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DIFFERENTIAL OPERATORS ARISING FROM TRANSLATION OF POISSON FUNCTIONALS
Author(s) -
Ito Yoshifusa
Publication year - 1988
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1988.tb00480.x
Subject(s) - mathematics , operator theory , poisson distribution , operator (biology) , differential operator , fourier integral operator , constant coefficients , translation (biology) , differential (mechanical device) , spectral theorem , measure (data warehouse) , basis (linear algebra) , mathematical analysis , pure mathematics , computer science , physics , statistics , data mining , biochemistry , chemistry , geometry , repressor , messenger rna , transcription factor , gene , thermodynamics
summary A translation operator T η on Poisson functionals is defined by T η φ(x)=φ(x=y)dv η (y), where v η is a measure which defines a Poisson process with intensity η and independent of the basic Poisson process. By means of the translation operator the differential operators with respect to a Poisson white noise are redefined. This makes it possible to understand why the differential operators are actually difference operators when applied to suitable Poisson functionals, and provides a basis for applications of the differential operators.