
Damir
Kinzebulatov
Professeur agrégé à l'Université Laval Associate Professor at Laval University Ville de Québec, Canada damir.kinzebulatov@mat.ulaval.ca Bureau VCH1083, pavillon AlexandreVachon 


[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 KolmogorovFokkerPlanck 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 "Nonlocal parabolic equations with singular (Morrey) timeinhomogeneous drift", Preprint, arXiv:2405.08652 2405.08652.pdf We obtain Sobolev regularity estimates for solutions of nonlocal 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 McKeanVlasov 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 repulsingattracting or general interactions, we prove global weak wellposedness 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 formbounded drifts. The corresponding strong wellposedness 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 (formbounded) drift via RoecknerZhao approach", Preprint, arXiv:2306.04825 2306.04825.pdf We use the approach of RoecknerZhao to prove strong wellposedness for SDEs with singular drift satisfying some minimal assumptions. [33] SURVEY D.Kinzebulatov "Formboundedness and SDEs with singular drift", INdAM Meeting 2022: Kolmogorov Operators and Their Applications, 2305.00146.pdf We survey and refine recent results on weak wellposedness 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 timeinhomogeneous singular drifts. [31] D.Kinzebulatov "Parabolic equations and SDEs with timeinhomogeneous 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 scalinginvariant Morrey class). [30] D.Kinzebulatov, R.Vafadar "On divergencefree (formbounded type) drifts", Discrete Contin. Dyn. Syst. Ser. S, 17 (2024), 20832107, 2209.04537.pdf We prove a posteriori Harnack inequality for elliptic equation with divergencefree 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 nonzero 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. formboundedness, 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 wellposednss result, reaching this critical threshold from below, for the entire class of formbounded drifts. [26] D. Kinzebulatov, K.R. Madou, Yu.A. Semenov "On the supercritical fractional diffusion equation with Hardytype 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 timedependent singular drift", J. Differential Equations, 337 (2022), 255293 2105.07312.pdf We prove existence and uniqueness of weak solution to SDEs under minimal assumptions on singular drift (formboundedness). [24] D. Kinzebulatov, Yu.A. Semenov "Heat kernel bounds for parabolic equations with singular (formbounded) vector fields", Math. Ann., 384 (2022), 18831929 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. [23] 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 wellposedness of stochastic transport and continuity equations with singular drift in a large class, which allows to address the problem of strong wellposedness of the corresponding SDEs. [22] D. Kinzebulatov, Yu.A. Semenov "Kolmogorov operator with the vector field in Nash class", Tohoku Math. J., 74(4) (2022), 569596 2012.02843.pdf We develop solution theory, and establish a posteriori Gaussian heat kernel bounds, for divergenceform 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 Hardytype drift by transferring it to appropriate weighted space with singular weight. [20] D. Kinzebulatov, K.R. Madou "On admissible singular drifts of symmetric alphastable process", Math. Nachr., 295(10) (2022), 20362064 2002.07001.pdf We prove weak wellposedness 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. 18611900 1904.07363.pdf We establish sharp twosided bounds on the heat kernel of the fractional Laplacian, perturbed by the attracting Hardytype drift having criticalorder 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 (formbounded) drift", Osaka J. Math., 58 (2021), 855–883 feller_sdes.pdf Preprints 1803.06033.pdf 1904.01268.pdf We establish weakposedness of Ito and Stratonovich SDEs under minimal assumptions on singular drift and discontinuous diffusion coefficients. [17] D. Kinzebulatov "Regularity theory of Kolmogorov operator revisited", Canadian Bull. Math. 64 (2021), p. 725736 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 wellposedness of the corresponding SDE. [16] D. Kinzebulatov, Yu.A. Semenov "Regularity of solutions to Kolmogorov equation with GilbargSerrin matrix", J. Evol. Equations, 22 (# 21) (2022) 1802.05167.pdf We consider divergenceform 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. 27372750 (arXiv:1710.06729 1710.06729.pdf) This is the first paper on weak wellposedness 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), 15731647
(arXiv: 1709.08598.pdf) This is a detailed regularity theory of divergenceform elliptic and parabolic operators with singular vector field and discontinuous diffusion matrix. 

[13] D.
Kinzebulatov "Feller
generators with measurevalued drifts", Potential Anal.,
48 (2018), p. 207222 measure_perturb.pdf We construct Feller process under very broad assumptions on singular measurevalued drift, using an interpla between two resolvent representations (one not allowing measurevalued drifts, but providing strong Sobolev regularity theory, and the other one providing minimal theory but allowing to handle measurevalued drifts). 

[12]
D.
Kinzebulatov "A new approach to the
L^ptheory of \Delta + b\grad,
and its applications to Feller
processes with general drifts", Ann. Sc. Norm. Super.
Pisa Cl. Sci. (5), 17
(2017), p. 685711 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 formbounded drifts). 

[11] D.
Kinzebulatov "Feller evolution families and
parabolic equations with formbounded vector fields", Osaka J. Math., 54 (2017),
p. 499516 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 KovalenkoSemenov. [10] I. Binder, D. Kinzebulatov and M.Voda "Nonperturbative localization with quasiperiodic potential in continuous time", Comm. Math. Phys., 351 (2017), p. 11491175 s0022001627237.pdf We establish Anderson localization in continuous onedimensional multifrequency Schrödinger operators. This is a cointinuous analogue of the result of BourgainGoldstein. 

[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. 167186 kohn.pdf We extend J.J. Kohn's Hodgetype decomposition to the (p,q) Dolbeault cohomology groups of spaces of differential forms taking values in certain (possibly infinitedimensional) holomorphic Banach vector bundles. 

[8] A.Brudnyi, D. Kinzebulatov "Towards OkaCartan theory for algebras of holomorphic functions on coverings of Stein manifolds I", Revista Mat. Iberoamericana, 31(4) (2015), p. 11671230 bru_kinz_1.pdf [7] A.Brudnyi, D. Kinzebulatov "Towards OkaCartan theory for algebras of holomorphic functions on coverings of Stein manifolds II", Revista Mat. Iberoamericana, 31(4) (2015), p. 9891032 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. 267300
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. 26622681 UC_jfa.pdf This is the strongest to date result on unique continuation for eigenfunctions of Schroedinger operators, strengthening the classical results of Stein, JerisonKoenig, ChanilloSawyer. 

[4] A.Brudnyi, D.
Kinzebulatov
"Holomorphic
semialmost periodic functions",
Integr.
Equ. Operat. Theory, 66
(2010), p.293325 0911.0954v1.pdf [3] A.Brudnyi, D. Kinzebulatov "On algebras of holomorphic functons with semialmost periodic boundary values", C. R. Math. Rep. Acad. Sci. Canada, 32 (2010), p.112 [2] A.Brudnyi, D. Kinzebulatov "On uniform subalgebras of L^\infty on the unit circle generated by almost periodic functions", St. Petersburg Math. J., 19 (2008), p.495518 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 firstkind discontinuities. However, there are functions \Phi such that \Phi(F)_{\partial D} can be firstkind 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 firstkind boundary discontinuities and Sarason's semialmost periodic functions on \partial D (a priori, there is nothing almost periodic about bounded holomorphic functions and firstkind 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, etc. Our methods admit extension to subalgebras determined by other functions \Phi. 

[1] D.
Kinzebulatov "A
note of
GagliardoNirenberg
type
inequalitities
on analytic
sets", C. R. Math. Rep.
Acad. Sci. Canada, 30 (2009), p.97105. 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 Sobolevtype inequalities relating norms of functions on these sets. 

Algorithmic trading (20122014):  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. 10461067 kinz_viab.pdf  V.Derr, D. Kinzebulatov "Dynamical generalized functions and the multiplication problem", Russian Math., 51 (2007), p.3243 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. 107128 brav_kinz.pdf 

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

CV 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) 