CS44: Stable-Type Processes

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

On mean exit time from a ball for symmetric stable processes

Michał Ryznar

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.

On ``dynamic’’ approximation scheme for L'evy processes

Victoria Knopova

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

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

Tomasz Klimsiak

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.