2:00 Julian Sieber

On the (Non-)stationary density of fractional SDEs

I will present a novel approach for studying the density of SDEs driven by additive fractional Brownian motion. It allows us to establish smoothness and Gaussian-type upper and lower bounds for both the non-stationary as well as the stationary density. While the stationary density has not been studied in any previous works, the former was the subject of multiple articles by Baudoin, Hairer, Nualart, Ouyang, Pillai, Tindel, among others. The common theme of all of these works is to obtain the results through bounds on the Malliavin derivative. The main disadvantage of this approach lies in the non-optimal regularity conditions on the SDE's coefficients. In case of additive noise, the equation is known to be well-posed if the drift is merely sublinear and measurable (resp. Holder continuous). Relying entirely on classical methods of stochastic analysis (avoiding any Malliavin calculus), we prove the aforementioned Gaussian-type bounds under optimal regularity conditions.

The talk is based on a joint work with Xue-Mei Li and Fabien Panloup.

2:45 Thomas Tendron

A central limit theorem for a spatial logistic branching process in the slow coalescence regime

We study the scaling limits of a spatial population dynamics model which describes the sizes of colonies located on the integer lattice, and allows for branching, coalescence in the form of local pairwise competition, and migration. When started near the local equilibrium, the rates of branching and coalescence in the particle system are both linear in the local population size - we say that the coalescence is slow. We identify a rescaling of the equilibrium fluctuations process under which it converges to an infinite dimensional Ornstein-Uhlenbeck process with alpha-stable driving noise if the offspring distribution lies in the domain of attraction of an alpha-stable law with alpha between one and two.

3:30 Break

4:00-5:30 Careers Discussion

2:30 -3.00 Will Turner (CDT Student, Imperial College London)

Topologies on unparameterised path space

The signature of a path is a non-commutative exponential introduced by K.T. Chen in the 1950s, and appears as a central object in the theory of rough paths developed by T. Lyons in the 1990s. For continuous paths of bounded variation, the signature may be realised as a sequence of iterated integrals, which provides a succinct summary for multimodal, irregularly sampled, time-ordered data. The terms in the signature act as an analogue to monomials for finite dimensional data: linear functionals on the signature uniformly approximate any compactly supported continuous function on unparameterised path space (Levin, Lyons, Ni 2013). Selection of a suitable topology on the space of unparameterised paths is then key to the practical use of this approximation theory. We present new results on the properties of several candidate topologies for this space. If time permits, we will relate these results to two classical models: the fixed-time solution of a controlled differential equation, and the expected signature model of Levin, Lyons, and Ni. This is joint work with Thomas Cass.

3.05 -3.35 Ross Zhang (CDT Student, University of Oxford)

Random vortex dynamics via functional stochastic differential equations

The talk focuses on the representation of the three-dimensional (3D) Navier-Stokes equations by a random vortex system. This new system could give us new numerical schemes to efficiently approximate the 3D incompressible fluid flows by Monte Carlo simulations. Compared with the 2D Navier-Stokes equation, the difficulty of the 3D Navier-Stokes equation lies in the stretching of vorticity. To handle the stretching term, a system of stochastic differential equations is coupled with a functional ordinary differential equation in the 3D random vortex system. Two main tools are developed to derive the new system: the first is the investigation of pinned diffusion measure, which describes the conditional distribution of a time reversal diffusion, and the second is a forward-type Feynman Kac formula for nonlinear PDEs, which utilizes the pinned diffusion measure to delicately overcome the time reversal issue in PDE. Although the main focus of the research is the Navier-stokes equation, the tools developed in this research are quite general. They could be applied to other nonlinear PDEs as well, thereby providing respective numerical schemes.

3.40 - 4.25pm Dr Cris Salvi (Imperial College London)

Signature kernel methods

Kernel methods provide a rich and elegant framework for a variety of learning tasks including supervised learning, hypothesis testing, Bayesian inference, generative modelling and scientific computing. Sequentially ordered information often arrives in the form of complex streams taking values in non-trivial ambient spaces (e.g. a video is a sequence of images). In these situations, the design of appropriate kernels is a notably challenging task. In this talk, I will outline how rough path theory, a modern mathematical framework for describing complex evolving systems, allows to construct a family of characteristic kernels on pathspace known as signature kernels. I will then present how signature kernels can be used to develop a variety of algorithms such as two-sample hypothesis and (conditional) independence tests for stochastic processes, generative models for time series and numerical methods for path-dependent PDEs.

4.30 Refreshments

We introduce so-called functional input neural networks defined on infinite dimensional weighted spaces, where we use an additive family as hidden layer maps and a non-linear activation function applied to each hidden layer. Relying on approximation theory based on Stone-Weierstrass and Nachbin type theorems on weighted spaces, we can prove global universal approximation results for (differentiable and) continuous functions going beyond approximation on compact sets. This applies in particular to approximation of (non-anticipative) path space functionals via functional input neural networks but also via linear maps of the signature of the respective paths. We apply these results in the context of stochastic portfolio theory to generate path dependent portfolios that are trained to outperform the market portfolio. The talk is based on joint works with Philipp Schmocker and Josef Teichmann.

In this talk, we present recent results on an anisotropic variant of the Kardar-Parisi-Zhang equation, the Anisotropic KPZ equation (AKPZ), in the critical spatial dimension d=2. This is a singular SPDE which is conjectured to capture the behaviour of the fluctuations of a large family of random surface growth phenomena but whose analysis falls outside of the scope not only of classical stochastic calculus but also of the theory of Regularity Structures and paracontrolled calculus. We first consider a regularised version of the AKPZ equation which preserves the invariant measure and prove the conjecture made in [Cannizzaro, Erhard, Toninelli, "The AKPZ equation at stationarity: logarithmic superdiffusivity"], i.e. we show that, at large scales, the correlation length grows like t1/2 (log t)1/4 up to lower order correction. Second, we prove that in the so-called weak coupling regime, i.e. the equation regularised at scale N and the coefficient of the nonlinearity tuned down by a factor (log N)-1/2, the AKPZ equation converges to a linear stochastic heat equation with renormalised coefficients. Time allowing, we will comment on how some of the techniques introduced can be applied to other SPDEs and physical systems at and above criticality.