We extend the popular Gibson and Schwartz (1990) and Schwartz and Smith (2000) two-factor models for the spot price of a commodity to include stochastic volatility and correlation. This generalization is based on the Wishart variance-covariance matrix process. For both of the extended models we present the joint characteristic functions of the two state variables. The original models are known to fit the term-structure of implied volatility in futures and options markets very well. However, the extended models are also able to match volatility smiles observed in these markets. Regarding the analysis of financial time series, the assumption of a constant correlation between the state variables is known to be too restrictive. Introducing time-varying correlation via the Wishart process allows us to study its empirical behaviour in commodity markets through the use of filtering techniques.

We consider a thin layer of an incompressible viscous Newtonian fluid coating the inner wall of a horizontal cylinder rotating with constant speed. Assuming the film height is small compared to the radius of the cylinder, we formally derive a closed equation for the height h(t, θ) > 0 of the liquid film by means of a lubrication approximation: ht + hθ + γ h3(hθθθ + hθ) θ = g h3 cos θ θ in (0, T ) × T This rimming flow equation is of fourth-order, degenerate-parabolic, and quasilinear. Compet- ing effects are observed between viscosity, the surface tension γ, and gravity g. For g = 0 and a fixed mass m, the two-dimensional manifold M(m) := m + a sin θ + b cos θ a2 + b2 < m2 is invariant. If 0 < g ∼ δ ≪ 1 is small, we show that solutions which are bounded away from zero converge exponentially fast to a δ-neighbourhood of M(m). Here, the existence of solutions on a large time scale t ∼ 1/δ2 can be shown. Moreover, such solutions evolve on two distinct time scales: On the fast time scale t they only rotate around the origin with the speed of the cylinder, while on the slow time scale τ = δ2t the dynamics are governed by an ODE in τ on M(m).

We study causality in systems that allow for feedback loops among the variables via models of cross-sectional data from a dynamical system. Specifically, we consider the set of distributions which appears as the steady-state distributions of a stochastic differential equation (SDE) where the drift matrix is parametrized by a directed graph. The nth-order cumulant of the steady state distribution satisfies the corresponding nth-order continuous Lyapunov equation. Under the assumption that the driving Lévy process of the SDE is not a Brownian motion (so the steady state distribution is non-Gaussian) and the coordinates are independent, we are able to prove generic identifiability for any connected graph from the second and third-order Lyapunov equations while allowing the cumulants of the driving process to be unknown diagonal. We propose a minimum distance estimator of the drift matrix, which we are able to prove is consistent and asymptotically normal by utilizing the identifiability result.

State-sum constructions have numerous applications in both mathematics and physics. In mathematics, they yield invariants for knots and manifolds and serve as a powerful organizing principle in representation theory. To illustrate this principle, we discuss equivariant Frobenius-Schur indicators. In the context of physics, we explain how state-sum models offer a conceptual framework for tensor network models, based on collaboration with Fuchs, Haegeman, Lootens, and Verstraete.

Structure recovery is at the heart of modern Data Science. Roughly put, the goal in structure recovery problems is to identify (or at least approximate) an unknown object using limited, random information – e.g., a random sample of the unknown object.

As it happens, key questions on recovery are fundamental (open) problems in Asymptotic Geometric Analysis and High Dimensional Probability. In this talk I will give one example (out of many) that exhibits the rather surprising ties between those seemingly unrelated areas.

I will explain why noise-free recovery is dictated by the geometry of natural random sets: for a class of functions 𝐹 and n i.i.d random variables 𝜎 = (𝑋_1,…,𝑋_n), the random sets are 𝑃_𝜎 (𝐹) = { (𝑓(𝑋_1),….,𝑓(𝑋_n)) : 𝑓 ∈ 𝐹 }.

I will outline a (sharp) estimate on the structure of a typical 𝑃_𝜎 (𝐹) that leads to the solution of the noise-free recovery problem under minimal assumptions. I will explain why the same estimate resolves various questions in high dimensional probability (e.g., the smallest singular values of certain random matrices) and high dimensional geometry (e.g., the Gelfand width of a convex body).

The optimality of the solution is implied by a exposing a “hidden extremal structure” contained in 𝑃_𝜎 (𝐹), which in turn is based on a complete answer to Talagrand’s celebrated entropy problem. _____________________________________________

Invited by Prof. Holger Rauhut.

This talk shall be concerned with the surface quasi-geostrophic equation driven by a generic additive noise process W. By means of convex integration techniques, we establish existence of weak solutions whenever the stochastic convolution z associated with W is well defined and fulfils certain regularity constraints. Quintessentially, we show that the so constructed solutions to the non-linear equation are controlled by z in a linear fashion. This allows us to deduce further properties of the so constructed solutions, without relying on structural probabilistic properties such as Gaussianity, Markovianity or a martingale property of the underlying noise W. This is joint work with Florian Bechtold (University of Bielefeld) and Jörn Wichmann (Monash University) (cf. [1]). This activity receives partial funding from the European Research Council (ERC) under the EU-HORIZON EUROPE ERC-2021-ADG research and innovation programme (project „Noise in Fluids“, grant agreement no. 101053472). [1] F. Bechtold, T. Lange, J. Wichmann, "On convex integration solutions to the surface quasi-geostrophic equation driven by generic additive noise", Electronic Journal of Probability 29 (2024): 1-38

Incorporating microscopic details into fluctuating hydrodynamics has long been a challenge, as conventional top-down approaches often fail to capture the full spectrum of hydrodynamic interactions and fluctuations arising from microscopic physics. Dean and Kawasaki tackled this challenge by deriving a stochastic partial differential equation for density operators from microscopic Brownian processes; however, its mathematically ill-defined nature has posed significant obstacles for numerical implementations to sample field-theoretic dynamics. In this talk, I will introduce a mathematically regularized framework that overcomes these challenges and present a mesoscopic coarse-graining methodology to construct mesoscopic fields from microscopic molecules in terms of interactions and correlations. I will also demonstrate numerical implementation for weakly interacting molecular liquids at the mesoscopic field level. Even though mesoscopic interactions and correlations are integrated away during coarse-graining, the proposed approach effectively captures the temporal evolution of coarse-grained microscopic density profiles.

We consider a stochastic process on the graph $\mathds{Z}^d$. Each $x\in \mathds{Z}^d$ starts with a cluster of size 1 with probability $p \in (0,1]$ independently. Each cluster $C$ of performs a continuous time SRW with rate $\abs{C}^{-\alpha}$. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. Focusing on dimension $d=1$, we show that for $\alpha>-2$, at time $t$, the cluster size is of order $t^\frac{1}{\alpha + 2}$, and for $\alpha \le -2$ we get an infinite component. Additionally, for $\alpha = 0$ we show convergence in distribution of the scaling limit.

Let W be a conservative, ergodic Markov diffusion on some arbitrary state space M, converging exponentially fast to equilibrium. We consider: (1) Systems of up to countably many massive particles in M, with finite total mass. Each particle is subject to an independent instance of the noise W, with volatility the inverse mass carried by the particle. We prove that the corresponding infinite system of SDEs has a unique solution, for every starting configuration and every distribution of the masses in the infinite simplex. (2) Solutions to the Dean--Kawasaki SPDE with singular drift, driven by the generator L of W. We prove that the equation may be given rigorous meaning on M, and that it has a unique `distributional’ solution. This extends Konarovskyi--Lehmann--von Renesse's `ill-posedness vs. triviality' to the case of infinitely many massive particles. (3) Diffusions with values in the space P of all probability measures on M, driven by the geometry induced by L. (4) In the case when M is a manifold, differential-geometric and metric-measure Brownian motions on P induced by the geometry of optimal transportation and reversible for a normalized completely random measure. We show that all these objects coincide.

Most modern machine learning applications are based on overparameterized neural networks trained by variants of stochastic gradient descent. To explain the performance of these networks from a theoretical perspective (in particular the so-called "implicit bias"), it is necessary to understand the random dynamics of the optimization algorithms. Mathematically this amounts to the study of random dynamical systems with manifolds of equilibria. In this talk, I will give a brief introduction to machine learning theory and explain how the Lyapunov exponents of random matrix products can be used to characterize the set of possible limit points for stochastic gradient descent. The talk is based on joint work with Maxmilian Engel.

Multiscale dynamical systems are an active area of research with numerous real-world applications, characterized by the interaction of variables operating on different time scales—fast and slow. In this work, we focus on a two-dimensional multiscale stochastic dynamical system derived from the Van der Pol equation with random perturbations. To analyze the quantitative behavior of this system, we convert the stochastic differential equation into a Fokker-Planck equation, accounting for the distinct time scales involved. In this work, we look at the stochastic system with linear shifting term. We apply an asymptotic ansatz to obtain the explicit asymptotic solution of the Fokker-Planck equation. First, we use a direct asymptotic analysis. This method involves expanding the solution in powers of the time-scale separation parameter, yielding a system of equations at different orders. The explicit solution for each equation combines a normal distribution for the fast variable with a shift of the original distribution for the slow variable. Second, we decompose the multiscale Fokker-Planck equation into two equations that correspond to different time scales. One equation governs the behavior on the fast time scale, while the other addresses the dynamics in the space orthogonal to the fast solution. Asymptotic analysis is then applied to this decomposed system. The solutions obtained from this decomposition match those from the direct asymptotic method, confirming that there is no stationary solution for this shifting system. The next result involves using the linear system to describe the behavior of the nonlinear stochastic Van der Pol (VdP) equation. In this work, we approach the problem by partitioning the critical manifold of the VdP equation into three segments and linearizing the system within each segment. This segmentation allows us to apply the same methods used for linear systems to analyze the local behavior of the stochastic VdP equation in each region. By performing asymptotic analysis on these linearized systems, we gain insights into the local dynamics of the VdP equation around the critical manifold. Finally, by combining the results from the different segments, we can construct the global dynamics of the system, including the stationary distribution.

See https://www.lmu.de/ai-hub/en/news-events/all-events/event/munich-ai-lectures-prof.-dr.-jean-luc-starck.html

In the field of algebraic statistics, we view statistical models as part of an algebraic variety and use tools from algebra, geometry, and combinatorics to learn statistically relevant information about these models. In this talk, we discuss the algebraic interpretation of likelihood inference for discrete statistical models. We present recent work on the iterative proportional scaling (IPS) algorithm, which is used to compute the maximum likelihood estimate (MLE), and give algebraic conditions under which this algorithm outputs the exact MLE in one cycle. Next, we introduce quasi-independence models, which describe the joint distribution of two random variables where some combinations of their states cannot co-occur, but they are otherwise independent. We combinatorially classify the quasi-independence models whose MLEs are rational functions of the data. We show that each of these has a parametrization which satisfies the conditions that guarantee one-cycle convergence of the IPS algorithm.

The abelian sandpile model introduced by Per Bak, Chao Tang and Kurt Wiesenfeld was the first discovered example of a dynamical system exhibiting self-organized criticality. Since its introduction in 1987, the model has seen widespread research interest from mathematicians and physicists alike, with a focus on explaining the complex global behaviour that emerges from the interplay of the local toppling rules.

In my talk, I will introduce the abelian sandpile model, its toppling rules and how we use these dynamics to define the abelian sandpile Markov chain. We will cover the most important aspects of the sandpile Markov chain, and discuss how these apply to modern research questions on the abelian sandpile model about the distribution of particles and avalanche sizes, with a focus on how the model behaves on fractal state spaces.

The frog model is a classical branching model for the spread of an infection. In this model, sleeping frogs are placed on the vertices of a graph and initially one vertex is activated. Active frogs move as simple random walks and wake up all sleeping frogs they encounter. Motivated by the addition of an immunological response, we present an extension of this model in which sleeping frogs must be visited a random number of times, i.i.d. as some $I$, before they awaken, and the active frogs that attempt to wake them up are killed.

We examine the propagation speed and demonstrate that the frogs spread ballistically for a $2+\varepsilon$-moment assumption on $I$, but sublinearly for a more heavy-tailed $I$. By constructing a series of renewal times, at which the front becomes independent of the past, we are able to derive a shape theorem under more technical assumptions.