# Lévy ProcessesandAnomalous Diffusion

## The University of Manchester

This summer, the School of Mathematics at the University of Manchester will host a one-day meeting on the twin topics of Lévy processes and anomalous diffusion, with invited speakers from Imperial, Oxford and Bath. The meeting celebrates the recent appointments in the School of Mathematics of Yanghong Huang and Alex Watson.

## Speakers

Pedro Aceves Sanchez (Imperial)
Fractional diffusion limit of a linear kinetic transport equation in a bounded domainAbstract. In recent years, the study of evolution equations featuring a fractional Laplacian has received many attention due the fact that they have been successfully applied into the modelling of a wide variety of phenomena, ranging from biology, physics to finance. The stochastic process behind fractional operators is linked, in the whole space, to an $\alpha$-stable processes as opposed to the Laplacian operator which is linked to a Brownian stochastic process. // In addition, evolution equations involving fractional Laplacians offer new interesting and very challenging mathematical problems. There are several equivalent definitions of the fractional Laplacian in the whole domain, however, in a bounded domain there are several options depending on the stochastic process considered. // In this talk we shall present results on the rigorous passage from a velocity jumping stochastic process in a bounded domain to a macroscopic evolution equation featuring a fractional Laplace operator. More precisely, we shall consider the long-time/small mean-free path asymptotic behaviour of the solutions of a re-scaled linear kinetic transport equation in a smooth bounded domain.
Sergei Fedotov (Manchester)
Anomalous diffusion on scale-free networksAbstract. We model transport of individuals across a heterogeneous scale-free network where a few weakly connected nodes exhibit heavy-tailed residence times. Using the empirical law of the axiom of cumulative inertia and fractional analysis, we show that “anomalous cumulative inertia” overpowers highly connected nodes in attracting network individuals. This fundamentally challenges the classical result that individuals tend to accumulate in high-order nodes. The derived residence time distribution has a non-trivial U shape which we encounter empirically across human residence and employment times.
Christina Goldschmidt (Oxford)
Critical random graphs with i.i.d. random degrees having power-law tails Abstract. Consider a graph with label set $\{1,2, \ldots,n\}$ chosen uniformly at random from those such that vertex i has degree $D_i$, where $D_1, D_2, \ldots, D_n$ are i.i.d. strictly positive random variables. The threshold for the emergence of a giant component in this setting is well known to be $E[D^2] = 2 E[D]$, and we assume additionally that $P(D = k) \sim c k^{-(\alpha + 2)}$ as $k$ tends to infinity, for some $\alpha \in (1,2)$ and some constant $c >0$. In this situation, it turns out that the largest components have sizes on the order of $n^{\alpha/(\alpha+1)}$. Building on earlier work of Joseph, we show that the components have scaling limits constructed from spanning trees which are absolutely continuous with respect to the corresponding limits for critical Galton-Watson trees with offspring distribution in the domain of attraction of an $\alpha$-stable law, plus some vertex-identifications. This gives a natural generalisation of the scaling limit for the Erd\H{o}s-Renyi random graph, and complements recent work on random graph scaling limits of various authors including Bhamidi, Broutin, Duquesne, van der Hofstad, van Leeuwaarden, Riordan, Sen, M. Wang and X. Wang. This is joint work in progress with Guillaume Conchon-Kerjan (ENS Paris).
Yanghong Huang (Manchester)
Finite difference methods for fractional LaplacianAbstract. The fractional Laplacian is the prototypical example of non-local diffusion, and has been employed in many models with long range interactions. In this talk, several finite difference methods in one dimension are reviewed or derived. The general form of the scheme has many discrete counterparts of its continuous definition: discrete convolution, random walk, and multiplier (or symbol) in semi-discrete Fourier transform. Despite the non-locality, the accuracy of different schemes can be obtained from the symbol, and is verified numerically. The schemes are also compared under different criteria, and can be chosen according to the applications. This is a joint work with Adam Oberman from McGill University.
Andreas Kyprianou (Bath)
Sphere stepping algorithms for Dirichlet-type problems with the fractional Laplacian
Sara Merino-Aceituno (Imperial)
Anomalous energy transport in one-dimensional crystalsAbstract. One-dimensional solid crystals are modelled through a chain of oscillators called Fermi-Pasta-Ulam chain. When atoms in the crystal are assumed to interact through a quartic potential, we prove that the macroscopic equation describing energy transport corresponds to a fractional heat equation rather than the standard heat equation. The proof uses an intermediate step between the discrete chain model and the macroscopic one; the phonon-Boltzmann equation, which describes vibrations through the chain. (This is a joint work with Dr Mellet from the University of Maryland).
Alex Watson (Manchester)
A probabilistic approach to spectral analysis of growth-fragmentation equationsAbstract. The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this talk, I discuss a probabilistic approach to the study of this asymptotic behaviour. The method is based on the Feynman-Kac formula and the identification of a driving Markov process. This is joint work with Jean Bertoin.

## Location

The meeting will take place in the School of Mathematics of the University of Manchester. The venue is the Frank Adams 1 seminar room on the first floor of Alan Turing Building.

Alan Turing Building is building 46 on the Campus Map. For more details on reaching the venue, please see Maps and Travel.

The room has stepless access.

## Programme

 09:30 Arrival & coffee 10:00 Sergei Fedotov 10:45 Christina Goldschmidt 11:30 Pedro Aceves Sanchez 12:15 Lunch 13:30 Andreas Kyprianou 14:15 Sara Merino-Aceituno 15:00 Coffee 15:30 Alex Watson 16:15 Yanghong Huang 17:00 Reception

Coffee and a reception for all participants will be provided.

## Registration

Registration has now closed.

## Contact

The organising committee is:

• Yanghong Huang
• Alex Watson

If you have any questions, please email Alex Watson.

This meeting is funded by a London Mathematical Society conference grant and a grant from the Manchester Institute for Mathematical Sciences.