Damir Kinzebulatov

Professeur agrégé à l'Université Laval
Associate Professor at Laval University

Ville de Québec, Canada


Bureau VCH-1083,
pavillon Alexandre-Vachon


Étudiants et postdocs


Bienvenue sur mon site web. Je suis mathématicien à l'Université Laval. Ma recherche se concentre sur les ÉDPs et les équations stochastiques singulières. Je travaille avec des singularités d'ordre critique qui ne sont que récemment devenues accessibles grâce aux nouvelles méthodes de théorie des opérateurs et d'analyse harmonique que je développe avec mes co-auteurs et mes étudiants. On peut faire une analogie avec les particules élémentaires entrant en collision à hautes énergies dans un accélérateur de particules : ce sont les singularités les plus fortes qui révèlent la structure profonde des principaux opérateurs de la Physique Mathématique.

Welcome to my website. I am a mathematician working at Laval University. My research centers on singular PDEs and singular stochastic differential equations. I work with critical order singularities that only recently became accessible, due to the new operator-theoretic and harmonic-analytic methods that I develop with my co-authors and my students. One can draw a loose analogy with elementary particles colliding at high energies in a particle acceleratior: it is the strongest singularities that reveal the deep structure of the principal operators of Mathematical Physics.


[38] D.Kinzebulatov, Yu.A. Semenov "Feller generators with drifts in the critical range", Preprint, arXiv:2405.12332 2405.12332.pdf

We construct strongly continuous Feller semigroup associated to Kolmogorov-Fokker-Planck operator covering for the first time the entire range of strengths of critical singularities of the drift. Our main instrument is De Giorgi's method in Lp.

[37] D.Kinzebulatov "Non-local parabolic equations with singular (Morrey) time-inhomogeneous drift", Preprint, arXiv:2405.08652 2405.08652.pdf

We obtain Sobolev regularity estimates for solutions of non-local parabolic equations with locally unbounded drift satisfying some minimal assumptions. These results yield Krylov bound for the corresponding Feller stable process as well as some a priori regularity estimates on solutions of McKean-Vlasov equations. A key element of our arguments is a parabolic operator norm inequality that we prove using some ideas of Adams and Krylov.

[36] D.Kinzebulatov "On particle systems and critical strengths of general singular interactions", Preprint, arXiv:2402.17009 2402.17009.pdf

For finite interacting particle systems with strong repulsing-attracting or general interactions, we prove global weak well-posedness almost up to the critical threshold of the strengths of attracting interactions (independent of the number of particles), and establish other regularity results, such as a heat kernel bound in the regions where strongly attracting particles are close to each other. Our main analytic instruments are a variant of De Giorgi's method in L^p and an abstract desingularization theorem.

[35] D.Kinzebulatov "Laplacian with singular drift in a critical borderline case", Preprint, arXiv:2309.04436 2309.04436.pdf

We reach the critical magnitude of form-bounded drifts. The corresponding strong well-posedness theory of the parabolic diffusion equation is developed in an Orlicz space that is, basically, dictated by the drift term.

[34] D.Kinzebulatov, K.R. Madou "Strong solutions of SDEs with singular (form-bounded) drift via Roeckner-Zhao approach", Preprint, arXiv:2306.04825 2306.04825.pdf

We use the approach of Roeckner-Zhao to prove strong well-posedness for SDEs with singular drift satisfying some minimal assumptions.

SURVEY D.Kinzebulatov "Form-boundedness and SDEs with singular drift", INdAM Meeting 2022: Kolmogorov Operators and Their Applications, 2305.00146.pdf

We survey and refine recent results on weak well-posedness of SDEs with singular drift satisfying some minimal assumptions.

[32] D.Kinzebulatov, Yu.A.Semenov "Remarks on parabolic Kolmogorov operator", Preprint, arXiv:2303.03993 2303.03993.pdf

We establish gradient estimates needed to study parabolic equations and SDEs with time-inhomogeneous singular drifts.

[31] D.Kinzebulatov "Parabolic equations and SDEs with time-inhomogeneous Morrey drift", Preprint, arXiv:2301.13805 2301.13805.pdf

We prove existence and uniqueness of weak solution of SDE with singular drift in a very large class (essentially, the largest scaling-invariant Morrey class).

[30] D.Kinzebulatov, R.Vafadar "On divergence-free (form-bounded type) drifts"
, Discrete Contin. Dyn. Syst. Ser. S, 17 (2024), 2083-2107, 2209.04537.pdf

We prove a posteriori Harnack inequality for elliptic equation with divergence-free singular drift. A key step in our proofs is a new iteration procedure used in addition to the classical De Giorgi's iterations and Moser's method.

[29] D.Kinzebulatov, Yu.A.Semenov "Regularity for parabolic equations with singular non-zero divergence vector fields", J. Differential Equations, to appear  2205.05169.pdf

We prove Gaussian bounds on the heat kernel of parabolic equation with singular drift having singular divergence, taking into account possible cancellation phenomena.

[28] R. Gibara, D.Kinzebulatov "On the vanishing of Green's function, desingularization and Carleman's method", St. Petersburg Math. J. (Algebra i Analiz), 35(3) (2023) 2202.10528.pdf

We establish quantiative estimates on the order of vanishing of Green's function of Schroedinger operator under minimal assumptions on the potential (i.e. form-boundedness, which ensures that the Schroedinger operator is well defined in L2). The proofs use Carleman's method.

[27] D. Kinzebulatov, Yu.A. Semenov "Sharp solvability for singular SDEs", Electron. J. Probab., 28 (2023), article no. 69, 1–15. 2110.11232.pdf

The attracting Hardy drift provides a counterexample to weak solvability of SDEs if the coefficient of the drift is larger than a certain critical threshold. We prove a positive well-posednss result, reaching this critical threshold from below, for the entire class of form-bounded drifts.

D. Kinzebulatov, K.R. Madou, Yu.A. Semenov "On the supercritical fractional diffusion equation with Hardy-type drift", J. d'Analyse Mathématique, to appear 2112.06329.pdf (2021)

We consider fractional diffusion operator with a drift in the supercritical case, i.e. when the drift dominates the diffusion. We show that the heat kernel can vanish even if the drift is Holder continuous.

[25] D. Kinzebulatov, K.R. Madou "Stochastic equations with time-dependent singular drift", J. Differential Equations, 337 (2022), 255-293 2105.07312.pdf

We prove existence and uniqueness of weak solution to SDEs under minimal assumptions on singular drift (form-boundedness).

[24] D. Kinzebulatov, Yu.A. Semenov "Heat kernel bounds for parabolic equations with singular (form-bounded) vector fields", Math. Ann., 384 (2022), 1883-1929 2103.11482.pdf

We shed light on the effect of the positive and
negative parts of singular divergence of the drift on the Gaussian heat kernel bounds. In particular, we establish for the first time and for a large class of drifts a Gaussian lower bound in the situation where there is no Gaussian upper bound.

D. Kinzebulatov, Yu.A. Semenov and R. Song "Stochastic transport equation with singular drift", Ann. Inst. Henri Poincaré (B) Probab. Stat., to appear 2102.10610.pdf

We establish well-posedness of stochastic transport and continuity equations with singular drift in a large class, which allows to address the problem of strong well-posedness of the corresponding SDEs.

D. Kinzebulatov, Yu.A. Semenov "Kolmogorov operator with the vector field in Nash class", Tohoku Math. J., 74(4) (2022), 569-596 2012.02843.pdf

We develop solution theory, and establish a posteriori Gaussian heat kernel bounds, for divergence-form parabolic equation under minimal assumptions on the drift, such that there is no weak solution theory in L2, but there is nevertheless a strong solution theory in L1. The proofs use Nash's method.

[21] D. Kinzebulatov, Yu.A. Semenov "Fractional Kolmogorov operator and desingularizing weights", Publ. Res. Inst. Math., Kyoto, to appear 2005.11199.pdf 

This is a continuation of [19]. We establish sharp upper and lower bounds on the heat kernel of the fractional Laplace operator perturbed by repulsing Hardy-type drift by transferring it to appropriate weighted space with singular weight.

[20] D. Kinzebulatov, K.R. Madou "On admissible singular drifts of symmetric alpha-stable process", Math. Nachr., 295(10) (2022), 2036-2064 2002.07001.pdf

We prove weak well-posedness of SDE driven by stable process under minimal assumptions on singular drift

[19] D. Kinzebulatov Yu.A. Semenov and K. Szczypkowski "Heat kernel of fractional Laplacian with Hardy drift via desingularizing weights", J. London Math. Soc., 104 (2021), p. 1861-1900 1904.07363.pdf

We establish sharp two-sided bounds on the heat kernel of the fractional Laplacian, perturbed by the attracting Hardy-type drift having critical-order singularity, using the method of desingularizing weights. The proofs use some ideas of J.Nash.

[18] D. Kinzebulatov, Yu.A. Semenov "Feller generators and stochastic differential equations with singular (form-bounded) drift", Osaka J. Math., 58 (2021), 855–883 feller_sdes.pdf   Preprints 1803.06033.pdf 1904.01268.pdf

We establish weak-posedness of Ito and Stratonovich SDEs under minimal assumptions on singular drift and discontinuous diffusion coefficients.

D. Kinzebulatov "Regularity theory of Kolmogorov operator revisited", Canadian Bull. Math. 64 (2021), p. 725-736 afb.pdf Preprint 1807.07597.pdf

We develop a new approach to regularity theory of diffusion operator with singular drift, which allows to establish weak well-posedness of the corresponding SDE.

[16] D. Kinzebulatov, Yu.A. Semenov "Regularity of solutions to Kolmogorov equation with Gilbarg-Serrin matrix", J. Evol. Equations, 22 (# 21) (2022) 1802.05167.pdf

We consider divergence-form elliptic equations having critical discontinuity in the matrix, and establish a quantiative dependence between this discontinuity and the regularity theory of these equations.

[15] D. Kinzebulatov, Yu.A. Semenov "Brownian motion with general drift", Stoch. Proc. Appl. 130 (2020), p. 2737-2750 (arXiv:1710.06729 1710.06729.pdf)

This is the first paper on weak well-posedness of SDE with singular drift that went beyond the popular Ld class of drifts.

[14] D. Kinzebulatov, Yu.A. Semenov "On the theory of the Kolmogorov operator in the spaces $L^p$ and $C_\infty$", Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 21 (2020), 1573-1647 (arXiv: 1709.08598.pdf)

This is a detailed regularity theory of divergence-form elliptic and parabolic operators with singular vector field and discontinuous diffusion matrix.

[13] D. Kinzebulatov "Feller generators with measure-valued drifts", Potential Anal., 48 (2018), p. 207-222 measure_perturb.pdf

We construct Feller process under very broad assumptions on singular measure-valued drift, using an interpla between two resolvent representations (one not allowing measure-valued drifts, but providing strong Sobolev regularity theory, and the other one providing minimal theory but allowing to handle measure-valued drifts).

[12] D. Kinzebulatov "A new approach to the L^p-theory of -\Delta + b\grad, and its applications to Feller processes with general drifts", Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 17 (2017), p. 685-711 diff_singular.pdf

Tl;dr: This is a major advancement in the regularity theory of diffusion operator with singular drift, which allowed to go beyond the standard classes of singular drifts (such as Kato class or form-bounded drifts).

[11] D. Kinzebulatov "Feller evolution families and parabolic equations with form-bounded vector fields"Osaka J. Math., 54 (2017), p. 499-516 feller.pdf

We construct Feller evolution family for parabolic equation with singular drift in a large class. The key element of the proof is new gradient bounds and the parabolic variant of the iteration procedure of Kovalenko-Semenov.

I. Binder, D. Kinzebulatov and M.Voda "Non-perturbative localization with quasiperiodic potential in continuous time", Comm. Math. Phys., 351 (2017), p. 1149-1175 s00220-016-2723-7.pdf

We establish Anderson localization in continuous one-dimensional multifrequency Schrödinger operators. This is a cointinuous analogue of the result of Bourgain-Goldstein.

[9] A.Brudnyi, D. Kinzebulatov "Kohn decomposition for forms on coverings of complex manifolds constrained along the fibres", Trans. Amer. Math. Soc., 369 (2017), p. 167-186 kohn.pdf

We extend J.J. Kohn's Hodge-type decomposition to the (p,q) Dolbeault cohomology groups of spaces of differential forms taking values in certain (possibly infinite-dimensional) holomorphic Banach vector bundles.

[8] A.Brudnyi, D. Kinzebulatov "Towards Oka-Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds I", Revista Mat. Iberoamericana, 31(4) (2015), p. 1167-1230 bru_kinz_1.pdf

[7] A.Brudnyi, D. Kinzebulatov "Towards Oka-Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds II", Revista Mat. Iberoamericana, 31(4) (2015), p. 989-1032 bru_kinz_2.pdf

We develop complex function theory for some algebras of holomorphic functions on coverings of Stein manifolds, including holomorphic almost periodic functions.

[6] A.Brudnyi, D. Kinzebulatov "Holomorphic almost periodic functions on coverings of complex manifolds", New York J. Math, 17a (2011), p. 267-300 nyjm.pdf

We establish some basic results of complex analytic geometry on the maximal ideal spaces of some algebras of holomorphic functions.

 [5] D. Kinzebulatov, L. Shartser "Unique continuation for Schroedinger operators. Towards an optimal result", J.Funct. Anal., 258 (2010), p. 2662-2681 UC_jfa.pdf  

This is the strongest to date result on unique continuation for eigenfunctions of Schroedinger operators,
strengthening the classical results of Stein, Jerison-Koenig, Chanillo-Sawyer.

[4] A.Brudnyi, D. Kinzebulatov "Holomorphic semi-almost periodic functions", Integr. Equ. Operat. Theory, 66 (2010), p.293-325 0911.0954v1.pdf

D. Kinzebulatov "On algebras of holomorphic functons with semi-almost periodic boundary values", C. R. Math. Rep. Acad. Sci. Canada, 32 (2010), p.1-12

D. Kinzebulatov "On uniform subalgebras of L^\infty on the unit circle generated by almost periodic functions", St. Petersburg Math. J., 19 (2008), p.495-518 BK - SPbJMath.pdf

Tl;dr: How
does fixing the type of discontinuity of boundary values of a bounded holomorphic function F on the unit disk D affects the properties of F inside of D? Lindelof theorem states that the boundary values f=F|_{\partial D} of F can not have first-kind discontinuities. However, there are functions \Phi such that \Phi(F)|_{\partial D} can be first-kind disconinuous. It turns out that in the case \Phi(z)=|z| there is a connection between bounded holomorphic functions on D whose moduli can have only first-kind boundary discontinuities and Sarason's semi-almost periodic functions on \partial D (a priori, there is nothing almost periodic about bounded holomorphic functions and first-kind discontinuities!).
This link allows us to apply, in our study, the results on almost periodic functions. In particular, we establish
for this algebra of bounded of holomorphic functions:
 - Grothendieck's approximation property (still a conjecture for H^\infty(D)).
 - Corona theorem.
 - Results on completion of matrices with entries in this algebra,
Our methods admit extension to subalgebras determined by other functions \Phi.

[1] D. Kinzebulatov "A note of Gagliardo-Nirenberg type inequalitities on analytic sets", C. R. Math. Rep. Acad. Sci. Canada, 30 (2009), p.97-105. kinzebulatovC351.pdf

We study a new local invariant of 
singularities of complex analytic sets. This invariant arises as the 'correcting exponent' in a family of Sobolev-type inequalities relating norms of functions on these sets.

Algorithmic trading (2012-2014):

A. Cartea, S. Jaimungal and D. Kinzebulatov "Algorithmic trading with learning", Int. J. Applied and Theoretical Finance, 19, no.4, 1650028 (2016) ( http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2373196)

We propose a model where an algorithmic trader takes a view on the distribution of prices at a 
future date and then decides how to trade in the direction of her predictions using the optimal mix of market and limit orders. Namely, we model the asset midquote price as a randomized Brownian bridge S_t=S_0+\sigma\beta_{tT}+ tD/T 
where D is a random variable that encodes the trader's prior belief on the asset's future price distribution, the noise term \beta_{tT} is a standard Brownian bridge over time interval independent of D. As time flows, the trader learns the realized value of D.

- S.Jaimungal, D. Kinzebulatov and D.Rubisov "Optimal accelerated share repurchase", Applied. Math. Finance, 24, no.3 (2017) (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2360394)

An accelerated share repurchase (ASR) allows a fiirm to repurchase a significant portion of its shares immediately, while shifting the burden of reducing the impact and uncertainty in the trade to an intermediary. The intermediary must then purchase the shares from the market over several days, weeks, or as much as several months. In this work, we address the intermediary's optimal execution and exit strategy taking into account the impact that trading has on the market. We demonstrate that it is optimal to exercise when the TWAP exceeds \zeta(t)S_t, where S_t is the fundamental price of the asset and \zeta(t) is deterministic. Moreover, we develop a dimensional reduction of the Stochastic Control and Stopping problem and implement an efficient numerical scheme to compute the optimal trading and exit strategies.

- S. Jaimungal,
D. Kinzebulatov "Optimal execution with a price limiter", RISK, July 2014, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2199889

Earlier research while studying for B.Sc and M.Sc:

- D. Kinzebulatov  "Systems with distributions and viability theorem", J. Math. Anal. Appl., 331 (2007), p. 1046-1067 kinz_viab.pdf

- V.Derr,
D. Kinzebulatov "Dynamical generalized functions and the multiplication problem", Russian Math., 51 (2007), p.32-43 0603351.pdf

We study some ordinary differential equations arising in singular optimal control problems.

- E. Braverman,
D. Kinzebulatov "Nicholson blowfiles equation with a distributed delay", Canadian Appl. Math. Q., 14  (2006), p. 107-128 brav_kinz.pdf


Maximal ideal space
The topological structure of the maximal ideal space of the algebra of
"semi-almost periodic" holomorphic functions in some papers with A. Brudnyi



I am on the Editorial Board of Annales Mathématiques du Québec

Associate Professor:
 - Université Laval, Québec (May 2022 - present)

Assistant Professor:
 - Université Laval, Québec (Jan 2017 - May 2022)

Visiting Assistant Professor:
 - Indiana University, Bloomington, USA (Jan - Dec 2016)

Postdoctoral Fellow:
 - McGill University and the CRM, Montréal (Sep - Dec 2015) (with I. Polterovich, D. Jakobson, J.Toth)
 - The Fields Institute (NSERC), U of Toronto (June 2012- Aug 2015) (with S. Jaimungal, E. Bierstone, I. Binder)
 - University of Calgary (Jan - May 2012) (with A. Brudnyi)

Ph.D, University of Toronto, 2012 (with P. Milman)
M.Sc, University of Calgary, 2006 (with E.Braverman)
B.Sc, Izhevsk State Technical University, Russia, 2004 (with V. Derr)