Convolution-type derivatives and transforms of Colombeau generalized stochastic processes

Abstract

In this paper, we introduce a theory of convolution-type (fractional) derivatives in the algebra of Colombeau generalized stochastic processes. Non-regularized and regularized Caputo fractional derivatives of Colombeau generalized stochastic processes are introduced and some of their properties are studied. As an example, a certain stochastic Cauchy problem is considered in two cases, with nonregularized and regularized Caputo fractional derivatives.

Keywords:

• convolution transform
• Colombeau generalized stochastic process
• fractional derivative
• regularization

AMS Subject Classifications :

• 26A33
• 33E20
• 44A05
• 44A35
• 60G20

Acknowledgements

This work was supported by the project Functional analysis, ODEs and PDEs with singularities, No. 144016, financed by the Ministry of Science, Republic of Serbia.