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Short Communications
On the smoothness and singularity of invariant measures and transition probabilities of infinite-dimensional diffusions
N. A. Tolmachev M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
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).
Keywords:
infinite-dimensional space, diffusion, transition probabilities, invariant measure, smoothness and singularity of measures, exceptional set, Hilbert space. Received: 22.01.1998
DOI:
10.4213/tvp2170
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