicity of fλ,
Since by [19, Lemma 1 and Proposition 1],
we obtain
Hence, λUλ(1) → 0 as λ → 0+. The extension to arbitrary x0 is immediate. □
Lemma A.2
Let \(x_{0} \in {\mathbb {R}^{d}} \setminus \{0\}\) be an arbitrary fixed point. For all \(x \in {\mathbb {R}^{d}} \setminus \{0\}\) we have \({\int \limits }_{0}^{\infty } \left \lvert p_{t}(x)-p_{t}(x_{0}) \right \rvert \mathrm {d} t<\infty \).
Proof
Let \(f \in C_{c}^{\infty }({\mathbb {R}^{d}})\) be symmetric and such that \(\mathbf {1}_{B_{\epsilon }} \leqslant f \leqslant \mathbf {1}_{B_{4\epsilon }}\), where 0 < 4𝜖 < 1. Denote, for λ > 0,
Let x0 = 1. Observe that
Note that the integrand has a positive sign. Indeed,
since 4𝜖 < 1. Furthermore,
Hence, by the Fourier inversion theorem,
By the monotone convergence theorem, Eq. A.1 and the fact that \( \left \lvert \widehat {f}(\xi ) \right \rvert \) decays faster than any polynomial,
Hence,
Since W1 is radially decreasing and positive for |x| < 1, Eq. A.2 implies that it may be infinite only for x = 0. It follows that W1 is well defined for \(0<|x|\leqslant 1\). Similarly \(0 \leqslant W_{x_{0}}<\infty \) for \(0<|x|\leqslant |x_{0}|\).
It remains to notice that for |x| > |x0| we have \(0 \leqslant \left \lvert W_{x_{0}}(x) \right \rvert =-W_{x_{0}}(x) = W_{x}(x_{0})<\infty \) by the first part of the proof. □
Lemma A.2 allows us to introduce, following [5], [10], [28], a compensated potential kernel by setting for \(x \in {\mathbb {R}^{d}} \setminus \{0\}\)
where \(x_{0} \in {\mathbb {R}^{d}} \setminus \{0\}\) is an arbitrary but fixed point. From the proof of Lemma A.2 we immediately obtain the following corollary.
Corollary A.3
For any \(x_{0} \in {\mathbb {R}^{d}} \setminus \{0\}\), \(W_{x_{0}}\)is locally integrable in \({\mathbb {R}^{d}}\).
Theorem A.4
Let \(x_{0} \in {\mathbb {R}^{d}} \setminus \{0\}\), \(d \leqslant 2\) and D be bounded. Then for x, y ∈ D,
Proof
Let x, y ∈ D. Fix \(x_{0} \in {\mathbb {R}^{d}} \{0\}\) and observe that for any λ > 0,
We want to pass with λ to 0+. The limit of left-hand side is well defined and is equal to GD(x, y), which, in view of [20, Theorem 1.3], is finite. From Lemma A.1 we get
Moreover, from Lemma A.2 we obtain that
It remains to show the convergence of the middle term of Eq. A.5. Since Uλ is radially decreasing, \(U^{\lambda }(y-X_{\tau _{D}})-U^{\lambda }(x_{0})\) is positive and increasing as λ → 0+ on the set \(\{y \in {\mathbb {R}^{d}}\colon |y-X_{\tau _{D}}|\leqslant |x_{0}|\}\), and non-positive on its complement. By Lemma A.2, and the Monotone Convergence Theorem,
Observe that the left-hand side of Eq. A.5 converges to GD so it is finite. The remaining integral on the right-hand side converges as well by the monotone convergence theorem, but since all the other terms are finite, it follows that the integral is also finite and we obtain
which ends the proof. □
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Grzywny, T., Kassmann, M. & Leżaj, Ł. Remarks on the Nonlocal Dirichlet Problem. Potential Anal 54, 119–151 (2021). https://doi.org/10.1007/s11118-019-09820-9
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DOI: https://doi.org/10.1007/s11118-019-09820-9