Quantum stochastic calculus with maximal operator domains
Stéphane Attal and J. Martin Lindsay
Source: Ann. Probab. Volume 32, Number 1A (2004), 488-529.
Abstract
Quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains. Operator adaptedness, conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of Fock space, together with sections of the adapted gradient operator. Free from exponential vector domains, our stochastic integrals may be satisfactorily composed yielding quantum Itô formulas for operator products as sums of stochastic integrals. The calculus has seen two reformulations since its discovery---one closely related to classical Itô calculus; the other to noncausal stochastic analysis and Malliavin calculus. Our theory extends both of these approaches and may be viewed as a synthesis of the two. The main application given here is existence and uniqueness for the Attal--Meyer equations for implicit definition of quantum stochastic integrals.
Primary Subjects: 81S25
Keywords: Quantum stochastic; Fock space; Itô calculus; noncausal; chaotic representation property; Malliavin calculus; noncommutative probability
Full-text: Access granted (open access)
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1078415843
Digital Object Identifier: doi:10.1214/aop/1078415843
Mathematical Reviews number (MathSciNet):
MR2040790
Zentralblatt MATH identifier:
02100726
References
Applebaum, D. B. and Hudson, R. L. (1984). Fermion Itô's formula and stochastic evolutions. Comm. Math. Phys. 96 473--496.
Mathematical Reviews (MathSciNet):
MR775042
Digital Object Identifier: doi:10.1007/BF01212531
Attal, S. (1994). An algebra of noncommutative bounded semimartingales: Square and angle quantum brackets. J. Funct. Anal. 124 292--332.
Mathematical Reviews (MathSciNet):
MR1289353
Digital Object Identifier: doi:10.1006/jfan.1994.1109
Attal, S. (2003). Extensions of quantum stochastic calculus. In Quantum Probability Communications XI. Proceedings of the Quantum Probability Summer School (S. Attal and J. M. Lindsay, eds.) 1--37. World Scientific, Singapore.
Mathematical Reviews (MathSciNet):
MR2032361
Zentralblatt MATH:
1077.81518
Attal, S. and Lindsay, J. M. (1996). Quantum Itô formula---the combinatorial aspect. In Proceedings of the Memorial Conference for Alberto Frigerio (C. Cecchini, ed.) 31--42. Udine Univ. Press.
Attal, S. and Meyer, P.-A. (1993). Interprétation probabiliste et extension des intégrales stochastiques non commutatives. Séminaire de Probabilités XXVII. Lecture Notes in Math. 1557 312--327. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR1308573
Barlow, M. T. and Imkeller, P. (1992). On some sample path properties of Skorohod integral processes. Séminaire de Probabilités XXVI. Lecture Notes in Math. 1526 70--80. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR1231984
Zentralblatt MATH:
0761.60047
Barnett, C., Streater, R. F. and Wilde, I. F. (1982). The Itô--Cliford integral. J. Funct. Anal. 48 172--212.
Mathematical Reviews (MathSciNet):
MR674057
Digital Object Identifier: doi:10.1016/0022-1236(82)90066-0
Barnett, C., Streater, R. F. and Wilde, I. F. (1983). Quasifree quantum stochastic integrals for the CAR and CCR. J. Funct. Anal. 52 19--47.
Mathematical Reviews (MathSciNet):
MR705990
Digital Object Identifier: doi:10.1016/0022-1236(83)90089-7
Belavkin, V. P. (1991). A quantum nonadapted Itô formula and stochastic analysis in Fock scale. J. Funct. Anal. 102 414--447.
Mathematical Reviews (MathSciNet):
MR1140634
Digital Object Identifier: doi:10.1016/0022-1236(91)90129-S
Biane, Ph. (1995). Calcul stochastique non-commutatif. Séminaire de Probabilités XXIX. Lecture Notes in Math. 1608 1--96. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR1383121
Zentralblatt MATH:
0878.60041
Biane, Ph. and Speicher, R. (1998). Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Probab. Theory Related Fields 112 373--409.
Mathematical Reviews (MathSciNet):
MR1660906
Digital Object Identifier: doi:10.1007/s004400050194
Clark, J. M. C. (1970). The representation of functionals of Brownian motion by stochastic integrals. Ann. Math. Statist. 41 1281--1295.
Mathematical Reviews (MathSciNet):
MR270448
Clark, J. M. C. (1971). Correction. Ann. Math. Statist. 42 1778.
Mathematical Reviews (MathSciNet):
MR341633
Diestel, J. and Uhl, J. (1977). Vector Measures. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet):
MR453964
Zentralblatt MATH:
0369.46039
Emery, M. (1989). On the Azéma martingales. Séminaire de Probabilités XXVIII. Lecture Notes in Math. 1372 66--87. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR1022899
Evans, M. P. (1989). Existence of quantum diffusions. Probab. Theory Related Fields 81 473--483.
Mathematical Reviews (MathSciNet):
MR995806
Fagnola, F. (1993). Characterisation of isometric and unitary weakly differentiable cocycles in Fock space. In Quantum Probability and Related Topics VIII (L. Accardi, ed.) 143--164. World Scientific, Singapore.
Mathematical Reviews (MathSciNet):
MR1258327
Gaveau, B. and Trauber, P. (1982). L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel. J. Funct. Anal. 46 230--238.
Mathematical Reviews (MathSciNet):
MR660187
Digital Object Identifier: doi:10.1016/0022-1236(82)90036-2
Goswami, D. and Sinha, K. B. (1999). Hilbert modules and stochastic dilation of a quantum dynamical semigroup on a von Neumann algebra. Comm. Math. Phys. 205 377--403.
Mathematical Reviews (MathSciNet):
MR1712583
Digital Object Identifier: doi:10.1007/s002200050682
Guichardet, A. (1972). Symetric Hilbert Spaces and Related Topics. Lecture Notes in Math. 261. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR493402
Zentralblatt MATH:
0265.43008
Hitsuda, M. (1972). Formula for Brownian partial derivatives. In Proceedings of the Second Japan--USSR Symposium on Probability Theory 2 111--114. Kyoto Univ.
Huang, Z. Y. (1993). Quantum white noises---white noise approach to quantum stochastic calculus. Nagoya Math. J. 129 23--42.
Mathematical Reviews (MathSciNet):
MR1210001
Hudson, R. L. and Parthasarathy, K. R. (1984). Quantum Itô's formula and stochastic evolutions. Comm. Math. Phys. 93 301--323.
Mathematical Reviews (MathSciNet):
MR745686
Digital Object Identifier: doi:10.1007/BF01258530
Hudson, R. L. and Parthasarathy, K. R. (1986). Unification of boson and fermion stochastic calculus. Comm. Math. Phys. 104 457--470.
Mathematical Reviews (MathSciNet):
MR840747
Digital Object Identifier: doi:10.1007/BF01210951
Kümmerer, B. and Speicher, R. (1992). Stochastic integration on the Cuntz algebra $O\sb \infty$. J. Funct. Anal. 103 372--408.
Mathematical Reviews (MathSciNet):
MR1151553
Digital Object Identifier: doi:10.1016/0022-1236(92)90126-4
Lindsay, J. M. (1986). Fermion martingales. Probab. Theory Related Fields 71 307--320.
Mathematical Reviews (MathSciNet):
MR816708
Digital Object Identifier: doi:10.1007/BF00332314
Lindsay, J. M. (1993). Quantum and noncausal stochastic calculus. Probab. Theory Related Fields 97 65--80.
Mathematical Reviews (MathSciNet):
MR1240716
Digital Object Identifier: doi:10.1007/BF01199312
Lindsay, J. M. and Parthasarathy, K. R. (1989). Cohomology of power sets with applications in quantum probability. Comm. Math. Phys. 124 337--364.
Mathematical Reviews (MathSciNet):
MR1012630
Digital Object Identifier: doi:10.1007/BF01219655
Lindsay, J. M. and Wills, S. J. (2000). Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise. Probab. Theory Related Fields 116 505--543.
Mathematical Reviews (MathSciNet):
MR1757598
Digital Object Identifier: doi:10.1007/s004400050261
Maassen, H. (1985). Quantum Markov processes on Fock space described by integral kernels. Quantum Probability and Applications II. Lecture Notes in Math. 1136 361--374. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR819517
Meyer, P.-A. (1986). Eléments de probabilités quantiques. Séminaire de Probabilités XX. Lecture Notes in Math. 1204 186--312. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR942022
Meyer, P.-A. (1993). Quantum Probability for Probabilists, 2nd ed. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR1222649
Zentralblatt MATH:
0773.60098
Mohari, A. (1991). Quantum stochastic differential equations with unbounded coefficients and dilations of Feller's minimal solution. Sankhyā Ser. A 53 255--287.
Mathematical Reviews (MathSciNet):
MR1189771
Mohari, A. and Sinha, K. B. (1990). Quantum stochastic flows with infinite degrees of freedom and countable state Markov processes. Sankhyā Ser. A 52 43--57.
Mathematical Reviews (MathSciNet):
MR1176275
Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1344217
Zentralblatt MATH:
0837.60050
Nualart, D. and Vives, J. (1990). Anticipative calculus for the Poisson process based on the Fock space. Séminaire de Probabilités XXIV. Lecture Notes in Math. 1426 154--165. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR1071538
Zentralblatt MATH:
0701.60048
Parthasarathy, K. R. (1992). An Introduction to Quantum Stochastic Calculus. Birkhäuser, Basel.
Mathematical Reviews (MathSciNet):
MR1164866
Zentralblatt MATH:
0751.60046
Parthasarathy, K. R. and Sinha, K. B. (1986). Stochastic integral representation of bounded quantum martingales in Fock space. J. Funct. Anal. 67 126--151.
Mathematical Reviews (MathSciNet):
MR842607
Digital Object Identifier: doi:10.1016/0022-1236(86)90047-9
Parthasarathy, K. R. and Sunder, V. S. (1998). Exponential vectors of indicator functions are total in the boson Fock space $\Gamma (L^2([0, 1])$. In Quantum Probability Communications X (R. L. Hudson and J. M. Lindsay, eds.) 281--284. World Scientific, Singapore.
Mathematical Reviews (MathSciNet):
MR1689477
Pisier, G. and Xu, Q. (1997). Noncommutative martingale inequalities. Comm. Math. Phys. 189 667--698.
Mathematical Reviews (MathSciNet):
MR1482934
Digital Object Identifier: doi:10.1007/s002200050224
Skorohod, A. V. (1975). On a generalization of a stochastic integral. Theory Probab. Appl. 20 219--233.
Mathematical Reviews (MathSciNet):
MR391258
Stroock, D. W. (1990). A Concise Introduction to the Theory of Integration. World Scientific, Singapore.
Mathematical Reviews (MathSciNet):
MR1141626
Zentralblatt MATH:
0729.28001
Vincent-Smith, G. F. (1997). The Itô formula for quantum semimartingales. Proc. London Math. Soc. (3) 75 671--720.
Mathematical Reviews (MathSciNet):
MR1466665
Digital Object Identifier: doi:10.1112/S0024611597000476
D.B. Applebaum and R.L. Hudson, Fermion Itô's formula and stochastic evolutions, Comm. Math. Phys. 96 (1984), 473--496.
Mathematical Reviews (MathSciNet):
MR775042
Digital Object Identifier: doi:10.1007/BF01212531
S. Attal, An algebra of noncommutative bounded semimartingales. Square and angle quantum brackets, J. Funct. Anal. 124 (1994), 292--332.
Mathematical Reviews (MathSciNet):
MR1289353
Digital Object Identifier: doi:10.1006/jfan.1994.1109
S. Attal, Extensions of quantum stochastic calculus, in, ``Quantum Probability Communications XI'', Proceedings of Quantum Probability Summer School (Grenoble, 1998), eds. S. Attal and J.M. Lindsay, to appear.
Mathematical Reviews (MathSciNet):
MR2032361
Zentralblatt MATH:
1077.81518
S. Attal and J.M. Lindsay, Quantum Itô formula --- the combinatorial aspect, in, ``Proceedings of the Memorial Conference for Alberto Frigerio,'' ed. C. Cecchini Forum, Udine University Press (1996), 31--42.
S. Attal and P.-A. Meyer, Interprétation probabiliste et extension des intégrales stochastiques non commutatives, in, ``Séminaire de Probabilités XXVII,'' eds. J. Azéma, P.-A. Meyer and M. Yor, Lecture Notes in Mathematics 1557, Springer-Verlag, Berlin (1993), 312--327.
Mathematical Reviews (MathSciNet):
MR1308573
M.T. Barlow and P. Imkeller, On some sample path properties of Skorohod integral processes, in ``Séminaire de Probabiliés XXVI,'' eds. J. Azéma, P.-A. Meyer and M. Yor, Lecture Notes in Mathematics 1526, Springer-Verlag, Berlin (1992), 70--80.
Mathematical Reviews (MathSciNet):
MR1231984
Zentralblatt MATH:
0761.60047
C. Barnett, R.F. Streater and I.F. Wilde, The Itô-Cliford integral, J. Funct. Anal. 48 (1982), 172--212.
Mathematical Reviews (MathSciNet):
MR674057
Digital Object Identifier: doi:10.1016/0022-1236(82)90066-0
C. Barnett, R.F. Streater and I.F. Wilde, Quasifree quantum stochastic integrals for the CAR and CCR, J. Funct. Anal. 52 (1983), 19--47.
Mathematical Reviews (MathSciNet):
MR705990
Digital Object Identifier: doi:10.1016/0022-1236(83)90089-7
V.P. Belavkin, A quantum nonadapted Itô formula and stochastic analysis in Fock scale, J. Funct. Anal. 102 (1991), 414--447.
Mathematical Reviews (MathSciNet):
MR1140634
Digital Object Identifier: doi:10.1016/0022-1236(91)90129-S
Ph. Biane, Calcul stochastique non-commutatif, in, ``Lectures on probability theory: Lectures from Saint-Flour Summer School XXIII, 1993'' ed. P. Bernard, Lecture Notes in Mathematics 1608, Springer-Verlag, Berlin, 1995.
Mathematical Reviews (MathSciNet):
MR1383121
Zentralblatt MATH:
0878.60041
Ph. Biane and R. Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Probab. Theory Related Fields 112 (1998), 373--409.
Mathematical Reviews (MathSciNet):
MR1660906
Digital Object Identifier: doi:10.1007/s004400050194
J.M.C. Clark, The representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Statist. 41 (1970), 1281--1295; ibid Correction, 42 (1971), 1778.
Mathematical Reviews (MathSciNet):
MR270448
J. Diestel and J. Uhl, ``Vector measures'', Mathematical Surveys No 15, American Mathematical Society, Providence, R.I., 1977.
Mathematical Reviews (MathSciNet):
MR453964
Zentralblatt MATH:
0369.46039
M. Emery, On the Azéma martingales, in, ``Séminaire de Probabilités XXIII,'' eds. J. Azéma, P.-A. Meyer and M. Yor, Lecture Notes in Mathematics 1372, Springer-Verlag, Berlin (1989), 66--87.
Mathematical Reviews (MathSciNet):
MR1022899
M.P. Evans, Existence of quantum diffusions, Probab. Theory Related Fields 81 (1989), 473--483.
Mathematical Reviews (MathSciNet):
MR995806
F. Fagnola, Characterisation of isometric and unitary weakly differentiable cocycles in Fock space, in, ``Quantum Probability and Related Topics VIII'', ed. L. Accardi, World Scientific, Singapore (1993), 143--164.
Mathematical Reviews (MathSciNet):
MR1258327
B. Gaveau and P. Trauber, L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel, J. Funct. Anal. 46 (1982), 230--238.
Mathematical Reviews (MathSciNet):
MR660187
Digital Object Identifier: doi:10.1016/0022-1236(82)90036-2
D. Goswami and K.B. Sinha, Hilbert modules and stochastic dilation of a quantum dynamical semigroup on a von Neumann algebra, Comm. Math. Phys. 205 (1999), 377--403.
Mathematical Reviews (MathSciNet):
MR1712583
Digital Object Identifier: doi:10.1007/s002200050682
A. Guichardet, ``Symmetric Hilbert spaces and related topics'', Lecture Notes in Mathematics 261, Springer-Verlag, Berlin, 1972.
Mathematical Reviews (MathSciNet):
MR493402
Zentralblatt MATH:
0265.43008
M. Hitsuda, Formula for Brownian partial derivatives, in, ``Proceedings of the 2nd Japan-USSR Symposium on Probability Theory'' (Kyoto 1972), Vol. 2, Kyoto Univ. Kyoto (1972), 111--114.
Z.Y. Huang, Quantum white noises---white noise approach to quantum stochastic calculus, Nagoya Math. J. 129 (1993), 23--42
Mathematical Reviews (MathSciNet):
MR1210001
R.L. Hudson and K.R. Parthasarathy, Quantum Itô's formula and stochastic evolutions, Comm. Math. Phys. 93 (1984), 301--323.
Mathematical Reviews (MathSciNet):
MR745686
Digital Object Identifier: doi:10.1007/BF01258530
R.L. Hudson and K.R. Parthasarathy, Unification of Boson and Fermion stochastic calculus, Comm. Math. Phys. 104 (1986), 457--470.
Mathematical Reviews (MathSciNet):
MR840747
Digital Object Identifier: doi:10.1007/BF01210951
B. Kümmerer and R. Speicher, Stochastic integration on the Cuntz algebra $O\sb \infty$, J. Funct. Anal. 103 (1992), 372--408.
Mathematical Reviews (MathSciNet):
MR1151553
Digital Object Identifier: doi:10.1016/0022-1236(92)90126-4
J.M. Lindsay, Fermion martingales, Probab. Theory Related Fields 71 (1986), 307--320.
Mathematical Reviews (MathSciNet):
MR816708
Digital Object Identifier: doi:10.1007/BF00332314
J.M. Lindsay, Quantum and noncausal stochastic calculus, Probab. Theory Related Fields 97 (1993), 65--80.
Mathematical Reviews (MathSciNet):
MR1240716
Digital Object Identifier: doi:10.1007/BF01199312
J.M. Lindsay and K.R. Parthasarathy, Cohomology of power sets with applications in quantum probability, Comm. Math. Phys. 124 (1989), 337--364.
Mathematical Reviews (MathSciNet):
MR1012630
Digital Object Identifier: doi:10.1007/BF01219655
J.M. Lindsay and S.J. Wills, Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise, Probab. Theory Related Fields 116 (2000), 505--543.
Mathematical Reviews (MathSciNet):
MR1757598
Digital Object Identifier: doi:10.1007/s004400050261
H. Maassen, Quantum Markov processes on Fock space described by integral kernels, in, ``Quantum Probability & Related Topics II,'' eds. L. Accardi and W. von Waldenfels, Lecture Notes in Mathematics 1136, Springer-Verlag, Berlin (1985), 361--374.
P.-A. Meyer, ``Quantum Probability for Probabilists,'' 2nd Edition, Lecture Notes in Mathematics 1538, Springer-Verlag, Berlin, 1993.
Mathematical Reviews (MathSciNet):
MR1222649
Zentralblatt MATH:
0773.60098
P.-A. Meyer, Eléments de probabilités quantiques, in, ``Séminaire de Probabilités XX,'' eds. J. Azéma and M. Yor, Lecture Notes in Mathematics 1204, Springer-Verlag, Berlin (1986), 186--312.
Mathematical Reviews (MathSciNet):
MR942022
A. Mohari, Quantum stochastic differential equations with unbounded coefficients and dilations of Feller's minimal solution, Sankhyā, Series A 53 (1991), 255--287.
Mathematical Reviews (MathSciNet):
MR1189771
A. Mohari and K.B. Sinha, Quantum stochastic flows with infinite degrees of freedom and countable state Markov processes, Sankhyā, Series A 52 (1990), 43--57.
Mathematical Reviews (MathSciNet):
MR1176275
D. Nualart, ``The Malliavin calculus and related topics,'' Probability and its Applications, Springer-Verlag, New York, 1995.
Mathematical Reviews (MathSciNet):
MR1344217
Zentralblatt MATH:
0837.60050
D. Nualart and J. Vives, Anticipative calculus for the Poisson process based on the Fock space, in,``Séminaire de Probabilités XXIV,'' eds. J. Azéma, P.-A. Meyer and M. Yor, Lecture Notes in Mathematics 1426, Springer-Verlag, Berlin (1990), 154--165.
Mathematical Reviews (MathSciNet):
MR1071538
Zentralblatt MATH:
0701.60048
K.R. Parthasarathy, ``An introduction to quantum stochastic calculus,'' Monographs in Mathematics, Birkhäuser, Basel, 1992.
Mathematical Reviews (MathSciNet):
MR1164866
Zentralblatt MATH:
0751.60046
K.R. Parthasarathy and K.B. Sinha, Stochastic integral representation of bounded quantum martingales in Fock space, J. Funct. Anal. 67 (1986), 126--151.
Mathematical Reviews (MathSciNet):
MR842607
Digital Object Identifier: doi:10.1016/0022-1236(86)90047-9
K.R. Parthasarathy and V.S. Sunder, Exponential vectors of indicator functions are total in the boson Fock space $\Gamma (L^2([0, 1])$, in, ``Quantum Probability Communications X,'' eds. R.L. Hudson and J.M. Lindsay, World Scientific, Singapore (1998), 281--284.
G. Pisier and Q. Xu, Noncommutative martingale inequalities, Comm. Math. Phys. 189 (1997), 667--698.
Mathematical Reviews (MathSciNet):
MR1482934
Digital Object Identifier: doi:10.1007/s002200050224
D.W. Stroock, ``A concise introduction to the theory of integration'', Series in Pure Mathematics 12, World Scientific, Singapore, 1990.
Mathematical Reviews (MathSciNet):
MR1141626
Zentralblatt MATH:
0729.28001
A.V. Skorohod, On a generalization of a stochastic integral, Theor. Probability Appl. 20 (1975), 219--233.
Mathematical Reviews (MathSciNet):
MR391258
G.F. Vincent-Smith, The Itô formula for quantum semimartingales, Proc. London Math. Soc. 75 (1997), 671--720.
Mathematical Reviews (MathSciNet):
MR1466665
Digital Object Identifier: doi:10.1112/S0024611597000476
The Annals of Probability
