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Stanislav V. Shaposhnikov

Publications and source records attributed to Stanislav V. Shaposhnikov.

11 recordsLinked to original sources
arXiv2026

Infinite-dimensional nonlinear stationary Fokker-Planck-Kolmogorov equations

We prove existence of a probability solution to the nonlinear stationary Fokker-Planck-Kolmogorov equation on an infinite dimensional space with a centered Gaussian measure $γ$ with a unit diffusion operator and a drift of the form $-x+v(p,x)$, where $v$ is a bounded mapping with values in the Cameron-Martin space $H$ of $γ$ and $v$ is defined on the space $E\times X$, where is $E$ is the subset of $L^2(γ)$ consisting of probability densities. The equation has the form $L_{b(p,\bullet)} ^*(p\cdot γ)=0$ with $L_{b(p,\bullet)}φ=Δ_H φ+ (b(p,\bullet) , D_{_H}φ)_{_H}$, so that the drift coefficient depends on the unknown solution, which makes the equation nonlinear. This dependence is assumed to satisfy a suitable continuity condition. This result is applied to drifts of Vlasov type defined by means of the convolution of a vector field with the solution. In addition, we consider a more general situation where only the components of $v$ are uniformly bounded and prove the existence of a probability solution under some stronger continuity condition on the drift.

math.AP↗
arXiv2025

Estimates for the distances between solutions to Kolmogorov equations with diffusion matrices of low regularity

We obtain estimates for the weighted $L^1$-norm of the difference of two probability solutions to Kolmogorov equations in terms of the difference of the diffusion matrices and the drifts. Unlike the previously known results, our estimate does not involve Sobolev derivatives of solutions and coefficients. The diffusion matrices are supposed to be non-singular, bounded and satisfy the Dini mean oscillation condition.

math.AP↗
arXiv2022

Zvonkin's transform and the regularity of solutions to double divergence form elliptic equations

We study qualitative properties of solutions to double divergence form elliptic equations (or stationary Kolmogorov equations) on~$\mathbb{R}^d$. It is shown that the Harnack inequality holds for nonnegative solutions if the diffusion matrix $A$ is nondegenerate and satisfies the Dini mean oscillation condition and the drift coefficient $b$ is locally integrable to a power $p>d$. We establish new estimates for the $L^p$-norms of solutions and obtain a generalization of the known theorem of Hasminskii on the existence of a probability solution to the stationary Kolmogorov equation to the case where the matrix $A$ satisfies Dini's condition or belongs to the class VMO. These results are based on a new analytic version of Zvonkin's transform of the drift coefficient.

math.AP↗
arXiv2019

On the Ambrosio-Figalli-Trevisan superposition principle for probability solutions to Fokker-Planck-Kolmogorov equations

We prove a generalization of the known result of Trevisan on the Ambrosio-Figalli-Trevisan superposition principle for probability solutions to the Cauchy problem for the Fokker-Planck-Kolmogorov equation, according to which such a solution is generated by a solution to the corresponding martingale problem. The novelty is that in place of the integrability of the diffusion and drift coefficients $A$ and $b$ with respect to the solution we require the integrability of $(\|A(t,x)\|+|\langle b(t,x),x\rangle |)/(1+|x|^2)$. Therefore, in the case where there are no a priori global integrability conditions the function $\|A(t,x)\|+|\langle b(t,x),x\rangle |$ can be of quadratic growth. Moreover, as a corollary we obtain that under mild conditions on the initial distribution it is sufficient to have the one-sided bound $\langle b(t,x),x\rangle \le C+C|x|^2 \log |x|$ along with $\|A(t,x)\|\le C+C|x|^2 \log |x|$.

math.PR↗
arXiv2018

On $L^1$-estimates for probability solutions to Fokker-Planck-Kolmogorov equations

We prove two new results connected with elliptic Fokker-Planck-Kolmogorov equations with drifts integrable with respect to solutions. The first result answers negatively a long-standing question and shows that a density of a probability measure satisfying the Fokker-Planck-Kolmogorov equation with a drift integrable with respect to this density can fail to belong to the Sobolev class~$W^{1,1}(\mathbb{R}^d)$. There is also a version of this result for densities with respect to Gaussian measures. The second new result gives some positive information about properties of such solutions: the solution density is proved to belong to certain fractional Sobolev classes.

math.PR↗
arXiv2013

An analytic approach to infinite-dimensional continuity and Fokker-Planck-Kolmogorov equations

We prove a new uniqueness result for solutions to Fokker-Planck-Kolmogorov (FPK) equations for probability measures on infinite-dimensional spaces. We consider infinite-dimensional drifts that admit certain finite-dimensional approximations. In contrast to most of the previous work on FPK-equations in infinite dimensions, we include cases with non-constant coefficients in the second order part and also include degenerate cases where these coefficients can even be zero. Also a new existence result is proved. Some applications to Fokker-Planck-Kolmogorov equations associated with SPDEs are presented.

math.PR↗
arXiv2005

Global Regularity and Bounds for Solutions of Parabolic Equations for Probability Measures

Given a second order parabolic operator $$ Lu(t,x) :=\frac{\partial u(t,x)}{\partial t} + a^{ij}(t,x)\partial_{x_i}\partial_{x_j}u(t,x) + b^i(t,x)\partial_{x_i}u(t,x), $$ we consider the weak parabolic equation $L^{*}μ=0$ for Borel probability measures on $(0,1)\times\mathbb{R}^d$. The equation is understood as the equality $$ \int_{(0,1)\times\mathbb{R}^d} Lu dμ=0 $$ for all smooth functions $u$ with compact support in~$(0,1)\times\mathbb{R}^d$. This equation is satisfied for the transition probabilities of the diffusion process associated with~$L$. We show that under broad assumptions $μ$ has the form $μ=\varrho(t,x) dt dx$, where the function $x\mapsto \varrho(t,x)$ is Sobolev, $|\nabla_x \varrho(x,t)|^2/\varrho(t,x)$ is Lebesgue integrable over $[0,τ]\times\mathbb{R}^d$, and $\varrho\in L^p([0,τ]\times\mathbb{R}^d)$ for all $p\in [1,+\infty)$ and $τ<1$. Moreover, a sufficient condition for the uniform boundedness of $\varrho$ on $[0,τ]\times\mathbb{R}^d$ is given.

math.PR↗