Consider dY(t)=f(X(t))dX(t), where X(t) is a pure jump Levy process with finite p-variation norm, 1<= p < 2, and f is a Lipchitz continuous function. Following the geometric solution construction of Levy-driven stochastic differential equations in (Williams 2001), we develop a class of epsilon-strong simulation algorithms that allows us to construct a probability space, supporting both the geometric solution Y and a fully simulatable process Y_epsilon, such that Y_epsilon is within epsilon distance from Y under the uniform metric on compact time intervals with probability 1. Moreover, the users can adaptively choose epsilon’ < epsilon, so that Y_epsilon’ can be constructed conditional on Y_epsilon. This tolerance-enforcement feature allows us to easily combine our algorithm with Multilevel Monte Carlo for efficient estimation of expectations, and adding as a benefit a straightforward analysis of rates of convergence. This is joint with Jose Blanchet, Fei He and Offer Kella.

We present a support theorem for subcritical parabolic stochastic partial differential equations (SPDEs) driven by Gaussian noises. In the spirit of the classical theorem by Stroock and Varadhan for ordinary stochastic differential equations, we identify the support of the solution to singular SPDEs with the closure of the union of the support of solutions to approximate and renormalized equations. We implement our approach in the setting of regularity structures and obtain a general result covering a range of singular SPDEs (including $\Phi^4_3$, $\Phi^d_2$, KPZ, PAM (2D+3D), SHE, ...). As a Corollary to our result we obtain the uniqueness of invariant measures for various interesting SPDEs. This is a joint work with Martin Hairer.

We present a numerical investigation of stochastic transport for the damped and driven incompressible 2D Euler fluid flows. According to Holm (Proc Roy Soc, 2015) and Cotter et al. (2017), the principles of transformation theory and multi-time homogenisation, respectively, imply a physically meaningful, data-driven approach for decomposing the fluid transport velocity into its drift and stochastic parts, for a certain class of fluid flows. We develop a new methodology to implement this velocity decomposition and then numerically integrate the resulting stochastic partial differential equation using a finite element discretisation. We show our numerical method is consistent.
Numerically, we perform the following analyses on this velocity decomposition. We first perform uncertainty quantification tests on the Lagrangian trajectories by comparing an ensemble of realisations of Lagrangian trajectories driven by the stochastic differential equation, and the Lagrangian trajectory driven by the ordinary differential equation. We then perform uncertainty quantification tests on the resulting stochastic partial differential equation by comparing the coarse-grid realisations of solutions of the stochastic partial differential equation with the ``true solutions'' of the deterministic fluid partial differential equation, computed on a refined grid. In these experiments, we also investigate the effect of varying the ensemble size and the number of prescribed stochastic terms. Further experiments are done to show the uncertainty quantification results "converge" to the truth, as the spatial resolution of the coarse grid is refined, implying our methodology is consistent. The uncertainty quantification tests are supplemented by analysing the L2 distance between the SPDE solution ensemble and the PDE solution. Statistical tests are also done on the distribution of the solutions of the stochastic partial differential equation. The numerical results confirm the suitability of the new methodology for decomposing the fluid transport velocity into its drift and stochastic parts, in the case of damped and driven incompressible 2D Euler fluid flows. This is the first step of a larger data assimilation project which we are embarking on. This is joint work with Colin Cotter, Dan Crisan, Darryl Holm and Igor Shevchenko.

 

I will talk about recent work that uses recent ideas from stochastic analysis to develop robust and non-parametric statistical tests for stochastic processes. 

In this talk, we will show how sharp bounds on the moments of the solutions to some stochastic heat equations can lead to various qualitative properties of the solutions. A major part of the method consists of approximating the solution by “independent quantities”. These quantities together with the moments bounds give us sharp almost sure properties of the solution.

The study of the Geometry of random nodal domains has attracted a lot of attention in the recent past, in particular due to their connection with famous conjectures such as Yau's conjecture on the nodal volume of eigenfunctions of the Laplacian on compact manifolds, and Berry's conjecture on the relation between the geometry of the nodal sets associated to these eigenfunctions and the geometry of the nodal sets associated to toric random waves.

At first, the randomness involved in the definition of random nodal domains is often chosen of Gaussian nature. This allows in particular the use of explicit techniques, such as Kac--Rice formula, to derive the asymptotics of many observables of interest (nodal volume, number of connected components, Leray's measure etc.). In this talk, we will raise the question of the universality of these asymptotics, which consists in deciding if the asymptotic properties of random nodal domains do or do not depend on the particular nature of the randomness involved. Among other results, we will establish the local and global universality of the asymptotic volume associated to the set of real zeros of random trigonometric polynomials with high degree.

We shall explain, from variational point of view, why the  Laplaciian operator introduced by Ebin-Marsden using deformations is suitable to describe the fluid motion in a milieu with viscosity.

TBC

In this talk I will present three families of differential equations (SDEs, BSDEs and PDEs) and their links to each other. The novel fact is that some of the coefficients are generalised functions living in a fractional Sobolev space of negative order. I will discuss the appropriate notion of solution for each type of equation and show existence and uniqueness results. To do so, I will use tools from analysis like semigroup theory, pointwise products, theory of function spaces, as well as classical tools from probability and stochastic analysis. The link between these equations will play a fundamental role, in particular the results on the PDE are used to give a meaning and solve both the forward and the backward stochastic differential equations.  

Stochastic differential equations have Taylor expansions in terms of iterated Wiener integrals. The convergence of such expansion depends on the limiting behavior of the order-N iterated integrals as N tends to infinity. Recently, there has been increased interests in processes stopped at a random time. A breakthrough in the study of the iterated integrals of Brownian motion up to the exit time of a domain was included in the work of Lyons-Ni (2012). The paper leaves open an interesting question: what is the sharp rate of decay for the expected iterated integrals up to the exit time. We will review the state of the art in this problem and report some recent progress. Joint work with Ni Hao (UCL).