Infinite-dimensional analysis and quantum theory as semimartingale calculus

A. I. Kirillov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 49 (1994) no. 3, pp. 43-95

Bibliographie

[1] Averbukh V. I., Smolyanov O. G., Fomin S. V., “Obobschennye funktsii i differentsialnye uravneniya v lineinykh prostranstvakh. I: Differentsiruemye mery”, Trudy MMO, 24, 1971, 133–174 | MR | Zbl

[2] Anulova S. V., Veretennikov A. Yu., Krylov N. V., Liptser R. Sh., Shiryaev A. N., “Stokhasticheskoe ischislenie”, Itogi nauki i tekhn. Sovr. probl. matem. Fundam. napravleniya, 45, VINITI, 1989, 5–260 | MR

[3] Berezanskii Yu. M., Samosopryazhennye operatory v prostranstvakh funktsii beskonechnogo chisla peremennykh, Nauk. dumka, Kiev, 1978

[4] Berezanskii Yu. M., Kondratev Yu. G., Spektralnye metody v beskonechnomernom analize, Nauk. dumka, Kiev, 1988 | Zbl

[5] Billingsli P., Skhodimost veroyatnostnykh mer, Nauka, M., 1977

[6] Bogachev V. I., “Neskolko rezultatov o differentsiruemykh merakh”, Matem. sb., 127:3 (1985), 336–351 | MR | Zbl

[7] Bogachev V. I., “Podprostranstva differentsiruemosti gladkikh mer na beskonechnomernykh prostranstvakh”, DAN SSSR, 299:1 (1988), 18–22 | MR | Zbl

[8] Bogachev V. I., Smolyanov O. G., “Analiticheskie svoistva beskonechnomernykh raspredelenii”, UMN, 45:3 (1990), 3–83 | MR | Zbl

[9] Bogolyubov N. N., Logunov A. A., Oksak A. I., Todorov I. T., Obschie printsipy kvantovoi teorii polya, Nauka, M., 1987

[10] Burbaki N., Obschaya topologiya. Ispolzovanie veschestvennykh chisel v obschei topologii. Funktsionalnye prostranstva. Svodka rezultatov, Nauka, M., 1975

[11] Vatanabe S., Ikeda N., Stokhasticheskie differentsialnye uravneniya i diffuzionnye protsessy, Nauka, M., 1986

[12] Vakhaniya N. N., Tarieladze V. I., Chobanyan S. A., Veroyatnostnye raspredeleniya v banakhovykh prostranstvakh, Nauka, M., 1985 | Zbl

[13] Veretennikov A. Yu., Krylov N. V., “O yavnykh formulakh dlya reshenii stokhasticheskikh uravnenii”, Matem. sb., 100 (142):2 (6) (1976), 266–284 | MR | Zbl

[14] Gardiner K., Stokhasticheskie metody v estestvennykh naukakh, Mir, M., 1986 | Zbl

[15] Gelfand I. M., Vilenkin N. Ya., Nekotorye primeneniya garmonicheskogo analiza. Osnaschennye gilbertovy prostranstva, Obobschennye funktsii. Vyp. 4, Fizmatgiz, M., 1961

[16] Gikhman I. I., Skorokhod A. V., Teoriya sluchainykh protsessov, T. III, Nauka, M., 1975

[17] Glimm Dzh., Dzhaffe A., Matematicheskie metody kvantovoi fiziki. Podkhod s ispolzovaniem funktsionalnykh integralov, Mir, M., 1984 | Zbl

[18] Daletskii Yu. L., “Stokhasticheskaya differentsialnaya geometriya”, UMN, 38:3 (1983), 87–111 | MR

[19] Daletskii Yu. L., Belopolskaya Ya. I., Stokhasticheskie uravneniya i differentsialnaya geometriya, Vyscha shkola, K., 1989

[20] Daletskii Yu. L., Fomin S. V., Mery i differentsialnye uravneniya v beskonechnomernykh prostranstvakh, Nauka, Moskva, 1983

[21] Efimova E. I., Uglanov A. V., “Formuly Grina na gilbertovom prostranstve”, Mat. sb., 119:2 (1982), 225–232 | MR | Zbl

[22] Efimova E. I., Uglanov A. V., “Formuly vektornogo analiza na banakhovom prostranstve”, DAN SSSR, 271:6 (1983), 1302–1306 | MR

[23] Zavyalov O. I., Sushko V. N., “Kanonicheskie peremennye v fizike beskonechnykh sistem”, TMF, 1:2 (1969), 153–181 | MR

[24] Ignatyuk I. A., Malyshev V. A., Sidoravichyus V., “Skhodimost metoda stokhasticheskogo kvantovaniya, I”, Teoriya veroyatn. i ee primen., 37:2 (1992), 241–253 | MR

[25] Kirillov A. I., “O dvukh matematicheskikh problemakh kanonicheskogo kvantovaniya, I”, TMF, 87:1 (1991), 22–33 | MR

[26] Kirillov A. I., “O dvukh matematicheskikh problemakh kanonicheskogo kvantovaniya, II”, TMF, 87:2 (1991), 163–172 | MR

[27] Kirillov A. I., “O separabelnosti prostranstva $L^p$ v sluchae mer na lokalno vypuklykh prostranstvakh”, Izv. vuzov. Matematika, 1991, no. 7, 77 | MR | Zbl

[28] Kirillov A. I., “O dvukh matematicheskikh problemakh kanonicheskogo kvantovaniya. III: Stokhasticheskaya mekhanika vakuuma”, TMF, 91:3 (1992), 377–395 | MR

[29] Kirillov A. I., “O dvukh matematicheskikh problemakh kanonicheskogo kvantovaniya, IV”, TMF, 93:2 (1992), 249–263 | MR

[30] Kirillov A. I., “O zadanii mer na funktsionalnykh prostranstvakh s pomoschyu chislovykh plotnostei i kontinualnykh integralov”, Matem. zametki, 53:5 (1993), 152–155 | MR | Zbl

[31] Kirillov A. I., “Brounovskoe dvizhenie so snosom v gilbertovom prostranstve i ego primenenie v teorii integrirovaniya”, Teoriya veroyatn. i ee primenen., 38:3 (1993), 629–634 | MR

[32] Kirillov A. I., “Pole tipa sin-Gordon v prostranstve–vremeni proizvolnoi razmernosti: Suschestvovanie mery Nelsona”, TMF, 97:1 (1994), 12–28 | MR

[33] Kirillov A. I., “O vosstanovlenii mer po ikh logarifmicheskim proizvodnym”, Izv. RAN. Ser. matem., 1994 (to appear)

[34] Kolmogorov A. N., Teoriya veroyatnostei i matematicheskaya statistika, Sb. statei, Nauka, M., 1986

[35] Kondratev Yu. G., “Operatory Dirikhle i gladkost reshenii beskonechnomernykh ellipticheskikh uravnenii”, DAN SSSR, 282:2 (1985), 269–273 | MR | Zbl

[36] Konstruktivnaya kvantovaya teoriya polya, Sb. statei, Matematika. Novoe v zarubezhnoi nauke, 6, Mir, M., 1977

[37] Kornfeld I. P., Sinai Ya. G., Fomin S. V., Ergodicheskaya teoriya, Nauka, M., 1980 | Zbl

[38] Krylov N. V., Rozovskii B. L., “Ob evolyutsionnykh stokhasticheskikh uravneniyakh”, Itogi nauki i tekhn. Sovr. probl. matem., 14, VINITI, 1979, 71–146 | MR | Zbl

[39] Liptser R. Sh., Shiryaev A. N., Statistika sluchainykh protsessov, Nauka, M., 1974 | Zbl

[40] Liptser R. Sh., Shiryaev A. N., Teoriya martingalov, Nauka, M., 1986 | Zbl

[41] Migdal A. B., “Stokhasticheskoe kvantovanie v teorii polya”, UFN, 149:1 (1986), 3–44

[42] Mikhlin S. G., Kurs matematicheskoi fiziki, Nauka, M., 1968

[43] Neve Zh., Matematicheskie osnovy teorii veroyatnostei, Mir, M., 1969 | Zbl

[44] Norin N. V., “Stokhasticheskie integraly i differentsiruemye mery”, Teor. veroyatn. i ee primen., 32:1 (1987), 114–124 | MR

[45] Pich A., Yadernye lokalno vypuklye prostranstva, Mir, M., 1967

[46] Prokhorov Yu. V., Rozanov Yu. A., Teoriya veroyatnostei, Nauka, M., 1967 | Zbl

[47] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki. T. 1. Funktsionalnyi analiz, Mir, M., 1977

[48] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki. T. 2. Garmonicheskii analiz. Samosopryazhennost, Mir, M., 1978

[49] Saimon B., Model $P(\varphi )_2$ evklidovoi kvantovoi teorii polya, Mir, M., 1976

[50] Skorokhod A. V., Integrirovanie v gilbertovom prostranstve, Nauka, M., 1975

[51] Smolyanov O. G., “Potoki de Rama i formula Stoksa v gilbertovom prostranstve”, DAN SSSR, 286:3 (1986), 554–558 | MR | Zbl

[52] Smorodina N. V., “Formula Gaussa-Ostrogradskogo prostranstva konfiguratsii”, Teoriya veroyatn. i ee primenen., 35:4 (1990), 727–739 | MR | Zbl

[53] Tribel Kh., Teoriya funktsionalnykh prostranstv, Per. s angl., Mir, M., 1986 | Zbl

[54] Khasminskii R. Z., Ustoichivost sistem differentsialnykh uravnenii pri sluchainykh vozmuscheniyakh ikh parametrov, Nauka, M., 1969

[55] Khafizov M. U., “Neskolko novykh rezultatov o differentsiruemykh merakh”, Vestnik MGU. Ser. 1. Matem., mekh., 1990, no. 4, 63–65 | MR

[56] Khida T., Brounovskoe dvizhenie, Nauka, M., 1987 | Zbl

[57] Tsikalenko T. V., “Skhodimost metoda stokhasticheskogo kvantovaniya dlya reshetochnykh sistem”, Metody funkts. anal. v zadachakh mat. fiziki, In-t matematiki AN USSR, Kiev, 1990

[58] Shavgulidze E. T., “O pryamom uravnenii Kolmogorova dlya mer v gilbertovoi shkale prostranstv”, Vestnik MGU. Ser. 1. Matem., mekh., 1978, no. 3, 19–28 | MR | Zbl

[59] Shefer Kh., Topologicheskie vektornye prostranstva, Mir, M., 1971

[60] Shurenkov V. M., Ergodicheskie protsessy Markova, Nauka, M., 1989

[61] Emkh Zh., Algebraicheskie metody v statisticheskoi mekhanike i kvantovoi teorii polya, ed. G. A. Zaitsev, Mir, M., 1976

[62] Albeverio S., Blanchard Ph., Combe Ph., Hoegh-Krohn R., Sirugue M., “Local relativistic invariant flows for quantum fields”, Comm. Math. Phys., 90:3 (1983), 329–351 | DOI | MR | Zbl

[63] Albeverio S., Hida T., Potthoff J., Rockner M., Streit L., “Dirichlet forms in terms of white noise analysis. I: Construction QFT examples”, Rew. Math. Phys., 1:2 (1990), 291–323 | DOI

[64] Albeverio S., Hoegh-Krohn R., “Quasi-invariant measures, symmetric diffusion processes and quantum fields”, Colloque intern. sur les methodes mathematiques de la theorie quantique des champs, no. 248, CNRS, Paris, 1976, 11–69

[65] Albeverio S., Hoegh-Krohn R., “Dirichlet forms and diffusion processes on rigged Hilbert spaces”, Z. Wahr. Verw. Geb., 40:1 (1977), 1–57 | DOI | Zbl

[66] Albeverio S., Hoegh-Krohn R., “Hunt processes and analytic potential theory on rigged Hilbert spaces”, Ann. Inst. Henri Poincare. Sect. B, 13 (1977), 269–291 | MR | Zbl

[67] Albeverio S., Kusuoka S., “Maximality of infinite dimensional Dirichlet forms and Hoegh-Krohn's model of quantum fields”, Memorial volume for R. Hoegh-Krohn, eds. Albeverio S., Holden H., Cambridge Univ. Press, Cambridge, 1991

[68] Albeverio S., Kusuoka S., Rockner M., “On partial integration in infinite dimensional Dirichlet forms”, J. Lond. Math. Soc., 42 (1990), 122–136 | DOI | MR | Zbl

[69] Albeverio S., Rockner M., “Classical Dirichlet forms on topological vector spaces-Construction of an associated diffusion process”, Prob. Th. Rel. Fields, 83 (1989), 405–434 | DOI | MR | Zbl

[70] Albeverio S., Rockner M., “Classical Dirichlet forms on topological vector spaces-Closability and a Cameron–Martin formula”, Journ. Funct. Anal., 88:2 (1990), 395–437 | DOI

[71] Albeverio S., Rockner M., “Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms”, Prob. Th. Rel. Fields, 89:3 (1991), 347–386 | DOI | MR | Zbl

[72] Antonjuk A. V., Kondratiev Ju. G., Log-concave smooth measures on Hilbert space and some properties of corresponding Dirichlet operators, Preprint BiBoS 478/91, 1991

[73] Araki H., “Hamiltonian formalism and the canonical commutation relations in quantum field theory”, J. Math. Phys., 1:6 (1960), 492–504 | DOI | MR | Zbl

[74] Araki H., “On Representations of the Canonical Commutation Relations”, Comm. Math. Phys., 20:1 (1971), 9–25 | DOI | MR | Zbl

[75] Aronszajn N., “Differentiability of Lipschitzian mappings between Banach spaces”, Studia Math., 57 (1976), 147–190 | MR | Zbl

[76] Beurling A., Deny J., “Espaces de Dirichlet. I: Le car elementaire”, Acta Math., 99:3–4 (1958), 203–224 | DOI | MR | Zbl

[77] Beurling A., Deny J., “Dirichlet Spaces”, Proc. Nat. Acad. Sci., 45:2 (1959), 208–215 | DOI | MR | Zbl

[78] Blanchard Ph., Combe Ph., Zheng W., Mathematical and Physical Aspects of Stochastic Mechanics, Springer, New York, 1987

[79] Borkar B. S., Chari R. T., Mitter S. K., “Stochastic quantization of field theory in finite and infinite volume”, J. Funct. Anal., 81 (1988), 184–206 | DOI | MR | Zbl

[80] Bouleau N., Hirsch F., “Formes de Dirichlet generales et densite des variables aleatoires reelles sur l'espace de Wiener”, J. Funct. Anal., 69:2 (1986), 229–259 | DOI | MR | Zbl

[81] Carlen E., Proc. IX Congress on Math. Phys., eds. B. Simon, A. Truman, I. M. Davies, Adam Hilger, N.-Y., 1989

[82] Doob J. L., “Asymptotic properties of Markoff transition probabilities”, Trans. Amer. Math. Soc., 63:3 (1948), 393–421 | DOI | MR | Zbl

[83] Fukushima M., “Dirichlet spaces and strong Markov processes”, Trans. Amer. Math. Soc., 162 (1971), 185–224 | DOI | MR

[84] Fukushima M., Dirichlet Forms and Markov processes, Kodansha; North Holland, Amsterdam, 1980 | Zbl

[85] Fukushima M., “On stochastic calculus related to Dirichlet forms and distorted Brownian motion”, Phys. Rep., 77 (1981), 256–262 | DOI | MR

[86] Fukushima M., “Basic properties of Brownian motion and a capacity on the Wiener space”, J. Math. Soc. Japan, 36:1 (1984), 161–176 | DOI | MR | Zbl

[87] Gilerak R., J. Math. Phys., 27:12 (1986), 2892–2902 | DOI | MR

[88] Gross L., “Logarithmic Sobolev inequalities”, Amer. J. Math., 97:4 (1976), 1061–1083 | DOI | Zbl

[89] Hegerfeldt G., “On Canonical Commutation Relations and Infinite Dimensional Measures”, J. Math. Phys., 13:1 (1972), 45–50 | DOI | MR | Zbl

[90] Hegerfeldt G., “Garding Domains and Analytic Vectors for Quantum Fields”, J. Math. Phys., 13:6 (1972), 821–827 | DOI | MR | Zbl

[91] Herbst I., “On canonical quantum field theories”, J. Math. Phys., 17:7 (1976), 1210–1221 | DOI | MR

[92] Hida T., Kuo H-H., Potthoff J., Streit L. (eds.), White Noise Analysis: Mathematics and Applications, World Scientific, Syngapore, 1990

[93] Hida T., Potthoff J., “White Noise Analysis – An Overview”, White Noise Analysis, eds. T. Hida et al., World Scientific, Singapore, 1990

[94] Hida T., Streit L., “On quantum theory in terms of white noise”, Nagoya Math. J., 68:12 (1977), 21–34 | MR | Zbl

[95] Jona-Lasinio G., Mitter S. K., “On the stochastic quantization of field theory”, Comm. Math. Phys., 101 (1985), 409–436 | DOI | MR | Zbl

[96] Kirillov A. I., On determining the measures on the functional spaces by their logarithmic derivatives and by infinitely multiple integrals, Preprint BiBoS No 587/8, 1993

[97] Kuo H.-H., “Lectures on White Noise Analysis”, Proc. Preseminar Int. Conf. on Gaussian Random Fields, eds. T. Hida and K. Saito, Nagoya, 1991, 1–65

[98] Kusuoka S., “Dirichlet forms and diffusion processes on Banach spaces”, J. Fac. Sci. Univ. Tokyo Sect. IA, 29 (1982), 79–95 | MR | Zbl

[99] Lee Y.-L., “Calculus of Generalized White Noise Functionals”, An abstract Wiener space approach, Proc. of Preseminar for Int. Conf. on Gaussian Random Fields, eds. T. Hida and K. Saito, Nagoya, 1991, 66–125

[100] Lyons T., Rockner M., “A note on tightness of capacities associated with Dirichlet forms”, Bull. Lond. Math. Soc., 24:2 (1992), 181–184 | DOI | MR | Zbl

[101] Malliavin P., “Stochastic calculus of variations and hypo-elliptic operators”, Proc. Int. Symp. on Stochastic Differential Equations, ed. K. Ito, Kinokuniya, Tokyo; Wiley, New York, 1978

[102] Malliavin P., “Implicite functions in finite corank on the Wiener space”, Proc. Taniguchi Int. Symp. on Stoch. Anal., ed. K. Ito, Kinokuniya, Katata–Kyoto, 1983

[103] Metivier M., Viot M., “On weak solutions of stochastic partial differential equations”, Lect. Notes Math., 1322, 1988, 139–150 | MR | Zbl

[104] Namiki M., Stochastic Quantization, Springer, 1992

[105] Nelson E., “The adjoint Markoff processes”, Duke Math. J., 25:4 (1958), 671–690 | DOI | MR | Zbl

[106] Nelson E., “Derivation of the Schrodinger equation from Newtonian mechanics”, Phys Rew., 150:4 (1966), 1079–1085 | DOI

[107] Pantic D., “Stochastic calculus on distorted Brownian motion”, J. Math. Phys., 39:1 (1988), 207–210 | DOI | MR

[108] Parthasarathy K. R., Probability measures on metric spaces, Acad. Press, New York–London, 1967, 276 pp. | MR | Zbl

[109] J. Potthoff, “White-Noise Approach to Malliavin's Calculus”, J. Funct. Anal., 71:2 (1987), 207–217 | DOI | MR | Zbl

[110] Potthoff J., Rockner M., “On the contraction propety of energy forms on infinite dimensional space”, J. Funct. Anal., 92:1 (1990), 155–165 | DOI | MR | Zbl

[111] Potthoff J., Streit L., Invariant States on Random and Quantum Fields: $\varphi $-Bounds and WNA, Preprint BiBoS 422/1990, 1990

[112] Razafimanantena E. A., “Construction of general class of Dirichlet forme in terms of white noise analysis”, Stoch. Processes and Appl., 39:2 (1991), 263–276 | DOI | MR | Zbl

[113] Rockner M., “Specifications and Martin boundaries for $P(\varphi )_2$ -random fields”, Commun. Math. Phys., 106:1 (1986), 105–135 | DOI | MR

[114] Schmuland B., “An alternative compactification for classical Dirichlet forms on topological vector spaces”, Stochastics, 33:1 (1990), 75–90 | MR | Zbl

[115] Schwartz L., Radon measures on arbitrary topological spaces and cylindrical measures, Publ. Tata Inst. of Fund. Research; Oxford Univ. Press, 1973

[116] Shigekawa I., “Existence of invariant measures of diffusions on an abstract Wiener space”, Osaka J. Math., 24:1 (1987), 37–59 | MR | Zbl

[117] Silverstein M. L., “Symmetric Marcov Processes”, Lect. Notes Math., 426, 1974, 1–287 | MR

[118] Streater R. F., “Euclidean quantum mechanics and stochastic integrals”, Lect. Notes Math., 851, 1981, 371–393 | MR | Zbl

[119] Stroock D. W., “The Malliavin calculus and its applications”, Lect. Notes Math., 851, 1981, 394–432 | MR | Zbl

[120] Stroock D., “The Malliavin calculus, a functional analytic approach”, J. Funct. Anal., 44:2 (1981), 213–257 | DOI | MR

[121] Takeda M., “$(r,p)$-capacity on the Wiener space and properties of Brownian motion”, Z. Warsh. verw. Geb., 68:2 (1984), 149–162 | DOI | MR | Zbl

[122] Takeda M., “On the uniqueness of Markovian self-adjoint extension of diffusion operators in infinite-dimensional spaces”, Osaka J. Math., 22:4 (1985), 733–742 | MR | Zbl

[123] Wielens N., “On the essential self-adjointness of generalized Schrödinger operators”, J. Funct. Anal., 61:1 (1985), 98–115 | DOI | MR | Zbl

[124] Williams D., “To begin at the begining”, Lect. Notes in Math., 851, 1981, 1–55 | Zbl

[125] Yor M., “Existence et unicite de diffusions a valeurs dans un espace de Hilbert”, Ann. Inst. H. Poincare. Sect. B, 10:1 (1974), 55–88 | MR | Zbl

[126] Zakai M., “The Malliavin calculus”, Acta Appl. Math., 1985, no. 3, 175–207 | DOI | MR | Zbl

[127] Bogachev V. I., “Lokalno vypuklye prostranstva so svoistvom TsPT i nositeli mer”, Vestnik MGU. Ser. 1. Matem., mekh., 1986, no. 6, 16–20 | MR

[128] Brelo M., Osnovy klassicheskoi teorii potentsiala, Mir, M., 1964 | Zbl

[129] Iosida K., Funktsionalnyi analiz, Mir, M., 1967

[130] Mazya V. G., Khavin V. P., “Nelineinaya teoriya potentsiala”, UMN, 27 (1983), 71–148

[131] Meier P.-A., Veroyatnost i potentsialy, Mir, M., 1973

[132] Norin N. V., Smolyanov O. G., “Neskolko rezultatov o logarifmicheskikh proizvodnykh mer na lokalno vypuklom prostranstve”, Matem. zametki, 54:6 (1993), 135–137 | MR

[133] Uermer Dzh., Teoriya potentsiala, Mir, M., 1980

[134] Albeverio S., Ma Z. M., Rockner M., A Beurling–Deny type structure theorem for Dirichlet forms on general state space, Preprint BiBoS No 98, 1990

[135] Albeverio S., Ma Z. M., Rockner M., A remark on the support cadlag processes, Preprint SFB 237 No 136, 1991

[136] Albeverio S., Ma Z. M., Rockner M., Regularization of Dirichlet Spaces and Applications, Preprint R.F.-W. Univ. No 216, 1992

[137] Albeverio S., Ma Z. M., Rockner M., “Quasi-regular Dirichlet Forms and Markov Processes”, J. Funct. Anal., 111:1 (1993), 118–154 | DOI | MR | Zbl

[138] Albeverio S., Rockner M., Zhang T. S., “Markov uniqueness for a class of infinite dimensional Dirichlet operators”, Preprint S.I.S.S.A. No 223 FM, 1992

[139] Rockner M., Zhang T. S., “Uniqueness of generalized Schrodinger operators and applications”, J. Funct. Anal., 105:1 (1992), 187–231 | DOI | MR

[140] Takeda M., “The maximum Markovian self-adjoint extensions of generalized Schrodinger operators”, J. Math. Soc. Japan, 44 (1992), 113–130 | DOI | MR | Zbl