Mateusz Kwaśnicki
Department of Pure Mathematics
Wrocław University of Science and Technology
ul. Wybrzeże Wyspiańskiego 27
50370 Wrocław, Poland
email:
skype: mateusz.kwasnicki
ORCID: 0000000338968124
Research articles
For an author's copy of any of the following papers, please send me an email at .
Preprint articles
 Oscillatory attraction and repulsion from a subset of the unit sphere or hyperplane for isotropic stable Lévy processes with Andreas E. Kyprianou, Sandra Palau and Tsogzolmaa Saizmaa [arXiv]
 Boundary traces of shiftinvariant diffusions in halfplane [arXiv]
 Characterisation of the class of bellshaped functions with Thomas Simon [arXiv]
 Harmonic extension technique for nonsymmetric operators with completely monotone kernels [arXiv]
Publications
 Random walks are determined by their trace on the positive halfline, Ann. Henri Lebesgue 3 (2020): 1389–1397 [onlineMRzb]
 A new class of bellshaped functions, Trans. Amer. Math. Soc. 373(4) (2020): 2255–2280 [onlineMRzb]
 Minimal Hermitetype eigenbasis of the discrete Fourier transform with Alexey Kuznetsov, J. Fourier Anal. Appl. 25(3) (2019): 1053–1079 [onlineMRzb]

Fluctuation theory for Lévy processes with completely monotone jumps,
Electron. J. Probab. 24 (2019), no. 40
[onlineMRzb]
Notes: In the last lines of equations (5.8) and (5.10), $f^(i \xi_2)$ should read $f^(i \xi_2)$. In the expression for $I$ in line 7 on page 26, $ds$ is missing.
 In the displayed equation in line 14 on page 31, $\lambda(r)$ should read $\lambda_f(r)$.
 The assumption that $f(\xi)$ is nonconstant is missing in Lemma 5.4, Theorem 5.5 and Corollary 5.6. In Theorem 5.7 and Lemma 6.1, $f(\xi)$ should be assumed to be nonconstant rather than nonzero. Finally, in the proof of Theorem 1.1 the trivial case of constant Rogers functions should be treated separately.
 A factor $\pi$ is missing in the lefthand side of equation in line 14 on page 16: should be $\tfrac{\pi f'(\xi)}{f(\xi)}$.
 On page 29, line 22, formula (5.18) (rather than (5.17)) should be referenced.
(supersedes: Rogers functions and fluctuation theory [arXiv])  On the $\ell^p$ norm of the discrete Hilbert transform with Rodrigo Bañuelos, Duke Math. J. 168(3) (2019): 471–504 [onlineMRzb]
 Pólya’s conjecture fails for the fractional Laplacian with Richard S. Laugesen and Bartłomiej Siudeja, J. Spectral Theory 9(1) 2019: 127–135 [onlineMRzb]
 Fractional Laplace Operator and its Properties, in: A. Kochubei, Y. Luchko, Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory, De Gruyter Reference, De Gruyter, Berlin, 2019 [onlineMRgoogle]

Extension technique for complete Bernstein functions of the Laplace operator
with Jacek Mucha,
J. Evol. Equ. 18(3) (2018): 1341–1379
[onlineMRzb]
Notes: In Theorem A.2, when the string $A(ds)$ contains an atom of mass $b$ at $0$, the expression for $L f$ describes the operator $L = \psi(\Delta)  b \Delta$, not $L = \psi(\Delta)$. In order to get the latter one, it is necessary to artificially add $b \Delta$ in the expressions for $L f$ (and assume that $f$ is in the domain of $\Delta$).
 In the last display on page 1376, as well as in the second line on page 1377, $(\psi(\lambda))^{1}$ should read $\psi(\lambda)$; the correct definition is $\varphi_\lambda(s) = f_N(s)  \psi(\lambda) f_D(s)$.
 In the penultimate line on page 1376, $\varphi_\lambda(0) = 0$ should read $\varphi_\lambda(0) = 1$. Thanks to Sigurd Assing for pointing out the above errors!
 Integrability of the Fourier transform of $f_n$ in the third paragraph of Section 7 does not seem to be known. A correct proof of continuity of $u_n$ proceeds as follows:
 prove that for every $s$ the function $\varphi(\xi^2, s)$ is the Fourier transform of a subprobability measure (this is clear with the probabilistic picture; analytically, a possible argument is given in a lemma below);
 deduce that $u_n(s, \cdot)$ is continuous, and furthermore the family of functions $u_n(s, \cdot)$ is uniformly equicontinous;
 use $\mathscr{L}^2(\mathbf{R}^d)$continuity of $u_n(s, \cdot)$ with respect to $s$ to conclude joint continuity of $u_n(s, x)$.
Sketch of the proof: By considering the string $\tilde{A}(E) = A(E + s)$, we find that $\partial_s (\log \varphi(\lambda, s))$ is a complete Bernstein function of $\lambda$ for every $s$. Integration with respect to $s$ over $[0, s_0]$ proves that $\eta(\lambda) = \log \varphi(\lambda, s_0)$ is a complete Bernstein function of $\lambda$ for every $s_0$. Thus, $\varphi(\lambda, s_0) = \exp(\eta(\lambda))$ is the Laplace transform of a positive subprobability measure on $[0, \infty)$, the distribution at time $1$ of the subordinator with Laplace exponent $\eta(\lambda)$. It follows that $\varphi(\xi^2, s_0)$ is the Fourier transform of a positive subprobability measure on $\mathbb{R}^d$, namely the distribution at time $1$ of the subordinate Brownian motion with characteristic exponent $\eta(\xi^2)$. $\square$
 Spectral analysis of stable processes on the positive halfline with Alexey Kuznetsov, Electron. J. Probab. 23 (2018), no. 10 [onlineMRzb]
 Contractivity and ground state domination properties for nonlocal Schrödinger operators with Kamil Kaleta and József Lőrinczi, J. Spectr. Theory 8 (2018): 165–189 [onlineMRzb]
 Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes with Tomasz Grzywny, Stoch. Proc. Appl. 128(1) (2018): 1–38 [onlineMRzb]
 Martin kernels for Markov processes with jumps with Tomasz Juszczyszyn, Potential Anal. 47(3) (2017): 313–335 [onlineMRzb]
 Eigenvalues of the fractional Laplace operator in the unit ball with Bartłomiej Dyda and Alexey Kuznetsov, J. London Math. Soc. 95 (2017): 500–518 [onlineMRzb]

Fractional Laplace operator and Meijer Gfunction
with Bartłomiej Dyda and Alexey Kuznetsov,
Constructive Approx. 45(3) (2017): 427–448
[onlineMRzb]
Notes: In the list of explicit examples a reference to the result of Port is missing: he gave an explicit expression for the Poisson kernel of the complement of a sphere (or the complement of a hyperplane), see [S. C. Port, The First Hitting Distribution of a Sphere for Symmetric Stable Processes, Tran. Amer. Math. Soc. 133 (1969): 115–125]. In Proposition 1, reference should be given to Proposition 7.1, not 7.2.
 There are two typos in the last paragraph on page 437: the expression for the Bessel function should read ${_0\mathbf{F}_1}(a \,  \, {x^2}) = x^{1a} J_{a1}(2x)$, and the displayed formula should read \[ (\Delta)^{\alpha/2} f(x) = \sqrt{\pi} \, \Gamma(\tfrac{d+\alpha}{2}) \, {_1\mathbf{F}_2}\Bigl(\begin{array}{c} \tfrac{d+\alpha}{2} \\ \tfrac{1+\alpha}{2}, \, \tfrac{d}{2} \end{array} \;\Big\vert\; {\tfrac{1}{4} x^2}\Bigr) . \] Thanks to Yanzhi Zhang for pointing out the above errors!
 Factor $(2 \pi)^{\ell}$ is missing in the righthand side of Bochner's relation (45) (due to different definitions of the Fourier transform here and in Stein's book) and, consequently, in several places in the proof of Proposition 3 (after 1st and 2nd sign in the second display on page 442, in the RHS of the third display, before and after 1st equality sign in the fifth display, and $(2 \pi)^{2 \ell}$ is missing after the 2nd equality sign in the fifth display). The statement of Proposition 3 remains correct.

Ten equivalent definitions of the fractional Laplace operator,
Frac. Calc. Appl. Anal. 20(1) (2017): 7–51
[onlineMRzb]
Note: $p \in (1, \infty)$ should read $p \in (1, \tfrac{d}{\alpha})$ in page 22, line 20.  The shape of the fundamental sloshing mode in axisymmetric containers with Tadeusz Kulczycki and Bartłomiej Siudeja, J. Eng. Math. 99(1) (2015): 157–183 [onlineMRzb]
 Asymptotic estimate of eigenvalues of pseudodifferential operators in an interval with Kamil Kaleta and Jacek Małecki, J. Math. Anal. Appl. 439(2) (2016): 896–924 [onlineMRzb]
 Hitting times of points for symmetric Lévy processes with completely monotone jumps with Tomasz Juszczyszyn, Electron. J. Probab. 20 (2015), no. 48 [onlineMRzb]
 Boundary Harnack inequality for Markov processes with jumps with Krzysztof Bogdan and Takashi Kumagai, Trans. Amer. Math. Soc. 367(1) (2015): 477–517 [onlineMRzb]
 New families of subordinators with explicit transition probability semigroup with James Burridge, Alexey Kuznetsov and Andreas Kyprianou, Stoch. Proc. Appl. 124(10) (2014): 3480–3495 [onlineMRzb]
 The Legacy of Vladimir Andreevich Steklov with Nikolay Kuznetsov, Tadeusz Kulczycki, Alexander Nazarov, Sergey Poborchi, Iosif Polterovich and Bartłomiej Siudeja, Notices Amer. Math. Soc. 61(1) (2014): 9–23 [onlineMRzb]

Onedimensional quasirelativistic particle in the box
with Kamil Kaleta and Jacek Małecki,
Rev. Math. Phys. 25(8) (2013) 1350014
[onlineMRzb]
Note: In the first displayed formula in Lemma~4.2 the norm in the lefthand side should not be squared.  Suprema of Lévy processes with Jacek Małecki and Michał Ryznar, Ann. Probab. 41(3B) (2013): 2047–2065 [onlineMRzb]

First passage times for subordinate Brownian motions
with Jacek Małecki and Michał Ryznar,
Stoch. Proc. Appl. 123 (2013): 1820–1850
[onlineMRzb]
Note: In Prop. 4.5 the last displayed formula should read \[\begin{aligned}\vartheta_\lambda & \ge \tfrac{1}{\pi} (\mathrm{arcsin}(...))^2 \\ & \qquad + \tfrac{1}{\pi} (\mathrm{arcsin}(...))^2 \\ & \qquad \qquad  \tfrac{1}{\pi} (\mathrm{arcsin}(...))^2\end{aligned}\] (factor $\tfrac{1}{\pi}$ is missing); in formula (7.2) a minus sign in the exponent is missing, the correct form is $\cdots \exp(\tfrac{1}{\pi} \int_0^\infty \cdots ) \cdots$.  On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains with Tadeusz Kulczycki, Proc. London Math. Soc. 105(5) (2012): 921–952 [onlineMRzb]

Spectral theory for onedimensional symmetric Lévy processes killed upon hitting the origin,
Electron. J. Probab. 17 (2012), no. 83
[onlineMRzb]
Notes: In Sect. 1 every occurrence of $2 \xi \Psi''(\xi) \le \Psi'(\xi)$ should be $\xi \Psi''(\xi) \le \Psi'(\xi)$ (5 times).
 In Example 5.1, in the last line on page 24, $\sin(\pi \alpha)$ and $\cos(\pi \alpha)$ should read $\sin(\pi \alpha/2)$ and $\cos(\pi \alpha/2)$.

Eigenvalues of the fractional Laplace operator in the interval,
J. Funct. Anal. 262(5) (2012): 2379–2402
[onlineMRzb]
Note: Columns 3, 4, 6 and 7 in Table 3 are wrong! Corrected values: 0.541, 0.586, 0.745, 1.110, 1.655, 2.103, 2.278 (column 3); 1.084, 1.174, 1.491, 2.222, 3.311, 4.207, 4.557 (column 4); 0.552, 0.610, 0.824, 1.360, 2.243, 3.029, 3.348 (column 6); 1.106, 1.222, 1.650, 2.721, 4.488, 6.060, 6.698 (column 7). Thanks to Juan Pablo Borthagaray for pointing out this error! 
Spectral analysis of subordinate Brownian motions on the halfline,
Studia Math. 206(3) (2011): 211–271
[onlineMRzb]
Note: In displayed formula (4.14) $\frac{\lambda^2}{\lambda^2 + \xi^2}$ should be $\frac{\lambda}{\lambda^2 + \xi^2}$; in Example 6.1 in displayed formulae for $\gamma_\lambda(\xi)$ and $\gamma(s)$ a minus sign in the exponent is missing, the correct form is $\cdots \exp(\tfrac{1}{\pi} \int_0^\infty \cdots ) \cdots$; in the displayed formula for $a(y)$ in page 267 the factor $c_\alpha^2$ should be replaced with $c_\alpha^{2/\alpha}$; the definition of $g_\lambda(y)$ in the following line should be $C_\alpha (c_\alpha^{1} \lambda^{\alpha/2} y)^{1/2} K_{\alpha/2}((c_\alpha^{1} \lambda^{\alpha/2} y)^{1/\alpha})$.  Boundary Harnack inequality for α harmonic functions on the Sierpiński triangle with Kamil Kaleta, Probab. Math. Stat. 30(2) (2010): 353–368 [onlineMRzb]
 Spectral properties of the Cauchy process on halfline and interval with Tadeusz Kulczycki, Jacek Małecki and Andrzej Stós, Proc. London Math. Soc. 101(2) (2010): 589–622 [onlineMRzb]
 Eigenvalues of the Cauchy process on an interval have at most double multiplicity, Semigroup Forum 79(1) (2009): 183–192 [onlineMRzb]
 Intrinsic ultracontractivity for stable semigroups on unbounded open sets, Potential Anal. 31(1) (2009): 57–77 [onlineMRzb]
 Spectral gap estimate for stable processes on arbitrary bounded open sets, Probab. Math. Statist. 28(1) (2008): 163–167 [onlineMRzb]

Estimates and structure of $\alpha$harmonic functions
with Krzysztof Bogdan and Tadeusz Kulczycki,
Prob. Theory Rel. Fields 140(3–4) (2008): 345–381
[onlineMRzb]
Note: The first displayed formula in p. 352 should read $w = r^{2} (r^2  x^2) (r^2  v^2) / x  v^2$.
Papers of my PhD students
 Tomasz Juszczyszyn, Decay rate of harmonic functions fornonsymmetric strictly αstable Lévy processes, Studia Mathematica, in press.
 Jacek Mucha, Spectral theory for onedimensional (nonsymmetric) stable processes killed upon hitting the origin, Electron. J. Probab. 26 (2021), #14: 1–33. [onlineMR]
Selected conference presentations
 Harmonic extensions, operators with completely monotone kernels, and traces of 2D diffusions [pdf]
Nonlocal operators, probability and singularities webinar (2020)  Bellshaped functions [pdf]
7th Lévy conference (Karlovasi, Greece, 2019)  Discrete Hilbert transform [pdf]
Gordin Prize lecture (Vilnius, Lithuania, 2018)  Fractional Laplace operator and operators with unimodal and isotropic kernels [pdf]
Young PDEers at work (Warsaw, Poland, 2018)  Random walks are completely determined by their trace on the positive halfline [pdf]
MexicoPoland 1st Meeting in Probability (Guanajuato, Mexico, 2017)  Fractional Laplace operator in the unit ball [pdfvideo]
Stable processes conference (Oaxaca, Mexico, 2016)  Fractional Laplacian: explicit calculations and applications [pdf]
Nonlocal operators and PDEs (Będlewo, Poland, 2016)  Rogers functions and fluctuation theory [pdf]
7th Conference on Lévy Processes (Wrocław, Poland, 2013)  Recent progress in the study of suprema of Lévy processes [pdf]
German–Polish Conference on Probability and Mathematical Statistics (Toruń, Poland, 2013)  Twoterm asymptotics for Lévy operators in intervals [pdf]
6nd International Conference on Stochastic Analysis and Its Applications (Będlewo, Poland, 2012)  Spectral decomposition of integrodifferential operators related to onedimensional Lévy processes in domains [pdf]
Harmonic Analysis and Probability (Angers, France, 2012)  Boundary Harnack Inequality For JumpType Processes [pdf]
Foundations of Stochastic Analysis (Banff, Canada, 2011)  Spectral Theory for Subordinate Brownian Motions in Halfline [pdf]
Yangyang Summer School (Yangyang, Korea, 2011)  Cauchy process on halfline [pdf]
3rd International Conference on Stochastic Analysis and Its Applications (Beijing, China, 2009)  Boundary Harnack inequality for stable processes on the Sierpiński gasket [pdf]
Fractal Geometry and Stochastics 4 (Greifswald, Germany, 2008)  Intrinsic ultracontractivity for isotropic stable processes in unbounded domains [pdf]
2nd International Conference on Stochastic Analysis and Its Applications (Seoul, Korea, 2008)