Equivalence of the Brownian and energy representations

S. Albeverio   ; B. K. Driver   ; M. Gordina   ; A. M. Vershik

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 22, Tome 441 (2015), pp. 17-44

Résumé

We consider two unitary representations of the infinite-dimensional groups of smooth paths with values in a compact Lie group. The first representation is induced by quasi-invariance of the Wiener measure, and the second representation is the energy representation. We define these representations and their basic properties, and then we prove that these representations are unitarily equivalent.

Bibliographie

[1] S. Albeverio, R. Høegh-Krohn, D. Testard, A. Vershik, “Factorial representations of path groups”, J. Funct. Anal., 51:1 (1983), 115–131 | DOI | MR | Zbl

[2] S. Albeverio, R. Høegh-Krohn, D. Testard, A. Vershik, “Note: ‘Factorial representations of path groups’”, J. Funct. Anal., 52:1 (1983), 148 | DOI | MR

[3] S. S. Albeverio, R. Høegh-Krohn, “The energy representation of Sobolev–Lie groups”, Compositio Math., 36:1 (1978), 37–51 | MR | Zbl

[4] S. Albeverio, R. Høegh-Krohn, J. A. Marion, D. H. Testard, B. S. Torrésani, Noncommutative distributions, Monographs and Textbooks in Pure and Applied Mathematics, 175, Marcel Dekker Inc., New York, 1993 | MR | Zbl

[5] V. I. Bogachev, Gaussian measures, Mathematical Surveys and Monographs, 62, American Mathematical Society, Providence, RI, 1998 | DOI | MR | Zbl

[6] Bruce K. Driver, “A Cameron–Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold”, J. Funct. Anal., 110:2 (1992), 272–376 | DOI | MR | Zbl

[7] Bruce K. Driver, “A Cameron–Martin type quasi-invariance theorem for pinned Brownian motion on a compact Riemannian manifold”, Trans. Amer. Math. Soc., 342:1 (1994), 375–395 | DOI | MR | Zbl

[8] Bruce K. Driver, “Towards calculus and geometry on path spaces”, Stochastic analysis (Ithaca, NY, 1993), Proc. Sympos. Pure Math., 57, Amer. Math. Soc., Providence, RI, 1995, 405–422 | MR | Zbl

[9] Bruce K. Driver, Brian C. Hall, “The energy representation has no non-zero fixed vectors”, Stochastic processes, physics and geometry: new interplays (Leipzig, 1999), v. II, CMS Conf. Proc., 29, Amer. Math. Soc., Providence, RI, 2000, 143–155 | MR | Zbl

[10] Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995 | MR | Zbl

[11] I. M. Gel'fand, M. I. Graev, A. M. Veršik, “Representations of the group of smooth mappings of a manifold $X$ into a compact Lie group”, Compositio Math., 35:3 (1977), 299–334 | MR | Zbl

[12] I. M. Gel'fand, M. I. Graev, A. M. Veršik, “Representations of the group of functions taking values in a compact Lie group”, Compositio Math., 42:2 (1980/81), 217–243 | MR | Zbl

[13] M. Gordina, “Taylor map on groups associated with a $\mathrm{II}_1$-factor”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5:1 (2002), 93–111 | DOI | MR | Zbl

[14] M. Gordina, “Stochastic differential equations on noncommutative $L^2$”, Finite and infinite dimensional analysis in honor of Leonard Gross (New Orleans, LA, 2001), Contemp. Math., 317, Amer. Math. Soc., Providence, RI, 2003, 87–98 | DOI | MR | Zbl

[15] L. Gross, “Uniqueness of ground states for Schrödinger operators over loop groups”, J. Funct. Anal., 112:2 (1993), 373–441 | DOI | MR | Zbl

[16] A. Guichardet, Symmetric Hilbert spaces and related topics. Infinitely divisible positive definite functions. Continuous products and tensor products. Gaussian and Poissonian stochastic processes, Lecture Notes in Mathematics, 261, Springer-Verlag, Berlin, 1972 | MR | Zbl

[17] Brian C. Hall, Ambar N. Sengupta, “The Segal–Bargmann transform for path-groups”, J. Funct. Anal., 152:1 (1998), 220–254 | DOI | MR | Zbl

[18] T. Hida, Brownian motion, Applications of Mathematics, 11, Springer-Verlag, New York, 1980, Translated from the Japanese by the author and T. P. Speed | MR | Zbl

[19] Richard V. Kadison, John R. Ringrose, Fundamentals of the theory of operator algebras, v. II, Pure and Applied Mathematics, 100, Advanced theory, Academic Press Inc., Orlando, FL, 1986 | MR | Zbl

[20] M.-P. Malliavin, P. Malliavin, “Integration on loop groups. I. Quasi invariant measures”, J. Funct. Anal., 93:1 (1990), 207–237 | DOI | MR | Zbl

[21] J. Marion, “Wiener functionals on spaces of Lie algebra valued $1$-currents, and unitary representations of current groups”, Publ. Mat., 33:1 (1989), 99–121 | DOI | MR | Zbl

[22] Derek W. Robinson, Elliptic operators and Lie groups, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991 | MR | Zbl

[23] I. E. Segal, “The two-sided regular representation of a unimodular locally compact group”, Ann. of Math. (2), 51 (1950), 293–298 | DOI | MR | Zbl

[24] Ichirō Shigekawa, “Transformations of the Brownian motion on a Riemannian symmetric space”, Z. Wahrsch. Verw. Gebiete, 65:4 (1984), 493–522 | DOI | MR

[25] Daniel W. Stroock, S. R. Srinivasa Varadhan, Multidimensional diffusion processes, Classics in Mathematics, Springer-Verlag, Berlin, 2006, Reprint of the 1997 edition | MR | Zbl

[26] V. S. Sunder, An invitation to von Neumann algebras, Universitext, Springer-Verlag, New York, 1987 | DOI | MR | Zbl

[27] D. Testard, “Representations of the group of equivariant loops in $\mathrm{SU}(N)$”, Stochastic processes-mathematics and physics (Bielefeld, 1985), v. II, Lecture Notes in Math., 1250, Springer, Berlin, 1987, 326–341 | DOI | MR

[28] Nolan R. Wallach, “On the irreducibility and inequivalence of unitary representations of gauge groups”, Compositio Math., 64:1 (1987), 3–29 | MR | Zbl