We consider determinantal point processes $\Lambda_\phi$ arising from generalized Fock spaces $\mathcal F_\phi$, defined by a doubling subharmonic weight $\phi$. We provide a complete characterization of when these processes are almost surely separated with respect to the euclidean metric on the plane. As a consequence, we also obtain a characterization of when these processes are almost surely interpolating for the classical Fock spaces. Furthermore, we emphasize the role of intrinsic repulsion in determinantal processes by comparing $\Lambda_\phi$ with the Poisson process with the same first intensity. Additionally, we demonstrated that, with probability one, the point process $\Lambda_\phi$ fails to be separated with respect to the distance induced by the reproducing kernel $K_\phi$.
In this seminar, we will explore a fractional version of a semilinear Neumann problem studied by Lin–Ni–Takagi in the late 1980s. The problem arises when considering steady states of the Keller–Segel model with nonlocal diffusion of chemical concentration. In the fractional setting, there are different possible notions of 'nonlocal Neumann conditions'. We will study the system with both spectral conditions, following Stinga and Volzone, and integral conditions (introduced by Dipierro, Ros-Oton, and Valdinoci), as investigated by Cinti and Colasuonno. While an extension theorem analogous to that of Caffarelli and Silvestre exists for the spectral boundary condition, no such theorem is available for integral boundary conditions. This makes it more challenging to obtain Liouville-type theorems and to prove the nonexistence of non-trivial positive solutions when the concentration parameter is sufficiently large.
Based on joint projects with Eleonora Cinti (UNIBO), Marco Ghimenti (UNIPI), Juncheng Wei (CUHK), and Tobias Weth (GUF).
In this talk, we consider partial differential equations of parabolic $p$-Laplace type in noncylindrical domains $U = \bigcup_{t \in [0,T]} \{t\} \times \Omega_t$, where the underlying spatial set $\Omega_t \subset \mathbb{R}^n$ may change in time. Our goal is to characterize time-dependent boundary values that yield the existence of a solution, focusing on domains $U$ that satisfy a Lipschitz condition with respect to a parabolic metric. In particular, we discuss the extension to $U$ of certain functions that are only defined on the parabolic boundary of $U$, and conversely the trace space for functions defined on $U$ and contained in a suitable parabolic function space. The talk is based on joint work with Daniel Campbell and Sebastian Schwarzacher.
