Positivity theorem for a stochastic delay equation on a manifold (Q1415884)

Malliavin calculus has been used in the years 1980 to study Itô diffusion processes in order to get the results about the existence of density and its positivity. In this paper, the author uses this machinery to deal with stochastic delay equations on a manifold \(M\). More precisely, the author considers the equation \[ dx_t= \sum^m_{i=1} \tau_{t,t-\delta} X_i(x_{t-\delta})\,d\omega^i_t, \] where \(X_i\) are vector fields on \(M\) and \(\tau_{t,t-\delta}\) denotes the parallel transport along \(\{x_s;\,s\in [t-\delta, t]\}\). Clearly \(x_t\) is dependent of the path \(\{x_s;\,s\in [t-\delta, t]\}\). The main result of the paper is that if the vector fields \(X_i\) \(\text{span\,}T_x(M)\) in all points of \(M\), then \(x_t\) has a smooth density \(q> 0\).