In certain cases of (linear) partial differential equations random perturbations have been observed to cause regularizing effects, in some cases even producing the uniqueness of solutions. In view of the long-standing open problems of uniqueness of solutions for certain PDE arising in fluid dynamics such results are of particular interest. In this talk we will extend some known results concerning the well-posedness by noise for linear transport equations to the nonlinear case.

# Past Stochastic Analysis Seminar

Recent work in regularity structures has provided a robust solution theory for a wide class of singular SPDEs. While much progress has been made on understanding the analytic and algebraic aspects of renormalisation of the driving signal, the action of the renormalisation group on the equation still needed to be performed by hand. In this talk, we aim to give a systematic description of the renormalisation procedure directly on the level of the PDE, which allows for explicit computation of the form of the renormalised equation. Joint work with Yvain Bruned, Ajay Chandra, and Martin Hairer.

In this work we study a stochastic three-dimensional Landau-Lifschitz-Gilbert equation perturbed by pure jump noise in the Marcus canonical form. We show existence of weak martingale solutions taking values in a two-dimensional sphere $\mathbb{S}^3$ and discuss certain regularity results. The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. This is a joint work with Utpal Manna (Triva

This talk will address a new link from stochastic differential equations (SDEs) to nonlinear parabolic PDEs. Starting from the necessary and sufficient condition of the path-independence of the density of Girsanov transform for SDEs, we derive characterisation by nonlinear parabolic equations of Burgers-KPZ type. Extensions to the case of SDEs on differential manifolds and the case od SDEs with jumps as well as to that of (infinite dimensional) SDEs on separable Hilbert spaces will be discussed. A perspective to stochastically deformed dynamical systems will be briefly considered.

I will described how ideas from constructive quantum field theory can be adapted to produce a systematic approach for analytic renormalization in the theory of regularity structures.

The Hastings-Levitov models describe the growth of random sets (or clusters) in the complex plane as the result of iterated composition of random conformal maps. The correlations between these maps are determined by the harmonic measure density profile on the boundary of the clusters. In this talk I will focus on the simplest case, that of i.i.d. conformal maps, and obtain a description of the local fluctuations of the harmonic measure density around its deterministic limit, showing that these are Gaussian. This is joint work with James Norris.

If we fix a rectangle in the affine real space and if we choose at random a real polynomial with given degree d, the probability P(d) that a component of its vanishing locus crosses the rectangle in its length is clearly positive. But is P(d) uniformly bounded from below when d increases? I will explain a positive answer to a very close question involving real analytic functions. This is a joint work with Vincent Beffara.

The Ising model is one of the most classical statistical mechanics model, which has seen spectacular mathematical and physical developments for almost a century. The description of its scaling limit at the phase transition is at the center of a fascinating (conjectured) connection between statistical mechanics and field theories. I will discuss how recent mathematical progress allows one to make the connection between the two-dimensional Ising model and Conformal Field Theory rigorous. If time allows, I will discuss the insight this gives one into related models and field theories.

Based off joint works with S. Benoist, D. Chelkak, H. Duminil-Copin, R. Gheissari, K. Izyurov, F. Johansson-Viklund, K. Kytölä, S. Park and S. Smirnov

If a dynamical system has a conservation law, i.e. a constant along the trajectory of the motion, the study of its evolution along the trajectories of a perturbed system becomes interesting. Conservation laws can be seen everywhere, especially at the level of probability distributions of a reduced dynamic. We explain this with a number of models, in which we see a singular perturbation problem and identify a conservation law, the latter is used to seek out the correct scale to work with and to reduce the complexity of the system. The reduced dynamic consists of a family of ODEs with rapidly oscillating right hands side from which in the limit we obtain a Markov process. For stochastic completely integrable system, the limit describes the evolution of the level sets of the family of Hamiltonian functions over a very large time scale.

The Yang-Mills heat equation is the gradient flow corresponding to the Yang-Mills functional. It was initially introduced by S. K. Donaldson to study the existence of irreducible Yang-Mills connections on the projective plane. In this talk, we will consider this equation over compact three-manifolds with boundary. It is a nonlinear weakly parabolic equation, but we will see how one can prove long-time existence and uniqueness of solutions by gauge symmetry breaking. We will also demonstrate some strong regularization results for the solution and see how they lead to detailed short-time asymptotic estimates, as well as the long-time convergence of the Wilson loop functions.