
Damir Kinzebulatov Professeur adjoint à l'Université Laval Assistant Professor at Laval University Québec, Canada damir.kinzebulatov@mat.ulaval.ca 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 


Enseignement: MAT7225 Équations aux dérivées partielles (automne 2018) ... MAT3600 Projet de fin d'études (hiver 2019) 

Conférences:
2018 "Northeastern Analysis Meeting", SUNY New Paltz, USA (October 2018) "Semigroups of Operators: Theory and Applications", Kazimierz Dolny, Poland (October 2018) "Stochastic Analysis and nonlocal 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) 2017 Congress of the Americas, Montréal, Canada (July 2017) AMS Sectional Meeting, New York, USA (May 2017) Publications: Analysis and PDEs 



[18] "\mathcal W^{\alpha,
p} and C^{0,\gamma} regularity of solutions to (\mu 
\Delta + b \cdot \nabla)u=f with formbounded 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 formbounded vector fields (containing vector fields having criticalorder 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 formbound 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 nondivergence 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 formbounded (this includes the subcritical class $[L^d + L^\infty]^d$ as well as vector fields having criticalorder singularities, e.g. in the weak $L^d$, the CampanatoMorrey or the ChangWilsonWolff classes). We characterize quantitative dependence on $c$ and the values of the formbounds of the $L^q \rightarrow W^{1,qd/(d2)}$ regularity of the resolvents of the operator realizations of (i), (ii) in $L^q$, $q \geq 2 \vee ( d2)$ 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 GilbargSerrin matrix", Preprint, arXiv:1802.05167 1802.05167.pdf In $\mathbb R^d$, $d \geq 3$, consider the divergence and the nondivergence form operators \Delta  \nabla \cdot (aI) \cdot \nabla + b \cdot \nabla, (i)  \Delta  (aI) \cdot \nabla^2 + b \cdot \nabla, (ii) where the second order perturbations are given by the matrix $$aI=cx^{2}x \otimes x, \quad c>1.$$ The vector field $b:\mathbb R^d \rightarrow \mathbb R^d$ is formbounded with the formbound $\delta>0$ (this includes a subcritical class $[L^d + L^\infty]^d$, as well as vector fields having criticalorder singularities). We characterize quantitative dependence on $c$ and $\delta$ of the $L^q \rightarrow W^{1,qd/(d2)}$ regularity of the resolvents of the operator realizations of (i), (ii) in $L^q$, $q \geq 2 \vee ( d2)$ 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 formbounded vector fields, containing, as proper subclasses, a subcritical class $[L^d+L^\infty]^d$, as well as critical classes such as weak $L^d$ class, Kato class, CampanatoMorrey class, ChangWilsonT.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 criticalorder singularities as the generators of Markov semigroups in $L^p$ and $C_\infty$. 

[13] (with I.Binder and M.Voda) "Nonperturbative localization with quasiperiodic potential in continuous time", Comm. Math. Phys., 351 (2017), p. 11491175 s0022001627237.pdf We consider continuous onedimensional 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 measurevalued drifts", Potential Anal., 48 (2018), p. 207222 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 $L^p(\mathbb R^d)$, $p>d1$, on the value of the relative bound of $\sigma$ . 

[11] "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 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 operatorvalued 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 operatorvalued function yields crucial information about smoothness of the domain of the generator. 

[10] "Feller evolution families and parabolic equations with formbounded vector fields", Osaka J. Math., 54 (2017), p. 499516 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 timedependent vector fields capturing critical singularities, constitute a Feller evolution family and, thus, determine a Feller process. The proof uses a Mosertype 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. 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 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 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] (with A.Brudnyi) "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 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 OkaCartan theory to coherenttype 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. 267300 nyjm.pdf We develop the basic elements of Complex Analytic Geometry on the maximal ideal spaces of some Frechet algebras of holomorphic functions. 

[5] (with L. Shartser) "Unique continuation for Schroedinger operators. Towards an optimal result", J.Funct. Anal., 258 (2010), p. 26622681 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 formsum. We apply our result to the problem of absence of positive eigenvalues for selfadjoint Schroedinger operators with formbounded potentials V, i.e. potentials that admit critical singularities. Our result stengthens the classical results by E. Stein and D. JerisonC. Kenig. 

[4] (with A.Brudnyi) "Holomorphic semialmost periodic functions", Integr. Equ. Operat. Theory, 66 (2010), p.293325 0911.0954v1.pdf [3] (with A.Brudnyi) "On algebras of holomorphic functons with semialmost periodic boundary values", C. R. Math. Rep. Acad. Sci. Canada, 32 (2010), p.112. [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.495518 BK  SPbJMath.pdf 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] "A note on GagliardoNirenberg type inequalities 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 a 'correcting exponent' in a family of Sobolevtype 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. 10461067 kinz_viab.pdf [2] (with V.Derr) "Dynamical generalized functions and the multiplication problem", Russian Math., 51 (2007), p.3243 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 firstkind discontinuous functions. [3] (with E. Braverman) "Nicholson blowfiles equation with a distributed delay", Canadian Appl. Math. Q., 14 (2006), p. 107128 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 firm to repurchase a signicant 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 

The topological structure of a maximal ideal space in [24] 

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 (20122015) 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 