On the smoothness and singularity of invariant measures and transition probabilities of infinite-dimensional diffusions

We construct two examples of nondegenerate diffusion specified by the stochastic differential equation
$$ d\xi_t=\sigma (\xi_t)\,dW_t + B(\xi_t)\,dt $$
in a Hilbert space $X$, where $\sigma (x)=I+\sigma_0(x)$ and $B(x)=\Lambda x+v(x)$; here $\Lambda$ is a continuous linear operator on $X$ and $\sigma_0$ and $v$ are infinitely Fréchet differentiable mappings with values in the spaces of nuclear operators on $X$ and in $X$, respectively, derivatives of any order of which are bounded. These diffusions possess the following properties: (i) In the first example, $\Lambda x =-\frac12 x$ and $\xi_t$ has a (unique) invariant measure which, the same as its transition probabilities, has no directions along which it is differentiable (and even continuous); (ii) in the second example, $\xi_t$ has two different invariant probability measures $\nu_1$ and $\nu_2$ such that $\nu_1$ is equivalent to a Gaussian measure and is differentiable, whereas $\nu_2$ has no directions along which it is nonsingular (or even continuous). In addition, for any $\varepsilon >0$ one can select $\sigma_0$ and $v$ in such a way that they vanish out of the $\varepsilon$-ball and have norms not exceeding $\varepsilon$ (in the spaces of nuclear operators on $X$ and in $X$, respectively).