Brownian motion is the most important stochastic process: it appears universally as the scaling limit of random walks whose step sizes have finite variance (this is Donsker’s invariance principle); is intimately connected to partial differential equations arising in classical analysis, such as those describing heat flow and electrostatic potential; plays a central role in stochastic calculus; and is an important building block for more complex processes, such as the recently-developed Schramm-Loewner Evolution, which appears in the description of various random interfaces. Brownian motion also falls into the wider class of Levy processes, which are defined to be random processes with stationary, independent increments – a characterisation that means they can be thought of as the continuous time analogues of random walks. Note that, whereas Brownian motion is always a continuous process, Levy processes can in general admit jumps, and this extra generality makes them extremely attractive from a modelling point of view.

Although the properties of Brownian motion and Levy processes have already been well-studied, many more questions about these processes remain. Wilfrid Kendall, for example, has been working on foundational problems concerned with coupling of Brownian motions, such as whether one can co-adaptively couple Brownian motion on nilpotent Lie groups, and also when it is possible to generate so-called shy couplings (when the two coupled Brownian motions avoid each other). There are significant and surprising interactions with pursuit problems on the one hand, and modern metric geometry on the other. David Hobson has been working on solutions of the Skorokhod embedding problem and links with martingale optimal transport. Given a stochastic process and a target law (for example, real valued Brownian motion and a centred probability measure) the Skorokhod embedding problem is to give a stopping time such that the law of the stopped process is equal to the given target law. Since there are potentially many solutions to the problem, one can look for solutions with nice additional properties, or optimality properties. Others at Warwick working in the general area of Brownian motion and Levy processes include Larbi Alili and Jon Warren.

Stochastic simulation is increasingly important for understanding visualisation and estimation of of stochastic processes. These problems are also strongly motivated by problems in Statistics, Biology, Physics and many other areas. Warwick has a large and very active group working on problems in this area. Some of this work involves the study of algorithms (such as Markov chain Monte Carlo) which are prevalent in Computer Science and Statistics. Paul Jenkins, Adam Johansen, Wilfrid Kendall, Jere Koskela, Krzysztof Łatuszyński, Alex Mijatovic, Gareth Roberts, and Dario Spano are working on this topic.

There are many instances in the physical sciences when an insight into the properties of a disordered medium (that is, a material that admits certain defects or inhomogeneities) can be gained through the study of a mathematical model that incorporates an element of randomness. Although such models are often simple to define, they can nonetheless lead to a rich array of behaviour, and their understanding can present considerable research challenges. Those working in this area at Warwick include: Jon Warren, whose research on random matrices also has connections to random tilings and surface growth models and Nikos Zygouras, who studies the behaviour of polymers in random potentials.

Markov processes are a special class of stochastic processes for which a memoryless property, whereby the future only depends on the present, not the past, is a defining characteristic. Both Brownian motion and Levy processes are examples. Functional analytic and stochastic differential equation (SDE) methods for the analysis of Markov processes with pseudo-differential generators (including stable-like and nonlinear) have been developed by Vassili Kolokoltsov. He is also working on continuous time random walks and related processes with memory described by fractional differential equations. Another academic at Warwick with an interest in the latter area is Dario Spano, who also has a project focusing on orthogonal polynomials and related Markov processes.

Important classes of Markov processes are measure-valued and partition-valued stochastic processes. To be more specific, in many applied contexts, the object under study is the dynamics of an infinite-dimensional quantity represented by a time-dependent random measure. In Bayesian statistics, for example, measure-valued processes are used as prior measure in time-varying nonparametric statistical models, and in population genetics they are employed to describe the evolutionary behaviour of the allele frequencies in a population of genes that can take on infinitely many types. Important problems in this area are to understand their path and spectral properties, and to describe their transition function explicitly. To do so, it is often convenient to identify dual stochastic processes with a countable representation. Partition-valued processes are a powerful class of combinatorial stochastic processes which in many cases serve such a purpose. The analysis of the interplay between measure-valued processes and their duals also involves uses of other random combinatorial structures (such as coagulation-fragmentation processes) and material from spectral theory and special functions theory. The relationship between such diverse areas has not been fully understood yet and is a source of many exciting possible directions of research, some of which are being pursued by Dario Spano. Measure-valued processes are also naturally connected with interacting particle models, see Vassili Kolokoltsov.

Some of the most important open problems in modern probability theory, both from an applied and a theoretical perspective, require us to develop a good understanding of the geometry of large, random discrete structures. To mention a few examples, consider (on the applied side) large random networks in biology and urban planning and the famous random satisfiability problem (k-SAT) from computer science; or (on the pure side) the geometry of random graphs, planar surfaces (or planar maps) and quantum gravity. Often in this area, analysis of the object considered is made particularly difficult by its high-dimensional or fractal nature, which can lead to a complex geometry with counter-intuitive properties, and a key challenge is to construct a scaling limit that will add to its understanding. As a prototypical example, one could consider the uniform random combinatorial tree, about which much has been learnt from the results that demonstrate it converges to a particular random fractal with a high-degree of stochastic self-similarity – the Brownian continuum random tree (CRT). Work at Warwick that focuses on random geometries includes that of Wilfrid Kendall, who has recently been studying the transport properties of spatial networks, and whose work on central limit phenomena for Riemannian barycentres has, in a surprising recent development, led to an entirely new perspective on vector-valued forms of the definitive work of Lindeberg and Feller on the central limit theorem.

Rough path theory, initiated by Terry Lyons, is motivated by the study of stochastic differential equations. Stochastic differential equation can be thought of as differential equations driven by the sample paths of Brownian motion. Rough path theory considers the same differential equation that is driven by an arbitrary path. In order for a differential equation driven by a path to make sense, rough path theory requires the iterated integrals of the path to be pre-specified and to satisfy certain analytic and algebraic conditions. A key feature of the theory is that the solution to the differential equation depends continuously on the driving path. A particularly fundamental example of such differential equation is the equation for the signature of a rough path. The signature equation is the first differential equation considered in rough path theory. The signature equation shares many properties with other differential equations, but also has some distinct features. For instance, the signatures uniquely determine paths up to the tree-like equivalence relation. An interesting question has been how to deduce the geometric properties of a path from its signature. A number of methods have been put forward, though there remain many important conjectures that have not yet been resolved. Horatio Boedihardjo is interested in relating variational norms of a path to its signature, as well as other problems in the study of path signature.

The typical objective of stochastic control theory is to identify strategies for obtaining a desired output in a dynamical system that is driven by a stochastic process. An illustrative application might be in determining how an investor should manage his portfolio so as to maximise his utility when investing in a financial market. Within this area, there is currently a focus on questions of optimal stopping, which concern the case when the problem is to determine the best time to take an action. A particularly famous example of such a problem is the so-called `secretary problem’, in which an employer wishes to find a procedure for hiring the best possible candidate for a secretarial position, but is only allowed to interview one candidate at a time, and must make the decision to hire or reject each candidate immediately after their interview. Optimal stopping is also relevant when trying to determine the moment at which a financial contract should be exercised or an asset sold, for instance. Amongst others at Warwick, Sigurd Assing, David Hobson and Saul Jacka are actively working on problems of this type. A related area that is an interest of Vassili Kolokoltsov is game theory, which concerns competitive versions of control and has a variety of applications in finance, evolutionary biology, statistical physics and econophysics.

Larbi Alili, David Hobson, Saul Jacka, and Vassili Kolokoltsov are all probabilists in the Department with a research interest in finance. More about the activities of this group can be read here.

Stochastic models of evolution seek to predict patterns of genetic diversity arising from evolutionary forces such as random mating, mutation, natural selection, and fluctuations in population size. They form the foundations of statistical inference methods for the ever-increasing volume of DNA sequence data, and are also drivers of research in probability and stochastic processes. Typical models come in pairs: one process describing the forward-in-time dynamics of population frequencies of genetic variants, and a reverse-time process of coalescing lineages which describes the common ancestry of a sample of individuals from the population. Prototypical examples include the Wright-Fisher diffusion and the Kingman coalescent, as well as more general (potentially measure-valued) jump diffusions and branching-coalescing random graphs. Probabilistic advances motivated by such models include so-called lookdown constructions in which the pair of processes are constructed on the same probability space, the notion of "coming down from infinity" where the ancestral process of a countably infinite number of lineages can either stay infinite or coalesce to a finite number of common ancestors, and separation of timescales phenomena in which "fast" dynamics occurring on the timescale of generations can be approximated by simple, mean-field processes on the much longer timescale of genetic evolution. Researchers at Warwick with interests in these phenomena include Paul Jenkins, Jere Koskela, and Dario Spano.

Research in stochastic partial differential equations (SPDEs) lies at the interface of analysis and probability theory. For example, in trying to understand how solutions of macroscopic partial differential equations can be approximated by scaled particle systems at the microscopic level, one of the main questions is about the structure of the fluctuations of the approximating system around its hydrodynamic limit. This limit can often be described by an SPDE of linear type with a Gaussian solution. However, if the space explored by the particle system lacks enough degrees of freedom, as in the case of the one-dimensional asymmetric simple exclusion process for instance, then the fluctuations become non-Gaussian and can only be described (if indeed they can be described at all) by a non-linear SPDE. The so-called current fluctuations, crucial for the understanding of the non-linearity, are linked with the distribution of eigenvalues of random matrices. Other problems are the limiting behaviour of systems with multiple temporal and spatial scales, the numerical analysis of approximations to SPDEs and the construction and analysis of MCMC methods in high dimensional spaces, which provide a probabilistic tool for understanding statistical sampling techniques. Recent developments include mean-field games and nonlinear Markov processes. Sigurd Assing, Vassili Kolokoltsov, Jon Warren and Nikos Zygouras are amongst those at Warwick who are involved in studies in this area.