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.

We study mean field games with unbounded coefficients. The existence of a solution is proved. We propose a new approach based on Fokker-Planck-Kolmogorov equations, the Ambrosio-Figalli-Trevisan superposition principle, the method of doubling variables and a priory estimates with Lyapunov functions.

We obtain sufficient conditions for the uniqueness of a probability solution to the stationary Kolmogorov equation with a degenerate diffusion matrix. We employ the method of doubling variables known in stochastic analysis directly to the Kolmogorov equation.

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× X, where is E is the subset of L^2(γ) consisting of probability densities. The equation has the form L_b(p,∙) ^*(p· γ)=0 with L_b(p,∙)φ=Δ_H φ+ (b(p,∙) , D__Hφ)__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.

We show that under broad assumptions, a probability solution to the Cauchy problem for the Fokker–Planck–Kolmogorov equation with a given initial distribution enables one to uniquely determine the diffusion matrix and the drift coefficient. In particular, this can be done if the coefficients satisfy certain global integrability condition with respect to the solution and are continuous or if the diffusion matrix is nondegenerate and sufficiently regular. Actually, the main result is formulated in terms of a technical approximability condition introduced in order to cover different cases in a unified way. The suggested reconstruction method employs the superposition principle and is based on the corresponding martingale problem for a single initial condition.

We study the asymptotic behaviour at infinity of solutions to the Cauchy problem for the Fokker–Planck–Kolmogorov equation. Our main result is an estimate for the difference of two solutions with different initial conditions in the weighted total variation norm in the case when the coefficients depend on time.

We consider Kolmogorov operators with constant diffusion matrices and linear drifts, i.e., Ornstein–Uhlenbeck operators, and show that all solutions to the corresponding stationary Fokker–Planck–Kolmogorov equations (including signed solutions) are invariant measures for the generated semigroups. This also gives a relatively explicit description of all solutions.

The stationary Kolmogorov equation with partially degenerate diffusion matrix and discontinuous drift coefficient is studied. Sufficient conditions for the existence of a probability solution are obtained. Examples demonstrating the sharpness of these conditions are given.

We obtain broad sufficient conditions for reconstructing the coefficients of a Kolmogorov operator by means of a solution to the Cauchy problem for the corresponding Fokker–Planck–Kolmogorov equation.

В работе дан обзор недавних исследований по нелинейным уравнениям Фоккера-Планка-Колмогорова эллиптического и параболического типа и приведен ряд новых результатов. Подробно обсуждаются проблемы существования и единственности решений, различные оценки решений, связи с линейными уравнениями, сходимость решений параболических уравнений к стационарным решениям. Библиография: 116 названий.

. Nonlinear Fokker-Planck-Kolmogorov equations are investigated. Sufficient conditions are obtained for the existence and uniqueness of a nonnegative solution with a prescribed value of the integral. Convergence of solutions for the Cauchy problem to a solution of the stationary equation is shown. An important distinction from the known results is a very general form of the nonlinearity, which makes it possible to consider simultaneously a local and nonlocal dependence of coefficients on solutions.

This paper gives a survey of recent investigations on nonlinear Fokker-Planck-Kolmogorov equations of elliptic and parabolic types and contains a number of new results. We discuss in detail the problems of existence and uniqueness of solutions, various estimates of solutions, connections with linear equations, and the convergence of solutions of parabolic equations to stationary solutions. Bibliography: 116 items.

Исследуется стационарное уравнение Колмогорова с вырожденной матрицей диффузии и разрывным коэффициентом сноса. Получены достаточные условия существования вероятностного решения. Построены примеры, показывающие точность условий. Библиография: 15 названий.

Принцип суперпозиции доставляет вероятностное представление решения $\{\mu_t\}_{t\in[0, T]}$ уравнения Фоккера-Планка-Колмогорова $\partial_t\mu_t=L^{*}\mu_t$ через решение $P$ мартингальной задачи с оператором $L$. Мы обобщаем принцип суперпозиции на случай уравнений на области, исследуем преобразование меры $P$ и оператора $L$ при замене переменных и получаем новые условия справедливости принципа суперпозиции, когда для неограниченной части коэффициента сноса существует функция Ляпунова.

В статье дан обзор нескольких направлений исследований, связанных с работами А. Н. Колмогорова о параболических и эллиптических уравнениях Фоккера-Планка-Колмогорова для переходных и стационарных вероятностей диффузионных процессов. Приведены основные результаты о существовании решений, единственности, свойствах плотностей решений. Упомянуты открытые вопросы в этой области.

The paper gives a survey of several directions of research connected with the works of A.N. Kolmogorov on parabolic and elliptic Fokker--Planck--Kolmogorov equations for transition and stationary probabilities of diffusion processes. We present the fundamental results on existence of solutions, their uniqueness, and the properties of solution densities. Open questions in this area are mentioned.

We study qualitative properties of solutions to double divergence form elliptic equations (or stationary Kolmogorov equations) on 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 some 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.

— In this note we develop a new analytic version of Zvonkin’s transform of the drift coefficient of a stationary Kolmogorov equation and apply this transform to derive the Harnack inequality for nonnegative solutions in the case where the diffusion matrix is not locally Sobolev. We also obtain a generalization of the known theorem of Hasminskii on existence of a probability solution to the stationary Kolmogorov equation.

—In this note we study the stationary Kolmogorov equation and prove that, in the case where the diffusion matrix satisfies Dini’s condition and the drift coefficient is locally integrable to a power greater than the dimension, the ratio of two probability solutions belongs to the Sobolev class, and in the case of existence of a Lyapunov function or the global integrability of the coefficients with respect to the solution a probability solution is unique.