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.
KEYWORDS
Stochastic Gradient Operator, Skorohod Integral Operator, Localization, Ex-Ponential Vector, Guichardet Spaces
Stochastic Gradient Operator, Skorohod Integral Operator, Localization, Ex-Ponential Vector, Guichardet Spaces
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.
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
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[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.