Quantitative Asymptotic Expansions of the Probabilistic Representation Formulae for (C[sub(0)]) m-Parameter Operator Semigroups

In this chapter we study quantitative asymptotic expansions for the probabilistic representation formulae of discrete type and asymptotic formulae of continuous type. We also provide applications. Namely, for special semigroups, the results presented apply to asymptotic expansions for multivariate Feller operators. Also asymptotic expansions of univariate and multivariate Bernstein operators are obtained and studied. The methods used are deeply probabilistic. In the quantitative approach we employ a natural modulus of continuity. This treatment is based on [179].