CS44: Stable-Type Processes

date: 7/14/2025, time: 16:00-17:30, room: ICS 140

Organizer: Tadeusz Kulczycki (Wroclaw University of Science and Technology)

Chair: Tadeusz Kulczycki (Wroclaw University of Science and Technology)

Nodal sets of supersolutions to Schrödinger equations based on symmetric jump processes

Tomasz Klimsiak (Nicolaus Copernicus University)

We address the question posed by H. Brezis (see \([1]\),\([2]\)) concerning the structure of the set \(\{u=0\}\) for non-negative supersolutions to the equation \[\tag{1} -Lu+Vu=0\quad \text{in }\mathbb R^d,\] where \(V\) is a singular potential on \(\mathbb R^d\) and \(L\) is the operator on \(L^2(\mathbb R^d;m)\) related to the form \[\mathcal{E}(u, v)=\int_{\mathbb{R}^d \times \mathbb{R}^d \backslash \operatorname{\frak d}}(u(x)-u(y))(v(x)-v(y)) J(x, y) d x d y, \quad u, v \in \mathcal D(\mathcal{E}),\] with \(J: \mathbb{R}^d \times \mathbb{R}^d \backslash \operatorname{\frak d} \rightarrow \mathbb{R}^{+}\)(here \(\operatorname{\frak d}:=\left\{(x, x): x \in \mathbb{R}^d\right\}\)) satisfying \[\frac{c_1}{|x-y|^d \varphi(|x-y|)} \leq J(x, y) \leq \frac{c_2}{|x-y|^d \varphi(|x-y|)}, \quad(x, y) \in \mathbb{R}^d \times \mathbb{R}^d \backslash \operatorname{ \frak d }\] for \(c_1, c_2>0\), and a strictly increasing function \(\varphi: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\), with \(\varphi(0)=0\) that is subject to suitable scaling conditions. The class of admissible potentials \(V\) consists of positive smooth measures, which includes, in particular, locally quasi-integrable positive functions, as well as generalized potentials, i.e. positive Borel measures that may be concentrated on \(m\)-negligible sets. Using the Green function of \(L\), we characterize the minimal set - depending only on \(L\) and \(V\) - where all possible zeros of non-trivial supersolutions to \((1)\) must lie. The key ingredient in establishing this structure result is the Feynman-Kac-type representation for supersolutions to \((1)\). As a corollary, we provide a necessary and sufficient condition on the potential \(V\), under the sole assumption that \(V:\mathbb R^d\to [0,\infty]\) is Borel measurable, ensuring that the strong maximum principle holds for the operator \(-L+V\).

We present the results included in the paper \([3]\).

Bibliography

\([1]\) Ancona, A.: Une propriété d’invariance des ensembles absorbants par perturbation d’un opérateur elliptique. Comm. Partial Differential Equations 4 (1979) 321–337.

\([2]\) Bénilan, P., Brezis, H.: Nonlinear problems related to the Thomas-Fermi equation. J. Evol. Equ. 3 (2003) 673–770.

\([3]\) Klimsiak, T.: Location of zeros of non-trivial positive supersolutions to Schrödinger equations. Math. Ann. (2025) https://doi.org/10.1007/s00208-025-03176-9.

On dynamic approximation scheme for L'evy processes

Victoria Knopova (Kyiv Taras Shevchenko National University)

Due to infinite activity, simulating the trajectory a Lévy process can be technically challenging. In order to overcome the problem, one can use certain approximation tricks, e.g. the Asmussen-Rosiński procedure, which allows to approximate the trajectory of a Lévy process by that of a compoud Poisson process and a (scaled) Brownian motion by cutting out the small jumps on a fixed level. In my talk I will discuss the so-called “dynamic cutting procedure”, which extends this technique by means of time-dependent cutting. I will discuss the weak approximation rates for this scheme, the strong approximation rated for the Euler scheme for the Lévy driven SDE, and some numerical results.

Bibliography

\([1]\) D. Ivanenko, V. Knopova, D. Platonov. "On Approximation of Some Lévy Processes". Austrian Journal of Statistics , Vol.54, Iss.1 pp. 177 - 199, - 2025

\([2]\) D. Platonov, V. Knopova. "Strong Convergence Rates for Euler Schemes of Lévy-Driven SDE using Dynamic Cutting". https://arxiv.org/abs/2504.11988

On mean exit time from a ball for symmetric stable processes

Michał Ryznar (Wrocław University of Science and Technology)

Getoor in \([3]\) calculated the mean exit time from a ball for the standard isotropic \(\alpha\)-stable process in \(R^d\) starting from the interior of the ball. The purpose of this talk is to show that, up to a multiplicative constant, the same formula is valid for arbitrary symmetric \(\alpha\)-stable process.

Bibliography

\([1]\) R. K. Getoor, First Passage Times for Symmetric Stable Processes in Space, Trans. Amer. Math. Soc. 101 (1961), 75-90.