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Abstracts
D. Applebaum (University of Sheffield)
Infinite Dimensional Ornstein-Uhlenbeck Processes with Jumps
I will discuss how infinite dimensional Ornstein-Uhlenbeck processes with jumps
arise naturally as solutions of stochastic PDEs with Levy noise, and show how they
give rise to a fascinating class of operators forming the so-called generalised
Mehler semigroups. Natural questions to investigate are (a) when do these
semigroups have an invariant measure, and (b) if an invariant measure exists,
when is the semigroup self-adjoint in the corresponding L^{2} space? The first
problem was solved nearly 30 years ago by Anna Chojnowska-Michalik. The
second question is unsolved except in the Gaussian case; however Ben Goldys
(Sydney) and I have conjectured that there are no other possibilities for selfadjointness other than the known Gaussian ones. If time allows, I will present
some evidence to support this conjecture.
A. Baule (Queen Mary University of London)
Feynman-Kac equation for anomalous diffusion processes with general waiting
times
Many transport processes in nature exhibit anomalous diffusive properties with a
nontrivial scaling of the mean square displacement, e.g., diffusion of cells or of
biomolecules inside the cell nucleus, where typically a crossover between
different scaling regimes appears over time. Here, we investigate a class of
anomalous diffusion processes that is able to capture such complex dynamics by
virtue of a general waiting time distribution [1]. We obtain a complete
characterization of such generalized anomalous processes, including their
functionals and multipoint structure, using a representation in terms of a normal
diffusive process plus a stochastic time change with an inverse Levy subordinator.
We show that the Feynman-Kac equation contains a memory term that is related
to the Laplace exponent of the waiting time process.
Reference
[1] A. Cairoli and A. Baule, Physical Review Letters 115, 110601 (2015)
C. Chimisov ( University of Warwick)
Adapting the Gibbs Sampler
Cyril Chimisov 1, Krys Latuszynski 2, and Gareth O. Roberts 3
Department of Statistics, University of Warwick, UK
1 E{mail: [email protected]
2 E{mail: [email protected]
3 E{mail: [email protected]
The popularity of Adaptive MCMC has been fueled on the one hand by its
success in applications, and on the other hand, by mathematically appealing
and computationally straightforward optimisation criteria for the Metropolis
algorithm acceptance rate (and, equivalently, proposal scale). Similarly
principled and operational criteria for optimising the selection probabilities of
the Random Scan Gibbs Sampler have not been devised to date.
In the present work we close this gap and develop a general purpose Adaptive
Random Scan Gibbs Sampler that adapts the selection probabilities. The
adaptation is guided by optimising the L2�spectral gap for the target's Gaussian
analogue [1,3], gradually, as target's global covariance is learned by the sampler.
The additional computational cost of the adaptation represents a small fraction
of the total simulation effort.
We present a number of moderately- and high-dimensional examples, including
Truncated Normals, Bayesian Hierarchical Models and Hidden Markov
Models, where signi_cant computational gains are empirically observed for both,
Adaptive Gibbs, and Adaptive Metropolis within Adaptive Gibbs version of the
algorithm, and where formal convergence is guaranteed by [2]. We argue that
Adaptive Random Scan Gibbs Samplers can be routinely implemented and
substantial computational gains will be observed across many typical Gibbs
References
[1] Amit, Y. Convergence properties of the Gibbs sampler for perturbations of
Gaussians, "The Annals of Statistics", 1996.
[2] Latuszynski, K., Roberts, G. O., and Rosenthal, J. S. Adaptive Gibbs samplers and related MCMC methods, "The Annals of Applied Probability",
2013.
[3] Roberts, G. O., and Sahu, S. K. Updating Schemes, Correlation Structure,
Blocking and Parameterization for the Gibbs Sampler, "Journal of the Royal
Statistical Society", 1997.
M Foondun (Loughborough University)
Some recent results about fractional heat equations.
I will describe some recent results concerning the fractional heat equations
driven by both white and colored noise. I will begin by describing the main
driving force behind those results and then describe how the fractional operator
can produce significant change in the behaviour of these equations. And if time
permits, I will give an idea of some proofs.
N. Georgiou (University of Sussex)
Solvable non-Markovian dynamic network
With Istvan Z. Kiss1 and Enrico Scalas1
1 Department of Mathematics, University of Sussex, UK
Non-Markovian processes are widespread in natural and human-made systems,
yet explicit modeling and analysis of such systems is underdeveloped. We
consider a non-Markovian dynamic network with random link activation and
deletion (RLAD) and heavy-tailed Mittag-Leffler distribution for the interevent
times. We derive an analytically and computationally tractable system of
Kolmogorov-like forward equations utilizing the Caputo derivative for the
probability of having a given number of active links in the network and solve them.
Simulations for the RLAD are also studied for power-law interevent times and we
show excellent agreement with the Mittag-Leffler model. This agreement holds
even when the RLAD network dynamics is coupled with the susceptible-infectedsusceptible spreading dynamics. Thus, the analytically solvable Mittag-Leffler
model provides an excellent approximation to the case when the network
dynamics is characterized by power-law-distributed interevent times. We further
discuss possible generalizations of our result [1].
Reference
[1] Nicos Georgiou, Istvan Z. Kiss, and Enrico Scalas, Solvable non-Markovian
dynamic network, Phys. Rev. E 92, 042801.
N. Jacob (Swansea University)
Transition densities of Lévy and Lévy-type processes in a broader context
Some of our recent studies suggest that densities of certain Lévy processes, i.e.
heat kernels of certain pseudo-differential operators with negative definite
symbols, should be seen as belonging to a larger class of objects. There are
interesting relations to densities proportional to characteristic functions and
additive processes, a new context emerges when looking at higher order
(fractional) powers of the Laplacian, and even modified classical mechanics seems
to be helpful, maybe necessary for a better understanding of some properties. In
this talk we will indicate some ideas and discuss some first results in more detail.
R. Klages (Queen Mary University of London)
Fluctuation relations for correlated Gaussian stochastic processes
With Aleksei V. Chechkin1,2 , Peter Dieterich3 , F.Lenz4
1 Institute for Theoretical Physics NSC KIPT, Kharkov, Ukraine
2 Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
3 Institut fuer Physiologie, Technische Universit¨at Dresden, Germany
4 Queen Mary University of London, School of Mathematical Sciences
Fluctuation relations (FRs) emerged as a key concept in nonequilibrium statistical
physics for assessing fluctuations very far from equilibrium [1]. For stochastic
processes generating normal diffusion they have been found to exhibit a
characteristic large deviation form. We test them for stochastic processes
generating anomalous diffusion [2]. As an example, we consider correlated
Gaussian stochastic dynamics by using a Langevin approach with two different
types of additive noise: (i) internal noise where the fluctuation-dissipation
relation of the second kind (FDR II) holds, and (ii) external noise without FDR II.
For internal noise the existence of FDR II implies the existence of the fluctuationdissipation relation of the first kind (FDR I), which in turn leads to conventional
(normal) forms of transient work FRs. For systems driven by external noise we
obtain violations of normal FRs [3]. Similar violations of FRs are observed in
computer simulations of glassy dynamics and in experiments on biological cell
migration.
References
[1] R. Klages, W. Just, C. Jarzynski (Eds.), Nonequilibrium Statistical Physics of
Small Systems: Fluctuation relations and beyond, Wiley-VCH (2013).
[2] R. Klages, G.Radons, I.M.Sokolov (Eds.), Anomalous transport: Foundations and
applications, Wiley-VCH (2008) .
[3] A. V. Chechkin, F. Lenz, R. Klages, J. Stat. Mech. L11001 (2012).
V. Kolokoltsov (University of Warwick, UK)
Fractional differential equations with two-sided and multidimensional boundaries
The probabilistic approach for solving Caputo type fractional differential
equations, suggested recently by the author, will be used to extend the theory of
fractional differential equations to the domains with two-sided or multidimensional boundary.
J. Lörinczi (Loughborough University)
Embedded eigenvalues and zero-energy resonances for the relativistic Schrödinger
operator
A celebrated result in the classical theory of Schrödinger operators, due to von
Neumann and Wigner, says that by carefully choosing the potential an eigenvalue
embedded in the essential spectrum occurs. In this talk I will present a similar
situation for the relativistic operator with non-zero rest mass. For the massless
case I will present examples of bound states or resonances at zero energy, and by
using a probabilistic representation will explain some mechanisms lying behind
these phenomena.
F. Mainardi (University of Bologna, Italy)
Anomalous Diffusion: Stochastic Processes Based on Fractional Calculus
In recent decades processes of anomalous diffusion have won more and more
interest in applications to applied sciences, and even to finance. In this lecture we
give a basic view of these processes by using the essential tools of the so-called
fractional calculus. Indeed, the governing equations contain pseudo-differential,
non-local operators that can be interpreted as space and/or time derivatives of
non-integer order. Our analysis is limited to the most simple models to be
interpreted in probability theory as peculiar stochastic processes.
References
[1] R. Gorenflo, A.A. Kilbas, F. Mainardi, S. Rogosin, Mittag-Leffler Functions.
Related Topics and Applications, Springer, Berlin (2014), pp. XII+ 420 [Springer
Monographs in Mathematics]
[2] R Gorenflo and F Mainardi, Parametric Subordination in Fractional Diffusion
Processes, In: J. Klafter, S.C. Lim and R. Metzler (Editors): Fractional Dynamics,
Recent Advances, World Scientific, Singapore (2012), Chapter 10, pp 227–261. [Eprint http://arxiv.org/abs/1210.8414]
[3] F. Mainardi, Yu Luchko and G. Pagnini, The fundamental solution of the spacetime fractional diffusion equation, Fractional Calculus and Applied Analysis 4, No
2, 153–192 (2001). [E-print http://arxiv.org/abs/cond-mat/0702419]
Nikolai N Leonenko (Cardiff University, UK)
Fractional Poisson Random Fields
We present new properties for the Fractional Poisson process introduced by
Mainardi, Gorenflo and Scalas [3] and the Fractional Poisson fields on the plane
[2]. A martingale characterization for Fractional Poisson processes is given. We
extend this result to Fractional Poisson fields, obtaining some other
characterizations.
The fractional differential equations are studied. The covariance structure is
given. Finally, we give some simulations of the Fractional Poisson fields on the
plane.
Joint work with G.Aletti (University of Milan, Italy) and E. Merzbach (Bar Ilan
University, Israel).
References
[1] Aletti, G., Leonenko, N.N. and Marzbach, E. (2016) Fractional Poisson fields
and martingales, submitted, http://arxiv.org/pdf/1601.08136.pdf
[2] Leonenko, N.N. and Merzbach, E.(2015) Fractional Poisson
fields, Methodology and Computing in Applied Probability, 17, 155-168
[3] Mainardi, F., Gorenflo, R. and Scalas, E. (2004) A fractional generalization of
the Poisson processes, Vietnam J. Math. 32, 53--64.
Enrico Scalas (University of Sussex)
Continuous-time random statistics and anomalous relaxation
The relation between continuous-time random statistics and anomalous
relaxation equations is explored, where a non-local operator replaces the local
derivative. This generalise results presented by Mainardi, Raberto, Gorenflo and
Scalas [1] and, more recently, by Meerschaert and Toaldo [2]. Several examples of
this relation will be discussed.
References
[1] Francesco Mainardi, Marco Raberto, Rudolf Gorenflo, Enrico Scalas (2000),
Fractional calculus and continuous-time finance II: the waiting-time distribution,
Physica A 287, 468-481.
[2] Mark M. Meerschaert and Bruno Toaldo (2015), Relaxation patterns and semiMarkov dynamics: arXiv:1506.02951.
Mailan Trinh (University of Sussex)
The fractional non-homogeneous Poisson process
We introduce a non-homogeneous fractional Poisson process by replacing the
time variable in the fractional Poisson process of renewal type with an
appropriate function of time. We characterize the resulting process by deriving
its non-local governing equation. We further compute the first and second
moments of the process. Eventually, we derive the distribution of arrival times.
Constant reference is made to previous known results in the homogeneous case
and to how they can be derived from the specialization of the non-homogeneous
process.
Joint work with Nikolai Leonenko and Enrico Scalas
Preprint: arXiv:1601.03965