  This work is devoted to switching diffusions that have two components (a
continuous component and a discrete component). Different from the so-called
Markovian switching diffusions, in the setup, the discrete component (the
switching) depends on the continuous component (the diffusion process). The
objective of this paper is to provide a number of properties related to the
well posedness. First, the differentiability with respect to initial data of
the continuous component is established. Then, further properties including
uniform continuity with respect to initial data, and smoothness of certain
functionals are obtained. Moreover, Feller property is obtained under only
local Lipschitz continuity. Finally, an example of Lotka-Voterra model under
regime switching is provided as an illustration.
