# Lecture series by Ioannis Karatzas

Professor Ioannis Karatzas (Columbia University) will give a 5 hour lecture series and a talk at the Probability Seminar on 12-13 March 2024.

**Schedule:**

**Tuesday 12 March**

**11:00-12:00 - MS.05**

**13:00-15:00 - MB0.07**

**Wednesday 13 March**

**11:00-13:00 MS.01**

**16:00-17:00 MS.04 (Talk at the Probability Seminar)**

**Title of the lecture series: ****Competing diffusive particle systems**

**Abstract:**

We introduce and study stable multidimensional diffusions interacting through their ranks. The interactions give rise to invariant measures in broad agreement with stability properties of large equity markets observed over long time-periods. The probabilistic models we present assign drifts and variances that depend on both the name (identity) and the rank (according to capitalization) of each individual asset; they are able realistically to capture critical features of the stability in capital distribution, yet are simple enough to allow rather detailed analytical study.

The methodologies in this study touch upon the question of triple points for systems of competing diffusive particles; in particular, some choices of parameters may permit triple (or higher-order) collisions to occur. We show, however, that such multiple collisions have no effect on any of the stability properties of the resulting system. This is accomplished through a detailed analysis of collision local times.

The models have connections with the analysis of Queueing Networks in heavy traffic, with multi-dimensional diffusions reflected off the faces of the positive orthant, and with competing particle systems in Statistical Mechanics (e.g., Sherrington- Kirkpatrick model for spin-glasses). Their hydrodynamic-limit behavior is governed by generalized porous medium equations with convection, and the fluctuations around these limits by appropriate linear stochastic partial differential equations of parabolic type with additive noise. Whereas, limits of a different kind display phase transitions and are governed by Poisson-Dirichlet distributions. We survey progress on some of these fronts, and suggest open problems for further study.

Most of the models have invariant probability densities of the simple finite-sum-of-products-of exponentials type. We study also systems of interacting particles in which some of the variances-by-rank are allowed to degenerate, and for which the invariant measures of interest are not at all straightforward to compute. We report very recent work on such degenerate three-particle systems, for which very explicit invariant measure computations *are* possible; via appropriate Carleman-type boundary value problems for the associated Laplace transforms, and via Jacobi-type theta functions for the invariant densities themselves. The full extent and scope of these methodologies are yet to be explored.

(Joint work with Drs. E. Robert Fernholz, Tomoyuki Ichiba, Mykhaylo Shkolnikov, Vilmos Prokaj, Andrei Sarantsev, Sandro Francheschi and Killlian Rachel.)

**Title of the talk:** **Conservative diffusions and Markov chains as entropic flows of steepest descent**

**Abstract: **** **

We provide a detailed, probabilistic interpretation for the variational characterization of conservative diffusions as entropic flows of steepest descent. Jordan, Kinderlehrer and Otto showed in 1998, via a numerical scheme, that for diffusions of Langevin-Smoluchowski type the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in terms of the quadratic Wasserstein metric in the ambient space of configurations. Using a very direct perturbation analysis we obtain novel, stochastic-process versions of such features; these are valid along almost every trajectory of the motion in both the forward and, most transparently, the backward, directions of time. The original results follow then simply by “aggregating”, i.e., taking expectations. As a bonus we obtain a version of the HWI inequality of Otto and Villani, relating relative entropy, Fisher information, and Wasserstein distance. Similar results are outlined for reversible Markov Chains.

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This lecture series supported by the ESPRC grant EP/V009478/1.

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