Smooth perturbations of regular Dirichlet forms

Stollmann, Peter

doi:10.1090/S0002-9939-1992-1107277-3

Smooth perturbations of regular Dirichlet forms
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by Peter Stollmann

Proc. Amer. Math. Soc. 116 (1992), 747-752

DOI: https://doi.org/10.1090/S0002-9939-1992-1107277-3

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Abstract:

Given a regular Dirichlet form $\mathfrak {h}$, we prove that a measure $\mu$ is smooth iff the domain of $\mathfrak {h} + \mu$ is dense in the domain of $\mathfrak {h}$ with respect to the form norm. The latter condition is in turn equivalent to the convergence of $\mathfrak {h} + a\mu$ to $\mathfrak {h}$.

References

Sergio Albeverio, Johannes Brasche, and Michael Röckner, Dirichlet forms and generalized Schrödinger operators, Schrödinger operators (Sønderborg, 1988) Lecture Notes in Phys., vol. 345, Springer, Berlin, 1989, pp. 1–42. MR 1037315, DOI 10.1007/3-540-51783-9_{1}5
Sergio Albeverio and Zhi Ming Ma, Perturbation of Dirichlet forms—lower semiboundedness, closability, and form cores, J. Funct. Anal. 99 (1991), no. 2, 332–356. MR 1121617, DOI 10.1016/0022-1236(91)90044-6
Sergio Albeverio and Zhi Ming Ma, Nowhere Radon smooth measures, perturbations of Dirichlet forms and singular quadratic forms, Stochastic differential systems (Bad Honnef, 1988) Lect. Notes Control Inf. Sci., vol. 126, Springer, Berlin, 1989, pp. 32–45. MR 1236053, DOI 10.1007/BFb0043770

Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms

J. Baxter, G. Dal Maso, and U. Mosco, Stopping times and $\Gamma$-convergence, Trans. Amer. Math. Soc. 303 (1987), no. 1, 1–38. MR 896006, DOI 10.1090/S0002-9947-1987-0896006-7
Ph. Blanchard and Zhi Ming Ma, Semigroup of Schrödinger operators with potentials given by Radon measures, Stochastic processes, physics and geometry (Ascona and Locarno, 1988) World Sci. Publ., Teaneck, NJ, 1990, pp. 160–195. MR 1124210

Smooth measures and Schrödinger semigroups

Ph. Blanchard and Zhi Ming Ma, New results on the Schrödinger semigroups with potentials given by signed smooth measures, Stochastic analysis and related topics, II (Silivri, 1988) Lecture Notes in Math., vol. 1444, Springer, Berlin, 1990, pp. 213–243. MR 1078851, DOI 10.1007/BFb0083618
Masatoshi Fukushima, Dirichlet forms and Markov processes, North-Holland Mathematical Library, vol. 23, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1980. MR 569058

Perturbation theory for linear operators

Barry Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Functional Analysis 28 (1978), no. 3, 377–385. MR 500266, DOI 10.1016/0022-1236(78)90094-0
Peter Stollmann, Admissible and regular potentials for Schrödinger forms, J. Operator Theory 18 (1987), no. 1, 139–151. MR 912817
Peter Stollmann and Jürgen Voigt, A regular potential which is nowhere in $L_1$, Lett. Math. Phys. 9 (1985), no. 3, 227–230. MR 793000, DOI 10.1007/BF00402834
Theo Sturm, Schroedinger equations with discontinuous solutions, Potential theory (Prague, 1987) Plenum, New York, 1988, pp. 333–337. MR 986311, DOI 10.1007/978-1-4613-0981-9_{4}1

Störung von Hunt-Prozessen durch signierte additive Funktionale

Jürgen Voigt, Absorption semigroups, their generators, and Schrödinger semigroups, J. Funct. Anal. 67 (1986), no. 2, 167–205. MR 845197, DOI 10.1016/0022-1236(86)90036-4