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.

 
 
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