[6] In 1984, V. Bogachev resolved three Aronszajn's problems on infinite-dimensional probability distributions and answered a famous question of I. M. Gelfand posed about 25 years before that.
In 1992, Vladimir Bogachev proved T. Pitcher’s conjecture (stated in 1961) on the differentiability of the distributions of diffusion processes.
In 1995, he proved (with Michael Röckner) the famous Shigekawa conjecture on the absolute continuity of invariant measures of diffusion processes.
In 1999, in a joint work with Sergio Albeverio and Röckner, Professor Bogachev resolved the well-known problem of S. R. S. Varadhan on the uniqueness of stationary distributions, which had remained open for about 20 years.
A remarkable achievement of Vladimir Bogachev is the recently obtained (2021) answer to the question of Andrey Kolmogorov (posed in 1931) on the uniqueness of the solution to the Cauchy problem: it is shown that the Cauchy problem with a unit diffusion coefficient and locally bounded drift has a unique probabilistic solution on