Abstract: We consider random walks and Lévy processes in the free nilpotent Lie group as rough paths. For any p > 1, we completely characterise (almost) all Lévy processes whose sample paths have finite p-variation, provide a Lévy-Khintchine formula for the characteristic function of the signature of a Lévy process treated as a rough path, and give sufficient conditions under which a sequence of random walks converges weakly to a Lévy process in rough path topologies. At the heart of our analysis is a criterion for tightness of p-variation for a collection of càdlàg strong Markov processes. We demonstrate applications of our results to weak convergence of stochastic flows.

Abstract: A diffusion process on a Riemannian manifold whose generator is one half of the Laplacian is called a Brownian motion. The mean local time of Brownian motion on a hypersurface will be considered, as will the situation in which a Brownian motion is conditioned to arrive in a fixed submanifold at a fixed positive time. Doing so provides motivation for the remainder of the talk, in which a probabilistic formula for the integral of the heat kernel over a submanifold is proved and used to deduce lower bounds, an asymptotic relation and derivative estimates applicable to the conditioned process.

Abstract: The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuous-time martingale transport on the Skorokhod space of cadlag paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S-topology and the dynamic programming principle. This is a joint work with Gaoyue Guo and Nizar Touzi.

Abstract: Joint work with Oleg Zaboronsky (Warwick).

Some one dimensional nearest neighbour particle systems are examples of Pfaffian point processes - where all intensities are determined by a single kernel.In some cases these kernels have appeared in the random matrix literature (where the points are the positions of eigenvalues). We are attempting to use random matrix tools on the particle sytems, and particle tools on the random matrices.

Abstract:   Consider the square $[0,n]^2$ with points from a Poisson point process of intensity 1 distributed within it. In a seminal work, Baik, Deift and Johansson proved that the number of points $L_n$ (length) on a maximal increasing path (an increasing path that contains the most number of points), when properly centered and scaled, converges to the Tracy-Widom distribution. Later Johansson showed that all maximal paths lie within the strip of width $n^{\frac{2}{3} +\epsilon}$ around the diagonal with probability tending to 1 as $n \to \infty$. We shall discuss recent work on the Gaussian behaviour of the length $L_n^{(\gamma)}$ of a maximal increasing path restricted to lie within a strip of width $n^{\gamma}, \gamma< \frac{2}{3}$.

 

TBC

We consider a controlled system, in which an input $X: [0, T] \rightarrow E:= \mathbb{R}^{d}$ is a continuous but potentially highly oscillatory path and the corresponding output $Y$ is the line integral along $X$, for some unknown function $f: E \rightarrow E$. The rough paths theory provides a general framework to answer the question on which mild condition of $X$ and $f$, the integral $I(X)$ is well defined. It is robust enough to allow to treat stochastic integrals in a deterministic way. In this paper we are interested in identification of controlled systems of this type. The difficulty comes from the high dimensionality caused by the input of a function type. We propose novel adaptive and non-parametric algorithms to learn the functional relationship between the  input and the output from the data by carefully choosing the feature set of paths based on the rough paths theory and applying linear regression techniques. The algorithms is demonstrated on a financial application where the task is to predict the P$\&$L of the unknown trading strategy.

Blenders and food processors have been around for years.  However, detailed understanding of the fluid and particle dynamics going on with in the multi-phase flow of the processing chamber as well as the influence of variables such as the vessel geometry, blade geometry, speeds, surface properties etc., are not well understood.  SharkNinja would like Oxford Universities help in developing a model that can be used to gain insight into fluid dynamics within the food processing chamber with the goal being to develop a system that will produce better food processing performance as well as predict loading on food processing elements to enable data driven product design.

Many vacuum cleaners sold claim “no loss of suction” which is defined as having only a very small reduction in peak air power output over the life of the unit under normal operating conditions.  This is commonly achieved by having a high efficiency cyclonic separator combined with a filter which the user washes at regular intervals (typically every 3 months).  It has been observed that some vacuum cleaners show an increase in peak air watts output after a small amount of dust is deposited on the filter.  This effect is beneficial since it prolongs the time between filter washing.  SharkNinja are currently working on validating their theory as to why this occurs.  SharkNinja would like Oxford University’s help in developing a model that can be used to better understand this effect and provide insight towards optimizing future designs.

Although a very simple system from a construction standpoint, creating a drip coffee maker that can be produce a range of coffee sizes from a single cup to a multi-cup carafe presents unique problems.  Challenges within this system result from varying pressure heads on the inlet side, accurate measurement of relatively low flow rates, fluid motive force generated by boilers, and head above the boiler on the outlet side.  Getting all of these parameters right to deliver the proper strength, proper temp, and proper volume of coffee requires in depth understanding of the fluid dynamics involved in the system.  An ideal outcome from this work would be an adaptive model that enables a fluid system model to be created from building blocks.  This system model would include component models for tubing, boilers, flow meters, filters, pumps, check valves, and the like.

It has been conjectured that marine ice sheets (those that

flow into the ocean) are unconditionally unstable when the underlying

bed-slope runs uphill in the direction of flow, as is typical in many

regions underneath the West Antarctic Ice Sheet. This conjecture is

supported by theoretical studies that assume a two-dimensional flow

idealization. However, if the floating section (the ice shelf) is

subject to three-dimensional stresses from the edges of the embayments

into which they flow, as is typical of many ice shelves in Antarctica,

then the ice shelf creates a buttress that supports the ice sheet.

This allows the ice sheet to remain stable under conditions that may

otherwise result in collapse of the ice sheet. This talk presents new

theoretical and experimental results relating to the effects of

three-dimensional stresses on the flow and structure of ice shelves,

and their potential to stabilize marine ice sheets.