Searcharxiv⌕ Search

arXiv subjects

S. V. Shaposhnikov

Publications and source records attributed to S. V. Shaposhnikov.

5 recordsLinked to original sources
arXiv2026

The Dirichlet problem for double divergence form elliptic equations with measures as boundary conditions

We introduce and study the Dirichlet problem for double divergence form elliptic equations with coefficients of low regularity and boundary conditions given by general Borel measures. Under broad assumptions we establish the solvability of this problem. It is also shown that a solution to a double divergence form equation on a domain serves as a solution to the Dirichlet problem on inner subdomains. The obtained results are applied to the study of properties of solutions to stationary Fokker--Planck--Kolmogorov equations.

math.AP↗
arXiv2019

Log-Sobolev-type inequalities for solutions to stationary Fokker-Planck-Kolmogorov equations

We prove that every probability measure $μ$ satisfying the stationary Fokker-Planck-Kolmogorov equation obtained by a $μ$-integrable perturbation $v$ of the drift term $-x$ of the Ornstein-Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure $γ$ and for the density $f=dμ/dγ$ the integral of $f |\log (f+1)|^α$ against $γ$ is estimated via $\|v\|_{L^1(μ)}$ for all $α<1/4$, which is a weakened $L^1$-analog of the logarithmic Sobolev inequality. This means that stationary measures of diffusions whose drifts are integrable perturbations of $-x$ are absolutely continuous with respect to Gaussian measures.

math.PR↗
arXiv2018

Convergence in variation of solutions of nonlinear Fokker-Planck-Kolmogorov equations to stationary measures

We study convergence in variation of probability solutions of nonlinear Fokker-Planck-Kolmogorov equations to stationary solutions. We obtain sufficient conditions for the exponential convergence of solutions to the stationary solution in case of coefficients that can have an arbitrary growth at infinity and depend on the solutions through convolutions with unbounded discontinuous kernels. In addition, we study a more difficult case where the nonlinear equation has several stationary solutions and convergence to a stationary solution depends on initial data. Finally, we obtain sufficient conditions for solvability of nonlinear Fokker-Planck-Kolmogorov equations.

math.PR↗