# Seminars 2012/13

 08 Oct 2012 Peter Mörters University of Bath Clustering in spatial preferential attachment networks 15 Oct 2012 Martin Klimmek University of Oxford From inverse optimal stopping to BLJ embeddings 22 Oct 2012 Maren Eckhoff University of Bath Surviving near criticality in a preferential attachment network 29 Oct 2012 Matt Roberts University of Warwick Intermittency in branching random walk in random environment 05 Nov 2012 Edward Crane University of Bristol Antichains in random partial orders 26 Nov 2012 Jesse Goodman Leiden University The gaps left by Brownian motion on a torus 03 Dec 2012 Bálint Tóth University of Bristol Diffusivity of random walk in divergence free drift field in 2d 10 Dec 2012 Tony Shardlow University of Bristol Pathwise approximation of SDEs and adaptive time stepping 14 Jan 2013 Stefan Adams University of Warwick Random Field of Gradients 11 Feb 2013 Tiziano De Angelis University of Manchester Optimal stopping of a Hilbert space valued diffusion and applications to finance 18 Feb 2013 Stefan Grosskinsky University of Warwick Dynamics of condensation in inclusion processes 25 Feb 2013 Dafydd Evans University of Cardiff Fast anomaly detection in spatial point processes 04 Mar 2013 Johannes Ruf University of Oxford Nonnegative local martingales, Novikov's and Kazamaki's criteria, and the distribution of explosion times 11 Mar 2013 Antal Jarai University of Bath Electrical resistance of the low-dimensional critical branching random walk 20 Mar 2013 Charles Bordenave Toulouse University Spectrum of Markov generators on sparse random graphs 08 Apr 2013 Tim Rogers University of Bath Demographic noise leads to the spontaneous formation of species 15 Apr 2013 David Applebaum University of Sheffield Brownian motion, martingale transforms and Fourier multipliers on Lie groups 22 Apr 2013 Dafydd Evans University of Cardiff Fast anomaly detection in spatial point processes 29 Apr 2013 Yan Fyodorov Queen Mary, London Fluctuations and extreme values in multifractal patterns 13 May 2013 Denis Denisov University of Manchester Tail behaviour of stationary distribution for Markov chains with asymptotically zero drift 20 May 2013 Peter Mörters University of Bath Emergence of condensation in models of selection and mutation

## Abstracts

### 8th October 2012 - Peter Mörters (University of Bath)

Clustering in spatial preferential attachment networks
I define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favouring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limiting power law, and the average clustering coefficient of the networks converges to a positive limit. A phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths. The talk is based on joint work with Emmanuel Jacob (ENS Lyon).

### 15th October 2012 - Martin Klimmek (University of Oxford)

From inverse optimal stopping to BLJ embeddings
Connections between an old earthwork problem (transport material from a distribution at the depot to a target distribution) and the theory of model-independent pricing/hedging have led to new interest in Skorokhod embedding theory and generalized convex analysis. Underlying the burgeoning field of optimal martingale transport  is a classical problem in martingale theory/mathematical finance: find extremal martingales consistent with given marginals/call prices at fixed times. The two inverse problems considered here differ in two ways. The time-horizon is either random or infinite, and solutions must be diffusions: We begin by constructing diffusions consistent with given value functions for perpetual horizon stopping problems using generalized convex analysis and basic one-dimensional diffusion theory. We move on to characterize the diffusions with a given marginal at a random time, the solution leading us back to Skorokhod embeddings.

### 22nd October 2012 - Maren Eckhoff (University of Bath)

Surviving near criticality in a preferential attachment network
We study a dynamical network model in which at every time step a new vertex is added and attached to every existing vertex independently with a probability proportional to a concave function of its current degree. Locally, this network can be approximated by a typed branching random walk (BRW) with an absorbing barrier.
We investigate the size of the giant component in the network near criticality, or equivalently, the asymptotics of the survival probability of the BRW under a changing offspring distribution. The talk is based on joint work with Peter Mörters.

### 29th October 2012 - Matt Roberts (University of Warwick)

Intermittency in branching random walk in random environment
Over the last 20 years mathematicians have proved rigorously that the parabolic Andreson model shows the intermittency behaviour predicted by physicists. We shall see that a branching random walk in Pareto random environment displays the same qualitative behaviour, but with several important differences. This is work in progress with Marcel Ortgiese.

### 26th November 2012 - Jesse Goodman (Leiden University)

The gaps left by Brownian motion on a torus
Run a Brownian motion on a torus for a long time. How large are the random gaps left behind when the path is removed?
In three (or more) dimensions, we find that there is a deterministic spatial scale common to all the large gaps anywhere in the torus. Moreover, we can identify whether a gap of a given shape is likely to exist on this scale, in terms of a single parameter, the classical (Newtonian) capacity. I will describe why this allows us to identify a well-defined "component" structure in our random porous set.

### 10th December 2012 - Tony Shardlow (University of Bath)

Pathwise approximation of SDEs and adaptive time stepping
We give a pathwise convergence analysis of one step integrators of SDEs. The analysis is motivated by rough path theory and applies to a general class of one steps methods subject to a pathwise bound on the sum of the truncation errors. We show how the method is applied to the Euler method. We take particular interest in showing the pathwise convergence of an adaptive time stepping method based on bounded diffusions and establish that the constant in the pathwise error can be controlled.

### 14th January 2013 - Stefan Adams (University of Warwick)

Random fields of gradients are a class of model systems arising in the studies of random interfaces, random geometry, field theory, and elasticity theory. These random objects pose challenging problems for probabilists as even an a priori distribution involves strong correlations. Gradient fields are likely to be an universal class of models combining probability, analysis and physics in the study of critical phenomena. They emerge in the following three areas, effective models for random interfaces, Gaussian Free Fields (scaling limits), and mathematical models for the Cauchy-Born rule of materials, i.e., a microscopic approach to nonlinear elasticity. The latter class of models requires that interaction energies are non-convex functions of the gradients. Open problems include unicity of Gibbs measures and strict convexity of the free energy. We present in the talk a first break through for the free energy at low temperatures using Gaussian measures and rigorous renormalisation group techniques. In addition we show that the correlation functions have Gaussian decay properties despite the fact having non-convex interaction potentials. The key ingredient is a finite range decomposition for parameter dependent families of Gaussian measures. If time permits to discuss the connection of Gaussian Free Fields and interlacements.

### 11th February 2013 - Tiziano De Angelis (University of Manchester)

Optimal stopping of a Hilbert space valued diffusion and applications to finance
Pricing American Bond options in a market model with forward interest rates corre- sponds, in mathematical terms, to an optimal stopping problem of an infinite dimensional diffusion. Motivated by this financial application we analyse a finite horizon optimal stopping problem for an infinite dimensional diffusion $$X$$ by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space $$H$$ with a non-linear diffusion coefficient $$\sigma(X)$$ and a generic unbounded operator $$A$$ in the drift term. We show that when the gain function $$\Theta$$ is time-dependent and fulfils mild regularity assumptions, the value function $$U$$ of the optimal stopping problem solves an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. The solution of the variational problem is found in a suitable Banach space $$V$$ fully characterized in terms of a Gaussian measure $$\mu$$ associated to the coefficient $$\sigma(X)$$.

### 18th February 2013 - Stefan Grosskinsky (University of Warwick)

Dynamics of condensation in inclusion processes
The inclusion process is an interacting particle system where particles on connected sites attract each other in addition to performing independent random walks. The system has stationary product measures and exhibits condensation in the limit of strong interactions, where all particles concentrate on a single lattice site. We study the equilibration dynamics on finite lattices in the limit of infinitely many particles, which, in addition to jumps of whole clusters, contains an interesting continuous mass exchange between clusters given by Wright-Fisher diffusions. During equilibration the number of clusters decreases monotonically, and the stationary dynamics consist of jumps of a single remaining cluster (the condensate). This is joint work with Frank Redig and Kiamars Vafayi.

### 4th March 2013 - Johannes Ruf (University of Oxford)

Nonnegative local martingales, Novikov's and Kazamaki's criteria, and the distribution of explosion times
I will give a new proof for the famous criteria by Novikov and Kazamaki, which provide sufficient conditions for the martingale property of a nonnegative local martingale. The proof is based on an extension theorem for probability measures that can be considered as a generalization of a Girsanov-type change of measure. In the second part of my talk I will illustrate how a generalized Girsanov formula can be used to compute the distribution of the explosion time of a weak solution to a stochastic differential equation.
Parts of this talk are based on joined working papers with Martin Larsson and Ioannis Karatzas.

### 11th March 2013 - Antal Jarai (University of Bath)

Electrical resistance of the low-dimensional critical branching random walk
We consider the trace of a critical branching random walk in $$d+1$$ dimensions conditioned to survive forever. We show that the electrical resistance between the origin and generation $$n$$ grows sublinearly in $$n$$ when $$d<6$$. In particular, it follows that in $$d=5$$ the spectral dimension of simple random walk on the trace is strictly larger than 4/3, answering a question of Barlow, Jarai, Kumagai and Slade. (Joint work with Asaf Nachmias.)

### 20th March 2013 - Charles Bordenave (Toulouse University)

Spectrum of Markov generators on sparse random graphs
In this talk, we will consider various probability distributions on the set of stochastic matrices with n states and on the set of Laplacian/Kirchhoff matrices on n states. They will arise naturally from the conductance model on n states with i.i.d conductances. With the help of random matrix theory, we will study the spectrum of these processes.

### 8th April 2013 - Tim Rogers (University of Bath)

Demographic noise leads to the spontaneous formation of species
In this talk I will discuss an evolutionary model of competition, which is a microscopic stochastic analogue of a famous population-level model in ecology. I will show how the effects of demographic noise in the stochastic model give rise to radically different macro-scale behaviour. The (non-rigorous) analysis uses an expansion in system size, coupled with a time-scale separation argument.

### 15th April 2013 - David Applebaum (University of Sheffield)

Brownian motion, martingale transforms and Fourier multipliers on Lie groups
We associate a space-time martingale to Brownian motion on a Lie group $$G$$ and transform it to obtain a family of "differentially subordinate" martingales. Using powerful inequalities dues to Burkholder, Banuelos and Wang we construct a family of linear operators which are bounded on $$L^{p}(G, \tau)$$ (where $$\tau$$ is a Haar measure) for all $$1 < p < \infty$$. When $$G$$ is compact, we can utilise non-commutative Fourier analysis to represent these operators as Fourier multipliers. Examples include second order Riesz transforms and operators of Laplace transform type.

### 29th April 2013 - Yan Fyodorov (Queen Mary, London)

Fluctuations and extreme values in multifractal patterns
The goal is to understand sample-to-sample fluctuations in disorder-generated multifractal intensity patterns. Arguably the simplest model of that sort is the exponential of an ideal periodic 1/f Gaussian noise. The latter process can be looked at as a one-dimensional "projection" of 2D Gaussian Free Field and inherits from it the logarithmic covariance structure. It most naturally emerges in the random matrix theory context, but attracted also an independent interest in statistical mechanics of disordered systems. I will determine the threshold of extreme values of 1/f noise and provide a rather compelling explanation for the mechanism behind its universality. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disorder-generated multifractal fields.
The presentation will be mainly based on the joint work with Pierre Le Doussal and Alberto Rosso, J Stat Phys: 149 (2012), 898-920 as well as on some related earlier works by the speaker.

### 13th May 2013 - Denis Denisov (University of Manchester)

Tail behaviour of stationary distribution for Markov chains with asymptotically zero drift
We consider a one-dimensional Markov chain with asymptotically zero drift and finite second moments of jumps which is positive recurrent. A power-like asymptotic behaviour of the invariant tail distribution is proven; such a heavy-tailed invariant measure happens even if the jumps of the chain are bounded. Our analysis is based on test functions technique and on construction of a harmonic function. This is a joint work with Korshunov and Wachtel.