Damir Kinzebulatov

Professeur adjoint à l'Université Laval
Assistant Professor at Laval University
Québec, Canada


Ph.D, University of Toronto, 2012   CV

Étudiant(e)s au doctorat:

Thierry Kouontchou Tchemb (avec J. Mashreghi)
Xiaoting Li (avec N. Doyon)

Aux étudiant(e)s potentiels / For prospective students



MAT-7225 Équations aux dérivées partielles (automne 2018)  
MAT-3600 Projet de fin d'études (hiver 2019)


"Northeastern Analysis Meeting", SUNY New Paltz, USA (October 2018)
"Semigroups of Operators: Theory and Applications", Kazimierz Dolny, Poland (October 2018)
"Stochastic Analysis and non-local PDEs", Joint Meeting of the Italian and the Polish Mathematical Societies, Wroclaw, Poland (September 2018)
CMS Summer Meeting "Advances in Harmonic Analysis and PDEs", Fredericton, Canada (June 2018)
"Complex Analysis and Spectal Theory. A conference in celebration of Thomas Ransford's 60th birthday", Quebec, Canada (May 2018)
"Qualitative theory of PDEs" AMS Sectional Meeting, Boston, USA (April 2018)

Congress of the Americas, Montréal, Canada (July 2017)
AMS Sectional Meeting, New York, USA (May 2017)


Analysis and PDEs


[18] "\mathcal W^{\alpha, p} and C^{0,\gamma} regularity of solutions to (\mu - \Delta + b \cdot \nabla)u=f with form-bounded vector fields", Preprint, arXiv:1807.07597 1807.07597.pdf

We consider the operator $-\Delta +b \cdot \nabla$ with $b:\mathbb R^d \rightarrow \mathbb R^d$ ($d \geq 3$) in the class of form-bounded vector fields (containing vector fields having critical-order singularities), and characterize quantitative dependence of the $\mathcal W^{1+\frac{2}{q},p}$ ($2 \leq p < q$) and the $C^{0,\gamma}$ regularity of solutions to the corresponding elliptic equation in $L^p$ on the value of the form-bound of $b$.

[17] (with Yu.A. Semenov) "W^{1,p} regularity of solutions to Kolmogorov equation and associated Feller semigroup", Preprint, arXiv:1803.06033 1803.06033.pdf

 In $\mathbb R^d$, $d \geq 3$, consider the divergence and the non-divergence form operators
- \nabla \cdot a \cdot \nabla + b \cdot \nabla    (i),
- a \cdot \nabla^2 + b \cdot \nabla    (ii),
where $a=I+c \mathsf{f} \otimes \mathsf{f}$, the vector fields $\nabla_i \mathsf{f}$ ($i=1,2,\dots,d$) and $b$ are form-bounded (this includes the sub-critical class $[L^d + L^\infty]^d$ as well as vector fields having critical-order singularities, e.g. in the weak $L^d$, the Campanato-Morrey or the Chang-Wilson-Wolff classes). We characterize quantitative dependence on $c$ and the values of the form-bounds 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. The latter allows to run an iteration procedure $L^p \rightarrow L^\infty$ that yields associated with (i), (ii) $L^q$-strong Feller semigroups.

[16] (with Yu.A. Semenov) "W^{1,p} regularity of solutions to Kolmogorov equation with Gilbarg-Serrin matrix", Preprint, arXiv:1802.05167 1802.05167.pdf

In $\mathbb R^d$, $d \geq 3$, consider the divergence and the non-divergence form operators 
-\Delta - \nabla \cdot (a-I) \cdot \nabla + b \cdot \nabla, (i)
- \Delta - (a-I) \cdot \nabla^2 + b \cdot \nabla, (ii)
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 $\delta>0$ (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 $\delta$ 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.

[15] (with Yu.A. Semenov) "Brownian motion with general drift", Preprint, arXiv:1710.06729 1710.06729.pdf

We construct and study the weak solution to stochastic differential equation $dX(t)=-b(X(t))dt+\sqrt{2}dW(t)$, $X_0=x$, for every $x \in \mathbb R^d$, $d \geq 3$, with $b$ in the class of weakly form-bounded vector fields, containing, as proper subclasses, a sub-critical class $[L^d+L^\infty]^d$, as well as critical classes such as weak $L^d$ class, Kato class, Campanato-Morrey class, Chang-Wilson-T.Wolff class.

[14] (with Yu.A. Semenov) "On the theory of the Kolmogorov operator in the spaces $L^p$ and $C_\infty.$ I", Preprint, arXiv:1709.08598 1709.08598.pdf

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 $C_\infty$.

[13] (with I.Binder 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 consider continuous one-dimensional multifrequency Schrödinger operators, with analytic potential, and prove Anderson localization in the regime of positive Lyapunov exponent for almost all phases and almost all Diophantine frequencies.

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

We construct a $L^p$-strong Feller process associated with the formal differential operator $-\Delta + \sigma \cdot \nabla$ on $\mathbb R^d$, $d \geqslant 3$, with drift $\sigma$ in a wide class of measures (e.g. the sum of a measure having density in weak $L^d$ space and a Kato class  measure), by exploiting a quantitative dependence of the smoothness of the domain  of an operator realization of $-\Delta + \sigma \cdot \nabla$ generating a holomorphic $C_0$-semigroup on $L^p(\mathbb R^d)$, $p>d-1$, on the value of the relative bound of $\sigma$

[11] "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

We develop a detailed regularity theory of $-\Delta +b \cdot \nabla$ in $L^p(\mathbb R^d)$, for  a wide class of vector fields. The $L^p$-theory allows us to construct associated strong Feller process in $C_\infty(\mathbb R^d)$.
Our starting object is an operator-valued function, which, we prove, determines the resolvent of an operator realization of $-\Delta + b\cdot \nabla$, the generator of a holomorphic $C_0$-semigroup on $L^p(\mathbb R^d)$. Then the very form of the operator-valued function yields crucial information about smoothness of the domain of the generator.

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

We show that the weak solutions of parabolic equation $\partial_t u - \Delta u + b(t,x) \cdot \nabla u=0$, $(t,x) \in (0,\infty) \times \mathbb R^d$, $d \geqslant 3$, for drift $b(t,x)$ in a wide class of time-dependent vector fields capturing critical singularities, constitute a Feller evolution family and, thus, determine a Feller process. The proof uses a Moser-type iterative procedure and an a priori estimate on the $L^p$-norm of the gradient of solution in terms of the $L^q$-norm of the gradient of initial function.

[9] (with A.Brudnyi) "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 on $D$. We apply this result to compute the $(p,q)$ Dolbeault cohomology groups of some regular coverings of $D$ defined by means of $C^\infty$ forms constrained along fibres of the coverings.

[8] (with A.Brudnyi) "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

(with A.Brudnyi) "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 a Complex Function Theory within some Frechet algebras of holomorphic functions on coverings of Stein manifolds, including Bohr's holomorphic almost periodic functions, holomorphic functions bounded along the fibres (arising e.g. in study of corona problem for H^\infty), etc. Our method is based on an extension of Oka-Cartan theory to coherent-type sheaves on the maximal ideal spaces of these algebras, topological spaces having some important features of complex analytic manifolds.

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

We develop the basic elements of Complex Analytic Geometry on the maximal ideal spaces of some Frechet algebras of holomorphic functions.

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

We prove the property of unique continuation for solutions of differential inequality |\Delta u| \leq |Vu|, with potential V belonging to a local analogue of a class for which
Schroedinger operator -\Delta+V is well defined in the sense of form-sum. We apply our result to the problem of absence of positive eigenvalues for self-adjoint Schroedinger operators with form-bounded potentials V, i.e. potentials that admit critical singularities. Our result stengthens the classical results by E. Stein and D. Jerison-C. Kenig.

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

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

[2] (with A.Brudnyi) "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

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] "A note on Gagliardo-Nirenberg type inequalities 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 a 'correcting exponent' in a family of Sobolev-type inequalities relating norms of functions on these sets.

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

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

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

We study some qualitative properties of ordinary differential equations arising in singular Optimal Control problems. To carry out this study, we had to introduce a new space of measures (distributions) with a continuous operation of multiplication by first-kind discontinuous functions.

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

We study properties of some cassical model of Population Dynamics.

Algorithmic and HF trading:

[1] (with A.Cartea, S.Jaimungal) "Algorithmic trading with learning", Preprint, http://ssrn.com/abstract=2373196 (2013), 20 p. 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.

[2] (with S.Jaimungal, D.Rubisov) "Optimal accelerated share repurchase"Preprint, SSRN eLibrary, http://ssrn.com/abstract=2360394 (2013), 28 p. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2360394

An accelerated share repurchase (ASR) allows a fi rm to repurchase a signi cant 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.

[3] (with S. Jaimungal) "Optimal execution with a price limiter", RISK, July 2014
, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2199889


Maximal ideal space
The topological structure of a maximal ideal space in [2-4]

Brief CV

Membre du CRM, du groupe de recherche en analyse à l'UL et de CIMMUL
Visiting Assistant Professor:
 - Indiana University, Bloomington, USA (2016)

Postdoctoral Fellow:
 - The Fields Institute (NSERC), U of Toronto (2012-2015)

Short visits:
 - McGill University and the CRM (Fall 2015)
 - University of Calgary (Spring 2012)

Ph.D, University of Toronto, 2012 (with Pierre Milman)

Detailed CV pdf