The Annals of Probability

Quantum stochastic calculus with maximal operator domains

Stéphane Attal and J. Martin Lindsay

Full-text: Open access

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.

Article information

Source
Ann. Probab., Volume 32, Number 1A (2004), 488-529.

Dates
First available in Project Euclid: 4 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1078415843

Digital Object Identifier
doi:10.1214/aop/1078415843

Mathematical Reviews number (MathSciNet)
MR2040790

Zentralblatt MATH identifier
1053.81053

Subjects
Primary: 81S25: Quantum stochastic calculus

Keywords
Quantum stochastic Fock space Itô calculus noncausal chaotic representation property Malliavin calculus noncommutative probability

Citation

Attal, Stéphane; Lindsay, J. Martin. Quantum stochastic calculus with maximal operator domains. Ann. Probab. 32 (2004), no. 1A, 488--529. doi:10.1214/aop/1078415843. https://projecteuclid.org/euclid.aop/1078415843


Export citation

References

  • Applebaum, D. B. and Hudson, R. L. (1984). Fermion Itô's formula and stochastic evolutions. Comm. Math. Phys. 96 473--496.
  • Attal, S. (1994). An algebra of noncommutative bounded semimartingales: Square and angle quantum brackets. J. Funct. Anal. 124 292--332.
  • 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.
  • 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.
  • 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.
  • Barnett, C., Streater, R. F. and Wilde, I. F. (1982). The Itô--Cliford integral. J. Funct. Anal. 48 172--212.
  • 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.
  • Belavkin, V. P. (1991). A quantum nonadapted Itô formula and stochastic analysis in Fock scale. J. Funct. Anal. 102 414--447.
  • Biane, Ph. (1995). Calcul stochastique non-commutatif. Séminaire de Probabilités XXIX. Lecture Notes in Math. 1608 1--96. Springer, Berlin.
  • 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.
  • Clark, J. M. C. (1970). The representation of functionals of Brownian motion by stochastic integrals. Ann. Math. Statist. 41 1281--1295.
  • Clark, J. M. C. (1971). Correction. Ann. Math. Statist. 42 1778.
  • Diestel, J. and Uhl, J. (1977). Vector Measures. Amer. Math. Soc., Providence, RI.
  • Emery, M. (1989). On the Azéma martingales. Séminaire de Probabilités XXVIII. Lecture Notes in Math. 1372 66--87. Springer, Berlin.
  • Evans, M. P. (1989). Existence of quantum diffusions. Probab. Theory Related Fields 81 473--483.
  • 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.
  • Gaveau, B. and Trauber, P. (1982). L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel. J. Funct. Anal. 46 230--238.
  • 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.
  • Guichardet, A. (1972). Symetric Hilbert Spaces and Related Topics. Lecture Notes in Math. 261. Springer, Berlin.
  • 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.
  • Hudson, R. L. and Parthasarathy, K. R. (1984). Quantum Itô's formula and stochastic evolutions. Comm. Math. Phys. 93 301--323.
  • Hudson, R. L. and Parthasarathy, K. R. (1986). Unification of boson and fermion stochastic calculus. Comm. Math. Phys. 104 457--470.
  • Kümmerer, B. and Speicher, R. (1992). Stochastic integration on the Cuntz algebra $O\sb \infty$. J. Funct. Anal. 103 372--408.
  • Lindsay, J. M. (1986). Fermion martingales. Probab. Theory Related Fields 71 307--320.
  • Lindsay, J. M. (1993). Quantum and noncausal stochastic calculus. Probab. Theory Related Fields 97 65--80.
  • Lindsay, J. M. and Parthasarathy, K. R. (1989). Cohomology of power sets with applications in quantum probability. Comm. Math. Phys. 124 337--364.
  • 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.
  • 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.
  • Meyer, P.-A. (1986). Eléments de probabilités quantiques. Séminaire de Probabilités XX. Lecture Notes in Math. 1204 186--312. Springer, Berlin.
  • Meyer, P.-A. (1993). Quantum Probability for Probabilists, 2nd ed. Springer, Berlin.
  • Mohari, A. (1991). Quantum stochastic differential equations with unbounded coefficients and dilations of Feller's minimal solution. Sankhyā Ser. A 53 255--287.
  • 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.
  • Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
  • 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.
  • Parthasarathy, K. R. (1992). An Introduction to Quantum Stochastic Calculus. Birkhäuser, Basel.
  • Parthasarathy, K. R. and Sinha, K. B. (1986). Stochastic integral representation of bounded quantum martingales in Fock space. J. Funct. Anal. 67 126--151.
  • 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.
  • Pisier, G. and Xu, Q. (1997). Noncommutative martingale inequalities. Comm. Math. Phys. 189 667--698.
  • Skorohod, A. V. (1975). On a generalization of a stochastic integral. Theory Probab. Appl. 20 219--233.
  • Stroock, D. W. (1990). A Concise Introduction to the Theory of Integration. World Scientific, Singapore.
  • Vincent-Smith, G. F. (1997). The Itô formula for quantum semimartingales. Proc. London Math. Soc. (3) 75 671--720.
  • D.B. Applebaum and R.L. Hudson, Fermion Itô's formula and stochastic evolutions, Comm. Math. Phys. 96 (1984), 473--496.
  • S. Attal, An algebra of noncommutative bounded semimartingales. Square and angle quantum brackets, J. Funct. Anal. 124 (1994), 292--332.
  • 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.
  • 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.
  • 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.
  • C. Barnett, R.F. Streater and I.F. Wilde, The Itô-Cliford integral, J. Funct. Anal. 48 (1982), 172--212.
  • 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.
  • V.P. Belavkin, A quantum nonadapted Itô formula and stochastic analysis in Fock scale, J. Funct. Anal. 102 (1991), 414--447.
  • 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.
  • 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.
  • 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.
  • J. Diestel and J. Uhl, ``Vector measures'', Mathematical Surveys No 15, American Mathematical Society, Providence, R.I., 1977.
  • 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.
  • M.P. Evans, Existence of quantum diffusions, Probab. Theory Related Fields 81 (1989), 473--483.
  • 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.
  • B. Gaveau and P. Trauber, L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel, J. Funct. Anal. 46 (1982), 230--238.
  • 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.
  • A. Guichardet, ``Symmetric Hilbert spaces and related topics'', Lecture Notes in Mathematics 261, Springer-Verlag, Berlin, 1972.
  • 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
  • R.L. Hudson and K.R. Parthasarathy, Quantum Itô's formula and stochastic evolutions, Comm. Math. Phys. 93 (1984), 301--323.
  • R.L. Hudson and K.R. Parthasarathy, Unification of Boson and Fermion stochastic calculus, Comm. Math. Phys. 104 (1986), 457--470.
  • B. Kümmerer and R. Speicher, Stochastic integration on the Cuntz algebra $O\sb \infty$, J. Funct. Anal. 103 (1992), 372--408.
  • J.M. Lindsay, Fermion martingales, Probab. Theory Related Fields 71 (1986), 307--320.
  • J.M. Lindsay, Quantum and noncausal stochastic calculus, Probab. Theory Related Fields 97 (1993), 65--80.
  • J.M. Lindsay and K.R. Parthasarathy, Cohomology of power sets with applications in quantum probability, Comm. Math. Phys. 124 (1989), 337--364.
  • 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.
  • 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.
  • 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.
  • A. Mohari, Quantum stochastic differential equations with unbounded coefficients and dilations of Feller's minimal solution, Sankhyā, Series A 53 (1991), 255--287.
  • 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.
  • D. Nualart, ``The Malliavin calculus and related topics,'' Probability and its Applications, Springer-Verlag, New York, 1995.
  • 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.
  • K.R. Parthasarathy, ``An introduction to quantum stochastic calculus,'' Monographs in Mathematics, Birkhäuser, Basel, 1992.
  • K.R. Parthasarathy and K.B. Sinha, Stochastic integral representation of bounded quantum martingales in Fock space, J. Funct. Anal. 67 (1986), 126--151.
  • 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.
  • D.W. Stroock, ``A concise introduction to the theory of integration'', Series in Pure Mathematics 12, World Scientific, Singapore, 1990.
  • A.V. Skorohod, On a generalization of a stochastic integral, Theor. Probability Appl. 20 (1975), 219--233.
  • G.F. Vincent-Smith, The Itô formula for quantum semimartingales, Proc. London Math. Soc. 75 (1997), 671--720.