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Vanja Wagner

Publications and source records attributed to Vanja Wagner.

9 recordsLinked to original sources
arXiv2026

Nonhomogeneous boundary condition for spectral non-local operators

We study semilinear non-local elliptic problems driven by spectral-type operators of the form $\psi(-L_{|D})$ in a bounded $C^{1,1}$ domain $D\subset \mathbb{R}^d$ with a nonhomogeneous boundary condition. Here $\psi$ is a Bernstein function satisfying a weak scaling condition at infinity, and $L_{|D}$ is the generator of a killed L\'evy process. This general framework covers and extends the theory of the interpolated fractional Laplacian. A key novelty in this setting is the analysis of the nonhomogeneous boundary condition formulated in terms of the Poisson potential with respect to the $d-1$ Hausdorff measure on $\partial D$. We establish sharp boundary estimates for Green and Poisson potentials, introduce a weak $L^1$ trace-like boundary operator, and provide existence results for solutions under quite general nonlinearities, including sign-changing and non-monotone cases. The methodology combines stochastic process techniques, potential theory, and spectral analysis, and expresses the boundary behavior of the solution in terms of the renewal function and the distance to the boundary, suggesting a possible unified treatment of semilinear boundary problems in non-local settings.

math.AP↗
arXiv2025

Large solutions for subordinate spectral Laplacian

We find a large solution to a semilinear Dirichlet problem in a bounded $C^{1,1}$ domain for a non-local operator $\phi(-\Delta\vert_{D})$, an extension of the infinitesimal generator of a subordinate killed Brownian motion. The setting covers and extends the case of the spectral fractional Laplacian. The upper bound for the explosion rate of the large solution is obtained, and is given in terms of the renewal function, distance to the boundary, and the Keller-Osserman-type transformation of the nonlinearity. Additionally, we prove interior higher regularity results for this operator.

math.AP↗
arXiv2025

Large solutions to semilinear equations for subordinate Laplacians in $C^{1,1}$ bounded open sets

We study the existence of a large solution to a semilinear problem in a bounded open $C^{1,1}$ set for a class of nonlocal operators obtained by an appropriate subordination of the Laplacian. These operators are classical generalisations of the fractional Laplacian. The existence result is shown under a nonlocal version of the Keller-Osserman condition, stated in terms of the subordinator and the source term $f$.

math.AP↗
arXiv2018

Nonlocal quadratic forms with visibility constraint

Given a subset $D$ of the Euclidean space, we study nonlocal quadratic forms that take into account tuples $(x,y) \in D \times D$ if and only if the line segment between $x$ and $y$ is contained in $D$. We discuss regularity of the corresponding Dirichlet form leading to the existence of a jump process with visibility constraint. Our main aim is to investigate corresponding Poincaré inequalities and their scaling properties. For dumbbell shaped domains we show that the forms satisfy a Poincaré inequality with diffusive scaling. This relates to the rate of convergence of eigenvalues in singularly perturbed domains.

math.AP↗
arXiv2018

Censored symmetric Lévy processes

We examine three equivalent constructions of a censored symmetric purely discontinuous Lévy process on an open set $D$; via the corresponding Dirichlet form, through the Feynman-Kac transform of the Lévy process killed outside of $D$ and from the same killed process by the Ikeda-Nagasawa-Watanabe piecing together procedure. By applying the trace theorem on $n$-sets for Besov-type spaces of generalized smoothness associated with complete Bernstein functions satisfying certain scaling conditions, we analyze the boundary behaviour of the corresponding censored Lévy process and determine conditions under which the process approaches the boundary $\partial D$ in finite time. Furthermore, we prove a stronger version of the 3G inequality and its generalized version for Green functions of purely discontinuous Lévy processes on $κ$-fat open sets. Using this result, we obtain the scale invariant Harnack inequality for the corresponding censored process.

math.PR↗
arXiv2018

A note on the trace theorem for Besov-type spaces of generalized smoothness on $d$-sets

The main goal of this paper is to give a complete proof of the trace theorem for Besov-type spaces of generalized smoothness associated with complete Bernstein functions satisfying certain scaling conditions on $d$-sets $D\subset\mathbb R^n$, $d\leq n$. The proof closely follows the classical approach by Jonsson and Wallin and the trace theorem for classical Besov spaces. Here, the trace space is defined by means of differences. When $d=n$, as an application of the trace theorem, we give a condition under which the test functions $C_c^\infty(D)$ are dense in the trace space on $D$.

math.FA↗
arXiv2017

On purely discontinuous additive functionals of subordinate Brownian motions

Let $A_t=\sum_{s\le t} F(X_{s-},X_s)$ be a purely discontinuous additive functional of a subordinate Brownian motion $X=(X_t, \mathbb P_x)$. We give a sufficient condition on the non-negative function $F$ that guarantees that finiteness of $A_{\infty}$ implies finiteness of its expectation. This result is then applied to study the relative entropy of $\mathbb P_x$ and the probability measure induced by a purely discontinuous Girsanov transform of the process $X$. We prove these results under the weak global scaling condition on the Laplace exponent of the underlying subordinator.

math.PR↗
arXiv2016

Boundary Harnack principle for the absolute value of a one-dimensional subordinate Brownian motion killed at $0$

We prove the Harnack inequality and boundary Harnack principle for the absolute value of a one-dimensional recurrent subordinate Brownian motion killed upon hitting $0$, when $0$ is regular for itself and the Laplace exponent of the subordinator satisfies certain global scaling conditions. Using the conditional gauge theorem for symmetric Hunt processes we prove that the Green function of this process killed outside of some interval $(a,b)$ is comparable to the Green function of the corresponding killed subordinate Brownian motion. We also consider several properties of the compensated resolvent kernel $h$, which is harmonic for our process on $(0,1)$.

math.PR↗