Markov uniqueness of degenerate elliptic operators

Robinson, Derek W.; Sikora, Adam

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 683-710.

Let $\Omega$ be an open subset of ${ℝ}^d$ and $H_{\Omega }=-{\sum }_{i,j=1}^d{\partial }_i{c}_{ij}{\partial }_j$ be a second-order partial differential operator on $L_2\left(\Omega \right)$ with domain $C_c^{\infty }\left(\Omega \right)$, where the coefficients $c_{ij}\in W^{1,\infty }\left(\Omega \right)$ are real symmetric and $C=\left(c_{ij}\right)$ is a strictly positive-definite matrix over $\Omega$. In particular, $H_{\Omega }$ is locally strongly elliptic. We analyze the submarkovian extensions of $H_{\Omega }$, i.e., the self-adjoint extensions that generate submarkovian semigroups. Our main result states that $H_{\Omega }$ is Markov unique, i.e., it has a unique submarkovian extension, if and only if ${\mathrm{cap}}_{\Omega }\left(\partial \Omega \right)=0$ where ${\mathrm{cap}}_{\Omega }\left(\partial \Omega \right)$ is the capacity of the boundary of $\Omega$ measured with respect to $H_{\Omega }$. The second main result shows that Markov uniqueness of $H_{\Omega }$ is equivalent to the semigroup generated by the Friedrichs extension of $H_{\Omega }$ being conservative.

Classification : 47B25, 47D07, 35J70

[1] T. Ariyoshi and M. Hino, Small-time asymptotic estimates in local Dirichlet spaces, Electron. J. Probab. 10 (2005), 1236–1259. | EuDML | MR | Zbl

[2] S. Albeverio, S. Kusuoka and M. Röckner, On partial integration in infinite-dimensional space and applications to Dirichlet forms, Proc. London Math. Soc. 42 (1990), 122–136. | MR | Zbl

[3] R. Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France 102 (1974), 193–240. | EuDML | Numdam | MR | Zbl

[4] N. Bouleau and F. Hirsch, “Dirichlet Forms and Analysis on Wiener Space”, de Gruyter Studies in Mathematics, Vol. 14, Walter de Gruyter & Co., Berlin, 1991. | MR | Zbl

[5] M. Biroli and U. Mosco, A Saint-Venant type principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl. 169 (1995), 125–181. | MR | Zbl

[6] M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum 57 (1998), 1–36. | MR | Zbl

[7] E.-B. Davies, $L^1$ properties of second order elliptic operators, Bull. London Math. Soc. 17 (1985), 417–436. | MR | Zbl

[8] E.-B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. Anal. Math. 58 (1992), 99–119. Festschrift on the occasion of the 70th birthday of Shmuel Agmon. | MR | Zbl

[9] A. Eberle, “Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators”, Lect. Notes in Math. 1718. Springer-Verlag, Berlin, 1999. | MR | Zbl

[10] A. F. M. ter Elst and D. W. Robinson, Conservation and invariance properties of submarkovian semigroups, J. Ramanujan. Math. Soc. 24 (2009), 1–13. | MR | Zbl

[11] A. F. M. ter Elst and D. W. Robinson, Uniform subellipticity, J. Operator Theory 62 (2009), 125–149. | MR | Zbl

[12] A. F. M. ter Elst, D. W. Robinson, A. Sikora and Y. Zhu, Dirichlet forms and degenerate elliptic operators, In: “Partial Differential Equations and Functional Analysis”, Koelink, E., Neerven, J. van, Pagter, B. de and Sweers, G. (eds.), Operator Theory: Advances and Applications, Vol. 168, Birkhäuser, 2006, 73–95. Philippe Clement Festschrift. | MR

[13] A. F. M. ter Elst, D. W. Robinson, A. Sikora and Y. Zhu, Second-order operators with degenerate coefficients, Proc. London Math. Soc. 95 (2007), 299–328. | MR | Zbl

[14] K. J. Falconer, “Fractal geometry”, Second edition, Mathematical Foundations and Applications, John Wiley & Sons Inc., Hoboken, NJ, 2003. | MR | Zbl

[15] M. Fukushima, Y. Oshima and M. Takeda, “Dirichlet forms and symmetric Markov processes”, de Gruyter Studies in Mathematics, Vol. 19, Walter de Gruyter & Co., Berlin, 1994. | MR | Zbl

[16] K. O. Friedrichs, Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren, I, Math. Ann. 109 (1934), 465–487. | EuDML | JFM | MR

[17] A. Friedman, “Partial Differential Equations of Parabolic Type”, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. | MR | Zbl

[18] M. P. Gaffney, The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math. 12 (1959), 1–11. | MR | Zbl

[19] D. Gatzouras, Lacunarity of self-similar and stochastically self-similar sets, Trans. Amer. Math. Soc. 352 (2000), 1953–1983. | MR | Zbl

[20] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Second edition, Grundlehren der mathematischen Wissenschaften 224. Springer-Verlag, Berlin, 1983. | MR | Zbl

[21] T. Kato, “Perturbation Theory for Linear Operators”, Second edition, Grundlehren der mathematischen Wissenschaften 132, Springer-Verlag, Berlin, 1980. | MR | Zbl

[22] M. G. Krein, The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications, I, Mat. Sbornik N.S. 20(62) (1947), 431–495. | EuDML | MR | Zbl

[23] Z. M. Ma and M. Röckner, “Introduction to the Theory of (Non Symmetric) Dirichlet Forms”, Universitext. Springer-Verlag, Berlin, 1992. | MR | Zbl

[24] E. M. Ouhabaz, “Analysis of Heat Equations on Domains”, London Mathematical Society Monographs Series, Vol. 31, Princeton University Press, Princeton, NJ, 2005. | MR | Zbl

[25] D. W. Robinson, Commutator theory on Hilbert space, Canad. J. Math. 37 (1987), 1235–1280. | MR | Zbl

[26] D. W. Robinson and A. Sikora, Degenerate elliptic operators: capacity, flux and separation, J. Ramanujan Math. Soc. 22 (2007), 385–408. | MR | Zbl

[27] D. W. Robinson and A. Sikora, Analysis of degenerate elliptic operators of Grušin type, Math. Z. 260 (2008), 475–508. | MR | Zbl

[28] D. W. Robinson and A. Sikora, Degenerate elliptic operators in one-dimension, J. Evol. Equ. 10 (2010), 731–759. | MR | Zbl

[29] K. T. Sturm, The geometric aspect of Dirichlet forms, In: “New Directions in Dirichlet Forms”, AMS/IP Stud. Adv. Math., Vol. 8, Amer. Math. Soc., Providence, RI, 1998, 233–277. | MR | Zbl

[30] M. Takeda, Two classes of extensions for generalized Schrödinger operators, Potential Anal. 5 (1996), 1–13. | MR | Zbl