On the essential selfadjointness of Dirichlet operators on group-valued path space

Acosta, Ernesto

doi:10.1090/S0002-9939-1994-1195711-4

On the essential selfadjointness of Dirichlet operators on group-valued path space
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Proc. Amer. Math. Soc. 122 (1994), 581-590 Request permission

Abstract:

Let G be a compact Lie group with Lie algebra $\mathcal {G}$. Consider the Wiener measure P on the space \[ {W_G} = \{ g:[0,1] \to G,g {\text {continuous}},g(0) = e\} \] For each h in the Cameron-Martin space H over $\mathcal {G}$, let ${\partial _h}$ be the associated right invariant vector field over ${W_G}$ and let $\partial _h^ \ast$ be its adjoint with respect to P. We prove for a particular h that the space of functions on ${W_G}$ generated by ${C^\infty }$-cylindrical functions on ${W_G}$ together with one Gaussian random variable is a core for the Dirichlet operator $\partial _h^ \ast {\partial _h}$. This is the first step in proving the essential selfadjointness of the Number operator over group-valued path spaces in the natural presumed core.

References

On the essential self-adjointness of Dirichlet operators on nonlinear path spaces

Hélène Airault, Differential calculus on finite codimensional submanifolds of the Wiener space—the divergence operator, J. Funct. Anal. 100 (1991), no. 2, 291–316. MR 1125228, DOI 10.1016/0022-1236(91)90113-J
H. Airault and P. Malliavin, Intégration géométrique sur l’espace de Wiener, Bull. Sci. Math. (2) 112 (1988), no. 1, 3–52 (French, with English summary). MR 942797
Hélène Airault and Johan Van Biesen, Le processus d’Ornstein-Uhlenbeck sur une sous-variété de l’espace de Wiener, Bull. Sci. Math. 115 (1991), no. 2, 185–210 (French, with English summary). MR 1101023
Sergio Albeverio and Raphael Høegh-Krohn, The energy representation of Sobolev-Lie groups, Compositio Math. 36 (1978), no. 1, 37–51. MR 515036
Sergio Albeverio and Michael Röckner, Classical Dirichlet forms on topological vector spaces—closability and a Cameron-Martin formula, J. Funct. Anal. 88 (1990), no. 2, 395–436. MR 1038449, DOI 10.1016/0022-1236(90)90113-Y
Ezra Getzler, Dirichlet forms on loop space, Bull. Sci. Math. (2) 113 (1989), no. 2, 151–174. MR 1002216
Leonard Gross, Logarithmic Sobolev inequalities on loop groups, J. Funct. Anal. 102 (1991), no. 2, 268–313. MR 1140628, DOI 10.1016/0022-1236(91)90123-M
Marie-Paule Malliavin and Paul Malliavin, Integration on loop groups. I. Quasi invariant measures, J. Funct. Anal. 93 (1990), no. 1, 207–237. MR 1070039, DOI 10.1016/0022-1236(90)90139-C
H. Airault and P. Malliavin, Integration on loop groups. II. Heat equation for the Wiener measure, J. Funct. Anal. 104 (1992), no. 1, 71–109. MR 1152460, DOI 10.1016/0022-1236(92)90091-V
H. P. McKean Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR 0247684
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
Michael Röckner and Tu Sheng Zhang, Uniqueness of generalized Schrödinger operators and applications, J. Funct. Anal. 105 (1992), no. 1, 187–231. MR 1156676, DOI 10.1016/0022-1236(92)90078-W
M. Röckner and N. Wielens, Dirichlet forms—closability and change of speed measure, Infinite-dimensional analysis and stochastic processes (Bielefeld, 1983) Res. Notes in Math., vol. 124, Pitman, Boston, MA, 1985, pp. 119–144. MR 865023
Masayoshi Takeda, On the uniqueness of the Markovian selfadjoint extension, Stochastic processes—mathematics and physics, II (Bielefeld, 1985) Lecture Notes in Math., vol. 1250, Springer, Berlin, 1987, pp. 319–325. MR 897814, DOI 10.1007/BFb0077364
Masayoshi Takeda, On the uniqueness of Markovian selfadjoint extension of diffusion operators on infinite-dimensional spaces, Osaka J. Math. 22 (1985), no. 4, 733–742. MR 815442

The maximum Markovian self-adjoint extension of generalized Schrödinger operators

The essential self-adjointness of generalized Schrödinger operators

61

Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252