Lectures
Bruno Toaldo, University of Turin
Modeling anomalous diffusion: The Role of CTRWs and non-local dynamics - lecture notes
In this course, we introduce the stochastic approach to anomalous diffusion processes
through continuous-time random walks (CTRWs) and non-local dynamics. We begin by
presenting the fundamentals of CTRWs and analyzing their semi-Markov properties. Next,
we examine scaling limits of CTRWs and discuss how these properties are preserved under
the limiting procedure. Finally, we establish the connection between the limiting processes
and time-nonlocal equations.
Vanja Wagner, University of Zagreb
Nonlocal operators of regional type and pure-jump processes on domains - lecture notes
We present an overview of several nonlocal operators of regional type, with the model
example being the regional operators generalising the fractional Laplacian. The focus will be
on the analysis of three classes of operators: the Dirichlet, spectral, and regional (censored). We will discuss their analytical properties, as well as associate them with appropriate pure-jump processes on domains.
Jaeyun Yi, Korea Institute For Advanced Study
Peaks of the parabolic Anderson model with white noise potential - slides day 1, slides day 2, slides day 3
In this lecture series, I will discuss the parabolic Anderson model (PAM), the Cauchy problem associated with the Anderson Hamiltonian operator. Here, the potential is given by white noise, a mean-zero, delta-correlated Gaussian random field. A key characteristic of the PAM is the emergence of tall peaks localized in small regions, a phenomenon inherited from the Anderson localization of the underlying operator. We will examine the asymptotic behavior of these peaks and their multifractal structure.
Due to the roughness of the noise in higher dimensions, the PAM in higher dimensions becomes a singular stochastic PDE, which requires modern solution theories. I will also introduce how to exploit heat kernel estimates and spectral analysis tailored to these singular settings. This is based on joint work with Promit Ghosal.
