Abstract: This talk is devoted to the geometry of stochastic flows. The geometry of homeomorphic flows generated by stochastic differential equations with smooth coefficients is determined by the properties of the Lie algebra generated by the coefficients of the equation. When a flow admits singularities (for example, particles can coalesce) the tools developed for equations with smooth vector fields do not work. It appears that the semigroup of random operators related to the flow gives an appropriate description of its geometry. In particular, for one-dimensional flows with coalescence such semigroup consists of operators with finite-dimensional images. We will consider the asymptotic behavior of widths for some compact sets with respect to these images. We will present also a general version of the Krylov-Veretennikov expansion for operator-valued multiplicative functionals from white noise, which covers the classical result related to stochastic differential equations with smooth coefficients.
