Back
 JAMP  Vol.3 No.7 , July 2015
Localization of Unbounded Operators on Guichardet Spaces
Abstract: As stochastic gradient and Skorohod integral operators, is an adjoint pair of unbounded operators on Guichardet Spaces. In this paper, we define an adjoint pair of operator , where with being the conditional expectation operator. We show that (resp.) is essentially a kind of localization of the stochastic gradient operators (resp. Skorohod integral operators δ). We examine that and satisfy a local CAR (canonical ani-communication relation) and forms a mutually orthogonal operator sequence although each is not a projection operator. We find that is s-adapted operator if and only if is s-adapted operator. Finally we show application exponential vector formulation of QS calculus.
Cite this paper: Zhang, J. , Wang, C. and Tian, L. (2015) Localization of Unbounded Operators on Guichardet Spaces. Journal of Applied Mathematics and Physics, 3, 792-796. doi: 10.4236/jamp.2015.37096.
References

[1]   Attal, S. and Lindsay, J.M. (2004) Quantum Stochastic Calculus with Maximal Operator Domains. The Annals of Probability, 32, 488-529. http://dx.doi.org/10.1214/aop/1078415843

[2]   Wang, C.S., Lu, Y.C. and Chai, H.F. (2011) An Alternative Approach to Privault’s Discrete-Time Chaotic Calculus. J.Math.Anal.Appl., 373, 643-654. http://dx.doi.org/10.1016/j.jmaa.2010.08.021

[3]   Hitsuda, M. (1972) Formula for Brownian Partial Derivatives. Proceedings of the 2nd Japan-USSR Symposium on Probability Theory Commun.Math.Phys., 2, 111-114.

[4]   Hudson, R.L. and Parthasarathy, K.R. (1984) Quantum Ito’s Formula and Stochastic Evolutions. Commun.Math.Phys., 93, 301-323. http://dx.doi.org/10.1007/BF01258530

[5]   Kuo, H.H. (1996) White Noise Distribution Theory. Probability and Stochastics Series, CRC Press.

[6]   Meyer, P.A. (1993) Quantum Probability for Probabilists. Lecture Notes in Mathematics, Spring-Verlag, Berlin. http://dx.doi.org/10.1007/978-3-662-21558-6

[7]   Privault, N. (2009) Moment Identities for Skorohod Integrals on the Wiener Space and Applications. Electronic Communications in Probability, 14, 116-121.

[8]   Privault, N. (2010) Random Hermite Polynomials and Girsanov Identities on the Wiener Space. Infinite Dimensional Analysis, 13, 663-675.

[9]   Skorohod, A.V. (1975) On a Generalization of a Stochastic Integral. Theory Probab. Appl., 20, 219-233.

 
 
Top