Geodesic
On $L^p$-uniqueness of symmetric diffusion operators on Riemannian manifolds
Sbornik. Mathematics, Tome 194 (2003) no. 7, pp. 969-978 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $M$ be a complete Riemannian manifold of dimension $d>1$, let $\mu$ be a measure on $M$ with density $\exp U$ with respect to the Riemannian volume, and let $\mathscr Lf=\Delta f+\langle b,\nabla f\rangle$, where $U\in H^{p,1}_{\mathrm{loc}}(M)$ and $b=\nabla U$. It is shown that in the case $p>d$ and $q\in[p',p]$ the operator $\mathscr L$ on the domain $C_0^\infty(M)$ has a unique extension generating a $C_0$-semigroup on $L^q(M,\mu)$, that is, the set $(\mathscr L-I)(C_0^\infty(M))$ is dense in $L^q(M,\mu)$. In particular, the operator $\mathscr L$ is essentially self-adjoint on $L^2(M,\mu)$. A similar result is proved for elliptic operators with non-constant second order part that are formally symmetric with respect to some measure. 
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     title = {On $L^p$-uniqueness of symmetric diffusion operators on {Riemannian} manifolds},
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     number = {7},
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     url = {http://geodesic.mathdoc.fr/item/SM_2003_194_7_a1/}
}
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V. I. Bogachev; M. Röckner. On $L^p$-uniqueness of symmetric diffusion operators on Riemannian manifolds. Sbornik. Mathematics, Tome 194 (2003) no. 7, pp. 969-978. http://geodesic.mathdoc.fr/item/SM_2003_194_7_a1/

[1] Albeverio S., Bogachev V., Röckner M., “On uniqueness of invariant measures for finite- and infinite-dimensional diffusions”, Comm. Pure Appl. Math., 52 (1999), 325–362 | 3.0.CO;2-V class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[2] Bogachev V. I., Krylov N. V., Röckner M., “Elliptic regularity and essential self-adjointness of Dirichlet operators”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24:3 (1997), 451–461 | MR | Zbl

[3] Bogachev V. I., Krylov N. V., Röckner M., “On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions”, Comm. Partial Differential Equations, 26:11–12 (2001), 2037–2080 | DOI | MR | Zbl

[4] Bogachev V. I., Röckner M., “Regularity of invariant measures on finite and infinite dimensional spaces and applications”, J. Funct. Anal., 133 (1995), 168–223 | DOI | MR | Zbl

[5] Bogachev V. I., Rëkner M., “Obobschenie teoremy Khasminskogo o suschestvovanii invariantnykh mer dlya lokalno integriruemykh snosov”, Teoriya veroyatnostei i ee prim., 45:3 (2000), 417–436 | MR | Zbl

[6] Bogachev V. I., Rëkner M., Shtannat V., “Edinstvennost reshenii ellipticheskikh uravnenii i edinstvennost invariantnykh mer diffuzii”, Matem. sb., 193:7 (2002), 3–36 | MR | Zbl

[7] Bogachev V. I., Röckner M., Wang F.-Y., “Elliptic equations for invariant measures on finite and infinite dimensional manifolds”, J. Math. Pures Appl. (9), 80:2 (2001), 177–221 | DOI | MR | Zbl

[8] Eberle A., Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators, Lecture Notes in Math., 1718, Springer-Verlag, New York, 1999 | MR | Zbl

[9] Eberle A., “$L^p$ uniqueness of non-symmetric diffusion operators with singular drift coefficients. I. The finite-dimensional case”, J. Funct. Anal., 173:2 (2000), 328–342 | DOI | MR | Zbl

[10] Zhikov V. V., Sirazhudinov M. M., “Usrednenie nedivergentnykh ellipticheskikh i parabolicheskikh operatorov vtorogo poryadka i stabilizatsiya resheniya zadachi Koshi”, Matem. sb., 116:2 (1981), 166–186 | MR | Zbl

[11] Stannat W., “(Nonsymmetric) Dirichlet operators on $L^1$: existence, uniqueness and associated Markov processes”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28:1 (1999), 99–140 | MR | Zbl

[12] Albeverio S., Høegh-Krohn R., Streit L., “Energy forms, Hamiltonians and distorted Brownian paths”, J. Math. Phys., 18:5 (1977), 907–917 | DOI | MR | Zbl

[13] Frehse J., “Essential self-adjointness of singular elliptic operators”, Bol. Soc. Brasil. Mat., 8:2 (1977), 87–107 | DOI | MR | Zbl

[14] Fukushima M., “On a stochastic calculus related to Dirichlet forms and distorted Brownian motion”, Phys. Rep., 77:3 (1981), 255–262 | DOI | MR

[15] Liskevich V. A., “On the uniqueness problem for Dirichlet operators”, J. Funct. Anal., 162 (1999), 1–13 | DOI | MR | Zbl

[16] Liskevich V. A., Semenov Yu. A., “Dirichlet operators: a priori estimates and the uniqueness problem”, J. Funct. Anal., 109 (1992), 199–213 | DOI | MR | Zbl

[17] Wielens N., “The essential self-adjointness of generalized Schrödinger operators”, J. Funct. Anal., 61 (1985), 98–115 | DOI | MR | Zbl

[18] Liskevich V. A., Tuv E. W., “A priori estimates for second order elliptic operators and their applications”, Israel J. Math., 81 (1993), 257–263 | DOI | MR | Zbl

[19] Liskevich V. A., “Smoothness estimates and uniqueness for the Dirichlet operator”, Mathematical results in quantum mechanics (Blossin, 1993), Oper. Theory Adv. Appl., 70, Birkhäuser, Basel, 1994, 149–152 | MR | Zbl

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

[21] Strichartz R. S., “Analysis of Laplacian on the complete Riemannian manifold”, J. Funct. Anal., 52 (1983), 48–79 | DOI | MR | Zbl

[22] Röckner M., Zhang T. S., “Uniqueness of generalized Schrödinger operators. II”, J. Funct. Anal., 119 (1994), 455–467 | DOI | MR | Zbl

[23] Cattiaux P., Fradon M., “Entropy, reversible diffusion processes, and Markov uniqueness”, J. Funct. Anal., 138:1 (1996), 243–272 | DOI | MR | Zbl