Stochastics Seminar

We present a new approach to developing an Itô–Stratonovich-type formula for rough sheets. Historically, planar integration for irregular paths has been notoriously difficult. The emergence of 'mixed differential' terms in 2D leads to overlapping iterated integrals, the construction of which previously required exhaustive combinatorial structures. One example of such a structure is the substantial 36-element planar signature introduced by K. Chouk and M. Gubinelli (Rough Sheets, arxiv.org/abs/1406.7748, 2014). In this work, we propose a simplified setting for rough sheet calculus that relies on elementary Taylor expansions in a more fundamental way. We claim that this simple trick significantly streamlines the complexity of planar algebraic integration. We illustrate the methodology in three regimes. Firstly, we present a special Rough-Young framework (joint work with S. Tindel,arxiv.org/abs/2606.20908) for paths with ‘’asymmetric'' Hölder regularity: $H_1 > 1/3$ in direction 1 and $H_2 > 1/2$ in direction 2. Relying on structured controlled path expansions, we rigorously minimize the set of iterated integrals required in the signature to 4 elements. We extend the classical planar change-of-variable formula to this regime and express it as an explicit limit of Riemann sums. The signature elements required for this are then constructed for the fractional Brownian sheet with $1/3 < H_1 < 1/2$ and $H_2 > 1/2$. Secondly, we consider the fully rough regime, where $1/3 < H_1, H_2 < 1/2$ (joint work with S. Tindel). Here, a symmetric second-order Taylor expansion produces a planar integral with a signature of 10 elements instead of 36 compared to Chouk-Gubinelli’s paper. Third, we demonstrate that the class of controlled sheets is not stable under integration. However, in the Rough-Young regime, the rough sheet integral can be interpreted as a one-parameter infinite-dimensional rough path integral (ongoing work with M. R. Lahwate). This allows us to interpret some stochastic Goursat PDEs as infinite-dimensional RDEs.

I will talk about the problem of determining whether a vector-valued stochastic process or the solution to an SPDE system can hit a given set with positive probability. This problem remains open in critical dimension for general processes. I will present some recent results on hitting probabilities for Gaussian random fields, including the case of critical dimension.

Consider the Gaussian unitary ensemble (GUE), the most fundamental class of complex random matrices. The expectation of its trace power can be calculated as a weighted sum over gluings of polygons into closed orientable surfaces of genus 0, 1, 2, and so on. A similar result holds for the GOE, the ensemble of real symmetric Gaussian matrices, when one also sums over non-orientable surfaces. This so-called topological expansion, together with its beta-version, yields many fascinating results in random matrix theory, geometry, and algebraic combinatorics. We use this idea to study the edge limit of the Jack processes, a class of discrete N-dimensional Markov processes analogous to Dyson Brownian motion, and derive a new multi-time formula for the limiting Airy beta process. The formula admits a clear expansion in terms of genus, and its beta equals 2 case leads to a new explicit expression for the Witten–Kontsevich intersection numbers.

Solutions of the stochastic heat equation (SHE) with multiplicative noise exhibit intermittency—extreme peaks concentrated in small pockets of space and time. This phenomenon can be demonstrated explicitly by comparing the asymptotic exponential growth rates of the first and second moments of the solution to the SHE. We develop a spectral decomposition formula for the second moment (as a function of time) of the SHE with multiplicative noise through a combination of the Bethe Ansatz and contour integral formulas. In this talk, I will provide details of the construction of the second moment formula and demonstrate intermittency as a consequence.

Comparison principles are a key tool for studying invariant measures and convergence of interacting particle systems, and their best known source is monotonicity. Kinetically constrained models, such as the Fredrickson–Andersen and East models, arise from independent Bernoulli updates by forbidding updates unless a local constraint holds. This destroys monotonicity. Does any comparison principle survive? We show that for a broad class of spin systems, which we call patch-positive, the dynamics preserve an order on probability laws that compares joint moments of the spins taken around a reference density. As an application, we prove that the East model perturbed by independent noise is uniformly ergodic, a question left open in earlier work. Joint work with Maciej Gluchowski (UCLA).