![Mateusz Kwaśnicki [photo]](img/photo.jpg)
Mateusz Kwaśnicki
Department of Pure Mathematics
Wrocław University of Science and Technology
ul. Wybrzeże Wyspiańskiego 27
50-370 Wrocław, Poland
e-mail:
skype: mateusz.kwasnicki
ORCID: 0000-0003-3896-8124
Research articles
For an author's copy of any of the following papers, please send me an email at
[MathSciNet | zbMATH | Scholar | Scopus | Web of Science | ORCID | MathOverflow]
Preprint articles
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Mateusz Kwaśnicki,
Fleming–Viot couples live forever
[arXiv] -
Tomasz Grzywny, Mateusz Kwaśnicki
Liouville’s theorems for Lévy operators
[arXiv] -
Mateusz Kwaśnicki
Suprema of Lévy processes with completely monotone jumps: spectral-theoretic approach
[arXiv] -
Jan Christoph Schlegel, Mateusz Kwaśnicki, Akaki Mamageishvili
Axioms for Constant Function Market Makers
[arXiv]
Extended abstract:
Proc. 24th ACM Conf. Econom. Comput. (EC ’23), July 9–12, 2023, London, UK. ACM, New York, NY, USA.
[online] -
Rodrigo Bañuelos, Mateusz Kwaśnicki,
The $\ell^p$ norm of the Riesz–Titchmarsh transform for even integer $p$
[arXiv] -
Rodrigo Bañuelos, Daesung Kim, Mateusz Kwaśnicki,
Sharp $\ell^p$ inequalities for discrete singular integrals
[arXiv]
Publications
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Mateusz Kwaśnicki, Jacek Wszoła,
Bell-shaped sequences,
Studia Math. 271(2) (2023): 151–185
[online]Note: Theorem 8.5.3 (instead of 11.5.3) in Karlin’s monograph should be referenced in Section 4. -
Mateusz Kwaśnicki,
Boundary traces of shift-invariant diffusions in half-plane,
Ann. Inst. Henri Poincaré Probab. Statist. 59(1) (2023): 411–436
[online | MR | zb]Note: In Definition 1.1 the process $Y(t)$ should be assumed transitive on $[0, R)$, as explained in the second paragraph of Section 3. -
Mateusz Kwaśnicki,
Harmonic extension technique for non-symmetric operators with completely monotone kernels,
Calc. Var. Partial. Differ. Equ. 61 (2022), no. 202: 1–40
[online | MR | zb] -
Mateusz Kwaśnicki, Thomas Simon,
Characterisation of the class of bell-shaped functions,
Math. Zeitschrift 301(3) (2022): 2659–2683
[online | MR | zb] -
Andreas E. Kyprianou, Mateusz Kwaśnicki, Sandra Palau, Tsogzolmaa Saizmaa,
Oscillatory attraction and repulsion from a subset of the unit sphere or hyperplane for isotropic stable Lévy processes,
In: L. Chaumont, A. E. Kyprianou (eds.), A Lifetime of Excursions Through Random Walks and Lévy Processes, Progress in Probability 78, Springer Nature Switzerland AG, 2021
[online | MR | zb] -
Mateusz Kwaśnicki,
Random walks are determined by their trace on the positive half-line,
Ann. Henri Lebesgue 3 (2020): 1389–1397 [online | MR | zb] -
Mateusz Kwaśnicki,
A new class of bell-shaped functions,
Trans. Amer. Math. Soc. 373(4) (2020): 2255–2280
[online | MR | zb] -
Alexey Kuznetsov, Mateusz Kwaśnicki,
Minimal Hermite-type eigenbasis of the discrete Fourier transform,
J. Fourier Anal. Appl. 25(3) (2019): 1053–1079
[online | MR | zb] -
Mateusz Kwaśnicki,
Fluctuation theory for Lévy processes with completely monotone jumps,
Electron. J. Probab. 24 (2019), no. 40: 1–40
[online | MR | zb]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 non-constant is missing in Lemma 5.4, Theorem 5.5 and Corollary 5.6. Similarly, in Theorem 5.7 and Lemma 6.1, $f(\xi)$ should be assumed to be non-constant rather than non-zero. Finally, in the proof of Theorem 1.1 the trivial case of constant Rogers functions should be treated separately.
- In the proof of Theorem 3.3, in the first displayed equation on page 10, a factor $i$ is missing in the right-hand side: the integrand should be $\operatorname{Im} \tfrac{i}{\xi + i s} \tilde{\varphi}(s)$.
- A factor $\pi$ is missing in the left-hand 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, unpublished [arXiv]) -
Rodrigo Bañuelos, Mateusz Kwaśnicki,
On the $\ell^p$ norm of the discrete Hilbert transform,
Duke Math. J. 168(3) (2019): 471–504
[online | MR | zb] -
Mateusz Kwaśnicki, Richard S. Laugesen, Bartłomiej Siudeja
Pólya’s conjecture fails for the fractional Laplacian,
J. Spectral Theory 9(1) (2019): 127–135
[online | MR | zb] -
Mateusz Kwaśnicki,
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
[online | MR | google] -
Mateusz Kwaśnicki, Jacek Mucha,
Extension technique for complete Bernstein functions of the Laplace operator,
J. Evol. Equ. 18(3) (2018): 1341–1379
[online | MR | zb]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 sub-probability 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 sub-probability 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 sub-probability measure on $\mathbb{R}^d$, namely the distribution at time $1$ of the subordinate Brownian motion with characteristic exponent $\eta(|\xi|^2)$. $\square$
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Alexey Kuznetsov, Mateusz Kwaśnicki,
Spectral analysis of stable processes on the positive half-line,
Electron. J. Probab. 23 (2018), no. 10: 1–29
[online | MR | zb] -
Kamil Kaleta, Mateusz Kwaśnicki, József Lőrinczi,
Contractivity and ground state domination properties for non-local Schrödinger operators,
J. Spectr. Theory 8 (2018): 165–189
[online | MR | zb] -
Tomasz Grzywny, Mateusz Kwaśnicki,
Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes,
Stoch. Proc. Appl. 128(1) (2018): 1–38
[online | MR | zb] -
Tomasz Juszczyszyn, Mateusz Kwaśnicki,
Martin kernels for Markov processes with jumps,
Potential Anal. 47(3) (2017): 313–335
[online | MR | zb] -
Bartłomiej Dyda, Alexey Kuznetsov, Mateusz Kwaśnicki,
Eigenvalues of the fractional Laplace operator in the unit ball,
J. London Math. Soc. 95 (2017): 500–518
[online | MR | zb]Note: In the first paragraph on page 509, the definition of the function $g$ should read $g(x) = \bigl(\int_{-1}^1 (f(x, y))^2 dy\bigr)^{1/2}$. Then indeed $Q(f) \geqslant Q(g)$, because $\|g\|_2 = \|f\|_2$, and $\mathscr{E}(f, f) \geqslant \mathscr{E}(g, g)$. The proof of the latter inequality goes as follows.
With the usual constant $C_{d, \alpha}$, we have $$C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{(f(x, y))^2}{|(x, y) - (x', y')|^{d + 1 + \alpha}} dy dy' = C_{d, \alpha} \frac{(g(x))^2}{|x - y|^{d + \alpha}} .$$ Furthermore, by Cauchy–Schwarz, $$\begin{aligned}&C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{f(x, y) f(x', y')}{|(x, y) - (x', y')|^{d + 1 + \alpha}} dy dy' \\ & \qquad \leqslant \biggl(C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{(f(x, y))^2}{|(x, y) - (x', y')|^{d + 1 + \alpha}} dy dy'\biggr)^{1/2} \times \\ & \qquad \qquad \biggl(C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{(f(x', y'))^2}{|(x, y) - (x', y')|^{d + 1 + \alpha}} dy dy'\biggr)^{1/2} \\ & \qquad = C_{d, \alpha} \frac{g(x) g(x')}{|x - x'|^{d + \alpha}}. \end{aligned}$$ It follows that $$C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{(f(x, y) - f(x', y'))^2}{|(x, y) - (x', y')|^{d + 1 + \alpha}} dy dy' \geqslant C_{d, \alpha} \frac{(g(x) - g(x'))^2}{|x - x'|^{d + \alpha}}.$$ Integration with respect to $x, x'$ leads to the desired inequality $\mathscr{E}(f, f) \geqslant \mathscr{E}(g, g)$. Thanks to Jiya Rose Johnson for pointing out this error! -
Bartłomiej Dyda, Alexey Kuznetsov, Mateusz Kwaśnicki,
Fractional Laplace operator and Meijer G-function,
Constructive Approx. 45(3) (2017): 427–448
[online | MR | zb]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 ${_0\mathbf{F}_1}(a \, | \, {-|x|^2}) = |x|^{1-a} J_{a-1}(2|x|)$, 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 right-hand 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.
- In formula (38) in Corollary 4, the factor $|x|^{\alpha - 2\rho}$ should be replaced by $|x|^{2\rho - \alpha}$. Same comment applies to the display after the statement of Corollary 4. Thanks to Vince Ervin for pointing out this error!
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Mateusz Kwaśnicki,
Ten equivalent definitions of the fractional Laplace operator,
Frac. Calc. Appl. Anal. 20(1) (2017): 7–51
[online | MR | zb]Note: $p \in (1, \infty)$ should read $p \in (1, \tfrac{d}{\alpha})$ in page 22, line 20. -
Kamil Kaleta, Mateusz Kwaśnicki, Jacek Małecki,
Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval,
J. Math. Anal. Appl. 439(2) (2016): 896–924
[online | MR | zb] -
Tadeusz Kulczycki, Mateusz Kwaśnicki, Bartłomiej Siudeja,
The shape of the fundamental sloshing mode in axisymmetric containers,
J. Eng. Math. 99(1) (2016): 157–183
[online | MR | zb] -
Tomasz Juszczyszyn, Mateusz Kwaśnicki,
Hitting times of points for symmetric Lévy processes with completely monotone jumps,
Electron. J. Probab. 20 (2015), no. 48: 1–24
[online | MR | zb] -
Krzysztof Bogdan, Takashi Kumagai, Mateusz Kwaśnicki,
Boundary Harnack inequality for Markov processes with jumps,
Trans. Amer. Math. Soc. 367(1) (2015): 477–517
[online | MR | zb] -
James Burridge, Alexey Kuznetsov, Andreas Kyprianou, Mateusz Kwaśnicki,
New families of subordinators with explicit transition probability semigroup,
Stoch. Proc. Appl. 124(10) (2014): 3480–3495
[online | MR | zb] -
Nikolay Kuznetsov, Tadeusz Kulczycki, Mateusz Kwaśnicki, Alexander Nazarov, Sergey Poborchi, Iosif Polterovich, Bartłomiej Siudeja,
The Legacy of Vladimir Andreevich Steklov,
Notices Amer. Math. Soc. 61(1) (2014): 9–23
[online | MR | zb] -
Kamil Kaleta, Mateusz Kwaśnicki, Jacek Małecki,
One-dimensional quasi-relativistic particle in the box,
Rev. Math. Phys. 25(8) (2013) 1350014
[online | MR | zb]Note: In the first displayed formula in Lemma 4.2 the norm in the left-hand side should not be squared. -
Mateusz Kwaśnicki, Jacek Małecki, Michał Ryznar,
Suprema of Lévy processes,
Ann. Probab. 41(3B) (2013): 2047–2065
[online | MR | zb] -
Mateusz Kwaśnicki, Jacek Małecki, Michał Ryznar,
First passage times for subordinate Brownian motions,
Stoch. Proc. Appl. 123 (2013): 1820–1850
[online | MR | zb]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 $\cdots \exp(-\tfrac{1}{\pi} \int_0^\infty \cdots ) \cdots$. -
Mateusz Kwaśnicki, Tadeusz Kulczycki,
On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains,
Proc. London Math. Soc. 105(5) (2012): 921–952
[online | MR | zb] -
Mateusz Kwaśnicki,
Spectral theory for one-dimensional symmetric Lévy processes killed upon hitting the origin,
Electron. J. Probab. 17 (2012), no. 83: 1–29
[online | MR | zb]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 $\cos(\pi \alpha/2)$.
-
Mateusz Kwaśnicki,
Eigenvalues of the fractional Laplace operator in the interval,
J. Funct. Anal. 262(5) (2012): 2379–2402
[online | MR | zb]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! -
Mateusz Kwaśnicki,
Spectral analysis of subordinate Brownian motions on the half-line,
Studia Math. 206(3) (2011): 211–271
[online | MR | zb]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 $C_\alpha (c_\alpha^{-1} \lambda^{\alpha/2} y)^{1/2} K_{\alpha/2}((c_\alpha^{-1} \lambda^{\alpha/2} y)^{1/\alpha})$. -
Kamil Kaleta, Mateusz Kwaśnicki,
Boundary Harnack inequality for $\alpha$-harmonic functions on the Sierpiński triangle,
Probab. Math. Stat. 30(2) (2010): 353–368
[online | MR | zb] -
Tadeusz Kulczycki, Mateusz Kwaśnicki, Jacek Małecki, Andrzej Stós,
Spectral properties of the Cauchy process on half-line and interval,
Proc. London Math. Soc. 101(2) (2010): 589–622
[online | MR | zb] -
Mateusz Kwaśnicki,
Eigenvalues of the Cauchy process on an interval have at most double multiplicity,
Semigroup Forum 79(1) (2009): 183–192
[online | MR | zb] -
Mateusz Kwaśnicki,
Intrinsic ultracontractivity for stable semigroups on unbounded open sets,
Potential Anal. 31(1) (2009): 57–77
[online | MR | zb] -
Mateusz Kwaśnicki,
Spectral gap estimate for stable processes on arbitrary bounded open sets,
Probab. Math. Statist. 28(1) (2008): 163–167
[online | MR | zb] -
Krzysztof Bogdan, Tadeusz Kulczycki, Mateusz Kwaśnicki,
Estimates and structure of $\alpha$-harmonic functions,
Prob. Theory Rel. Fields 140(3–4) (2008): 345–381
[online | MR | zb]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
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Krzysztof Bogdan, Michał Gutowski, Katarzyna Pietruska-Pałuba,
Polarized Hardy–Stein identity,
preprint, 2023
[arXiv] -
Tadeusz Kulczycki, Jacek Wszoła,
On the interior Bernoulli free boundary problem for the fractional Laplacian on an interval,
preprint, 2023
[arXiv] -
Michał Gutowski,
Hardy-Stein identity for pure-jump Dirichlet forms,
Bull. Pol. Acad. Sci. Math., in press
[online] -
Tomasz Juszczyszyn,
Decay rate of harmonic functions fornon-symmetric strictly α-stable Lévy processes,
Studia Mathematica 260 (2021): 141–165
[online|MR|zb] -
Jacek Mucha,
Spectral theory for one-dimensional (non-symmetric) stable processes killed upon hitting the origin,
Electron. J. Probab. 26 (2021), #14: 1–33
[online|MR|zb]
Selected conference presentations
- Discrete Hilbert transform [pdf]
Ryll-Nardzewski Day (Wrocław University of Science and Technology, Poland, 2023) - The interplay between spectral theory and Lévy processes [pdf]
Recent Developments in Stochastic Processes (Sofia, Bulgaria, 2023) - Liouville’s theorems for Lévy operators [pdf]
NOMP (Będlewo, Poland, 2023) - Discrete Hilbert transforms on $\ell^p$ [pdf]
ETA Seminar (online, Rutgers University, USA, 2022) - Harmonic extensions, operators with completely monotone kernels, and traces of 2-D diffusions [pdf]
Non-local operators, probability and singularities webinar (2020) - Bell-shaped 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 half-line [pdf]
Mexico-Poland 1st Meeting in Probability (Guanajuato, Mexico, 2017) - Fractional Laplace operator in the unit ball [pdf|video]
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) - Two-term asymptotics for Lévy operators in intervals [pdf]
6nd International Conference on Stochastic Analysis and Its Applications (Będlewo, Poland, 2012) - Spectral decomposition of integro-differential operators related to one-dimensional Lévy processes in domains [pdf]
Harmonic Analysis and Probability (Angers, France, 2012) - Boundary Harnack Inequality For Jump-Type Processes [pdf]
Foundations of Stochastic Analysis (Banff, Canada, 2011) - Spectral Theory for Subordinate Brownian Motions in Half-line [pdf]
Yangyang Summer School (Yangyang, Korea, 2011) - Cauchy process on half-line [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)