Searcharxiv⌕ Search

arXiv subjects

D. Kinzebulatov

Publications and source records attributed to D. Kinzebulatov.

At least 19 recordsLinked to original sources
arXiv2026

Uniform many-particle spectral gap inequality and heat kernel bounds for strong attractive interactions

We prove a spectral gap inequality for Gaussian measures modified by singular attractive pair interactions. The spectral gap constant is explicit and uniform in the number of particles and the regularization parameter. As an application, we use a corresponding spectral gap inequality for a cutoff interaction weight to obtain two-sided bounds on the transition density of the attractive logarithmic gas.

math.AP↗
arXiv2026

Strong solutions of SDEs with critical discontinuities in diffusion coefficients

We prove strong existence for Itô SDEs with diffusion coefficients that can introduce strong attraction to a submanifold. We extend and strengthen the Röckner-Zhao approach, which uses Malliavin calculus to establish compactness of the approximating solutions in Wiener-Sobolev space. At least when the diffusion coefficients are sufficiently regular in time, this provides an alternative to Krylov's recent proof of strong existence via analysis of the Itô-Duhamel series.

math.PR↗
arXiv2025

SDEs with critical general distributional drifts: sharp solvability and blow-ups

We establish weak well-posedness for SDEs having discontinuous diffusion coefficients and general distributional drifts that may introduce local blow up effects. Our drifts satisfy minimal assumptions, i.e.\,we assume only that the Cauchy problem for the Kolmogorov backward equation is well-posed in the standard Hilbert triple $W^{1,2} \hookrightarrow L^2 \hookrightarrow W^{-1,2}$. By a result of Mazya and Verbitsky, these assumptions are precisely those drifts that can be represented as the sum of a form-bounded component (encompassing, for example, Morrey or Chang-Wilson-Wolff drifts) and a divergence-free distributional component in the ${\rm BMO}^{-1}$ space of Koch and Tataru. We apply our results to finite particle systems with strong attracting interactions immersed in a turbulent flow. This includes particle systems of Keller-Segel type. Crucially, in dimensions $d \geq 3$, we cover almost the entire admissible range of attraction strengths, reaching nearly to the blow-up threshold. As a further application of our results for SDEs and of the theory of Bessel processes, we obtain an improved upper bound on the constant in the many-particle Hardy inequality. Consequently, the lower bound previously derived by Hoffmann-Ostenhof, Hoffmann-Ostenhof, Laptev, and Tidblom is shown to be close to optimal.

math.PR↗
arXiv2024

Feller generators with singular drifts in the critical range

We consider diffusion operator $-Δ+ b \cdot \nabla$ in $\mathbb R^d$, $d \geq 3$, with drift $b$ in a large class of locally unbounded vector fields that can have critical-order singularities. Covering the entire range of admissible magnitudes of singularities of $b$ (but excluding the borderline value), we construct a strongly continuous Feller semigroup on the space of continuous functions vanishing at infinity, thus completing a number of results on well-posedness of SDEs with singular drifts. The previous results on Feller semigroups employed strong elliptic gradient bounds and hence required the magnitude of the singularities to be less than a small dimension-dependent constant. Our approach is different and uses De Giorgi's method ran in $L^p$ for $p$ sufficiently large, hence the gain in the assumptions on singular drift. For the critical borderline value of the magnitude of singularities of $b$, we construct a strongly continuous semigroup in a ``critical'' Orlicz space on $\mathbb R^d$ whose local topology is stronger than the local topology of $L^p$ for any $2 \leq p<\infty$ but is slightly weaker than that of $L^\infty$.

math.PR↗
arXiv2024

Remarks on parabolic Kolmogorov operator

We obtain gradient estimates on solutions to parabolic Kolmogorov equation with singular drift in a large class. Such estimates allow to construct a Feller evolution family, which is used to construct unique weak solutions to the corresponding stochastic differential equation.

math.AP↗
arXiv2023

Parabolic equations and SDEs with time-inhomogeneous Morrey drift

We prove the unique weak solvability of stochastic differential equations with time-inhomogeneous drift in essentially the largest (scaling-invariant) Morrey class, i.e.\,with integrability parameter $q>1$ close to $1$. The constructed weak solutions constitute a Feller evolution family. The proofs are based on a detailed Sobolev regularity theory of the corresponding parabolic equation.

math.PR↗
arXiv2021

Stochastic equations with time-dependent singular drift

We prove unique weak solvability and Feller property for stochastic differential equations with drift in a large class of time-dependent vector fields. This class contains, in particular, the critical Ladyzhenskaya-Prodi-Serrin class, the weak $L^d$ class as well as some vector fields that are not even in $L^{2+\varepsilon}_{\rm loc}$, $\varepsilon>0$.

math.PR↗
arXiv2021

Heat kernel bounds for parabolic equations with singular (form-bounded) vector fields

We consider Kolmogorov operator $-\nabla \cdot a \cdot \nabla + b \cdot \nabla$ with measurable uniformly elliptic matrix $a$ and prove Gaussian lower and upper bounds on its heat kernel under minimal assumptions on the vector field $b$ and its divergence ${\rm div\,}b$. More precisely, we prove: (1) Gaussian lower bound, provided that ${\rm div\,}b \geq 0$, and $b$ is in the class of form-bounded vector fields (containing e.g.\,the class $L^d$, the weak $L^d$ class, as well as some vector fields that are not even in $L_{\rm loc}^{2+\varepsilon}$, $\varepsilon>0$); in these assumptions, the Gaussian upper bound is in general invalid; (2) Gaussian upper bound, provided that $b$ is form-bounded, and the positive part of ${\rm div\,}b$ is in the Kato class; in these assumptions, the Gaussian lower bound is in general invalid; (3) Gaussian upper and lower bounds, provided that $b$ is form-bounded, ${\rm div\,}b$ is in the Kato class; (4) A priori Gaussian upper and lower bounds, provided that $b$ is in a large class containing the class of form-bounded vector fields, ${\rm div\,}b$ is in the Kato class.

math.AP↗
arXiv2021

Kolmogorov operator with the vector field in Nash class

We consider divergence-form parabolic equation with measurable uniformly elliptic matrix and the vector field in a large class containing, in particular, the vector fields in $L^p$, $p>d$, as well as some vector fields that are not even in $L_{\rm loc}^{2+\varepsilon}$, $\varepsilon>0$. We establish Hölder continuity of the bounded soutions, sharp two-sided Gaussian bound on the heat kernel, Harnack inequality.

math.AP↗
arXiv2020

On admissible singular drifts of symmetric $α$-stable process

We consider the problem of existence of a (unique) weak solution to the SDE describing symmetric $α$-stable process with a locally unbounded drift $b:\mathbb R^d \rightarrow \mathbb R^d$, $d \geq 3$, $1<α<2$. In this paper, $b$ belongs to the class of weakly form-bounded vector fields. The latter arises as the class providing the $L^2$ theory of the non-local operator behind the SDE, i.e.\,$(-Δ)^{\fracα{2}} + b \cdot \nabla$, and contains as proper sub-classes the other classes of singular vector fields studied in the literature in connection with this operator, such as the Kato class, weak $L^{\frac{d}{α-1}}$ class and the Campanato-Morrey class (thus, $b$ can be so singular that it destroys the standard heat kernel estimates in terms of the heat kernel of the fractional Laplacian). We show that for such $b$ the operator $-(-Δ)^{\fracα{2}} - b \cdot \nabla$ admits a realization as a Feller generator, and that the probability measures determined by the Feller semigroup (uniquely in appropriate sense) admit description as weak solutions to the corresponding SDE. The proof is based on detailed regularity theory of $(-Δ)^{\fracα{2}} + b \cdot \nabla$ in $L^p$, $p>d-α+1$.

math.PR↗
arXiv2019

On the theory of the Kolmogorov operator in the spaces $L^p$ and $C_\infty.$ I

We obtain the basic results concerning the problem of constructing operator realizations of the formal differential expression $\nabla \cdot a \cdot \nabla - b \cdot \nabla$ with measurable matrix $a$ and vector field $b$ having critical-order singularities as the generators of Markov semigroups in $L^p$ and $C_\infty$.

math.AP↗
arXiv2019

$W^{1,p}$ regularity of solutions to Kolmogorov equation with Gilbarg-Serrin matrix

In $\mathbb R^d$, $d \geq 3$, consider the divergence and the non-divergence form operators \begin{equation} \tag{$i$} -Δ- \nabla \cdot (a-I) \cdot \nabla + b \cdot \nabla, \end{equation} \begin{equation} \tag{$ii$} - Δ- (a-I) \cdot \nabla^2 + b \cdot \nabla, \end{equation} where the second order perturbations are given by the matrix $$a-I=c|x|^{-2}x \otimes x, \quad c>-1.$$ The vector field $b:\mathbb R^d \rightarrow \mathbb R^d$ is form-bounded with the form-bound $δ>0$ (this includes a sub-critical class $[L^d + L^\infty]^d$, as well as vector fields having critical-order singularities). We characterize quantitative dependence on $c$ and $δ$ of the $L^q \rightarrow W^{1,qd/(d-2)}$ regularity of the resolvents of the operator realizations of ($i$), ($ii$) in $L^q$, $q \geq 2 \vee ( d-2)$ as (minus) generators of positivity preserving $L^\infty$ contraction $C_0$ semigroups.

math.AP↗