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We introduce a model of random walks on Z^3 with random orientations of lines. This model can be seen as a 3D-version of a model of diffusion in inhomogeneous porous medium that has been introduced by Matheron and de Marsily. This 3D-model is related to the new process of iterated random walk in random sceneries (PAPAPA in french). We establish a joint limit theorem for the random walk (PA in french), the random walk in random sceneries (PAPA in french), and the iterated random walk (PAPAPA). This result is a joint work with Nadine Guillotin-Plantard and Frédérique Watbled. We will explain the relation between this work and previous developments for random walks in random sceneries. We will also present a conjecture about iterated random walks of higher order, and discuss about the difficulties to establish this conjecture.

The talk is divided into two parts. In the first part, I will introduce structural equation processes as a model for causal inference in discrete-time stationary processes. A structural equation process (SEP) consists of a directed graph, an independent stationary (zero-mean) process for every vertex of the graph, and a filter (i.e., an absolutely summable sequence) for every link on the graph. Every structural vector autoregressive (SVAR) process, a commonly used linear time series model, admits a representation as a SEP. Furthermore, the Fourier-transformed SEP representation of an SVAR process is parameterized over the field of rational functions with real coefficients. Using this frequency domain parameterization, we will see that d- and t- separation statements about the causal graph (associated with the SVAR process) are generically characterized by rank conditions on the spectral density of the SVAR process. Here, the spectral density is considered as a matrix over the field of rational functions with real coefficients. Additionally, we will see that the Fourier-transformed SEP parameterization of an SVAR process comes with a notion of rational identifiability for the Fourier transformed link filters. This notion allows to reason about identifiability in the presence of latent confounding processes. For instance, the recent latent factor half-trek criterion can be used to determine if the effect (i.e., the associated link function) between two potentially confounded processes is a rational function of the spectral density of the observed processes. \[ \] In the second part of the talk, I will expand the SEP framework to include a specific class of non-stationary linear processes. This class of non-stationary SEPs includes SVAR processes with periodically changing coefficients. I will also demonstrate how this framework can be used to reason about identifiability in subsampled processes, i.e., when observations are gathered at a lower frequency than the frequency at which causal effects occur.

We present a multi-agent and mean-field formulation of a game between investors who receive private signals informing their investment decisions and who interact through relative performance concerns. A key tool in our model is a Poisson random measure which drives jumps in both market prices and signal processes and thus captures common and idiosyncratic noise. Upon receiving a jump signal, an investor evaluates not only the signal's implications for stock price movements but also its implications for the signals received by her peers and for their subsequent investment decisions. A crucial aspect of this assessment is the distribution of investor types in the economy. These types determine their risk aversion, performance concerns, and the quality and quantity of their signals. We demonstrate how these factors are reflected in the corresponding HJB equations, characterizing an agent's optimal response to her peers' signal-based strategies. The existence of equilibria in both the multi-agent and mean-field game is established using Schauder's Fixed Point Theorem under suitable conditions on investor characteristics, particularly their signal processes. Finally, we present numerical case studies that illustrate these equilibria from a financial-economic perspective. This allows us to address questions such as how much investors should care about the information known by their peers.

In our uncertain and ever-changing world, many systems face the danger of crossing tipping thresholds in the future. Therefore, there is a growing interest in developing swift and reliable early warning methods to signal such crossings ahead of time. Until now, most approaches have relied on critical slowing down, typically assuming white noise and neglecting spatial effects. We introduce a data-driven method that reconstructs the linearised reaction–diffusion dynamics directly from spatio-temporal data. From the inferred model, we compute the dispersion relation and analyse the stability of Fourier modes, allowing early detection of both homogeneous and spatial instabilities. By framing early detection as a data-driven stability analysis, this approach provides a unified and quantitative way to indicate whether and when a system is approaching a tipping point or a Turing-type transition.