DIFFERENTIAL OPERATORS ARISING FROM TRANSLATION OF POISSON FUNCTIONALS
Yoshifusa Ito · Australian Journal of Statistics · 1988
summary A translation operator Tη on Poisson functionals is defined by Tηφ(x)=φ(x=y)dvη(y), where vη 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.