Applications of the Integration by Parts Formula Involving Malliavin Derivative of the Solution Process and it's Inverse

In the present work we have established some applications of the integration by parts formula which enables to lay the foundations of the study of regularity properties of the distributions of the solution process of stochastic delay equation. In this work we recall our basic Delay SDE (1.1) and then we have defined clearly what we mean by the space flow of the solution process which is the Malliavin derivative of the solution process of the Delay SDE (1.1) and then we have formulated the corresponding Delay SDE of the space flow see equation (2.12) and it's inverse see equation (2.13).

For a concise formulation of the SDE of the space flow and it's inverse, we have introduce the Stochastic differentials in equations (2.6), (2.7) and (2.8). see also the Delay SDE's (2.16) and (2.17) in this work which concern the space flow and it's inverse respectively.