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We derive Mellin integral representations in terms of Macdonald functions for the squared absolute value s ↦ | Γ ⁢ ( a + i ⁢ s ) | 2 {s\mapsto|\Gamma(a+is)|^{2}} of the gamma function and its Fourier transform when a < 0 {a<0} is non-integer, generalizing known results in the case a > 0 {a>0}.This representation is based on a renormalization argument using modified Bessel functions of the second kind, and it applies to the representation of the solutions of a Fokker–Planck equation. [ABSTRACT FROM AUTHOR]

Proposes a stochastic calculus with respect to a class ... of anticipating processes which is wider than the class of Skorokhod integral processes. Notation and preliminaries on the Skorokhod integral operator; Extension of the Skorokhod integral as a continuous operator on Sobolev spaces; Conditions for the existence of pathwise integrals.

Discusses a study which examined the gradient of Riemannian manifold in configuration spaces. Details of probability and Poisson measures; Application of differential calculus on configuration space; Sample equation of integrals and stochastic integration.