Abstract:

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.

# Stochastic Analysis Seminar

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

We provide in this work a robust solution theory for random rough differential equations of mean field type

$$

dX_t = V\big( X_t,{\mathcal L}(X_t)\big)dt + \textrm{F}\bigl( X_t,{\mathcal L}(X_t)\bigr) dW_t,

$$

where $W$ is a random rough path and ${\mathcal L}(X_t)$ stands for the law of $X_t$, with mean field interaction in both the drift and diffusivity. Propagation of chaos results for large systems of interacting rough differential equations are obtained as a consequence, with explicit convergence rate. The development of these results requires the introduction of a new rough path-like setting and an associated notion of controlled path. We use crucially Lions' approach to differential calculus on Wasserstein space along the way. This is a joint work with I. Bailleul and R. Catellier.

Joint work with I. Bailleul (Rennes) and R. Catellier (Nice)

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals, and they are central to the theory of rough paths in stochastic analysis. For some special families of curves, such as polynomial paths and piecewise-linear paths, their parametrized signature tensors trace out algebraic varieties in the space of all tensors. We introduce these varieties and examine their fundamental properties, while highlighting their intimate connection to the problem of recovering a path from its signature. This is joint work with Peter Friz and Bernd Sturmfels.

Stochastic analysis on a Riemannian manifold is a well developed area of research in probability theory.

We will discuss some recent developments on stochastic analysis on a manifold whose Riemannian metric evolves with time, a typical case of which is the Ricci flow. Familiar results such as stochastic parallel transport, integration by parts formula, martingale representation theorem, and functional inequalities have interesting extensions from

time independent metrics to time dependent ones. In particular, we will discuss an extension of Beckner’s inequality on the path space over a Riemannian manifold with time-dependent metrics. The classical version of this inequality includes the Poincare inequality and the logarithmic Sobolev inequality as special cases.

The celebrated Black-Scholes theory shows that one can get a risk-neutral option price through hedging. The Cameron-Martin-Girsanov theorem for diffusion processes plays a key role in this theory. We show that one can get some statistical arbitrage from a sequence of well-designed repeated trading at their prices according to the ergodic theorem for stationary process. Our result is based on both theoretical model and the real market data.

Recently, Deya, Gubinelli, Hofmanova and Tindel ('16) (also Bailleul-Gubinelli '15) have provided a general approach in order to obtain a priori estimates for rough partial differential equations of the form

(*) du = Au dt + Bu dX

where X is a two-step rough path, A is a second order operator (elliptic), while B is first order. We will pursue the line of this work by presenting an L^p theory "à la Krylov" for generalized versions of (*). We will give an application of this theory by proving boundedness of solutions for a certain class

We present an algebraic formulation for the flow of a differential equation driven by a path in a Lie group. The formulation is motivated by formal differential equations considered by Chen.

Based on classical invariant theory, I describe a complete set of elements of the signature that is invariant to the general linear group, rotations or permutations.

A geometric interpretation of some of these invariants will be given.

Joint work with Jeremy Reizenstein (Warwick).

In this paper, we investigate the limit properties of frequency of empirical averages when random variables are described by a set of probability measures and obtain a law of large numbers for upper-lower probabilities. Our result is an extension of the classical Kinchin's law of large numbers, but the proof is totally different.

keywords: Law of large numbers,capacity, non-additive probability, sub-linear expectation, indepence

paper by: Zengjing Chen School of Mathematics, Shandong University and Qingyang Liu Center for Economic Research, Shandong University

I will present a mathematical model for the genetic evolution of a population which is divided in discrete colonies along a linear habitat, and for which the population size of each colony is random and constant in time. I will show that, under reasonable assumptions on the distribution of the population sizes, over large spatial and temporal scales, this population can be described by the solution to a stochastic partial differential equation with constant coefficients. These coefficients describe the effective diffusion rate of genes within the population and its effective population density, which are both different from the mean population density and the mean diffusion rate of genes at the microscopic scale. To do this, I will present a duality technique and a new convergence result for coalescing random walks in a random environment.