Eindhoven Stochastics Seminars

(STO)

From September 2011, the Eindhoven STOchastics seminar series will replace the seminars of QPA - RSS - SIM - MRM
and the Informal Meetings of Statisticians &Probabilists series.

Venue: Green Room at Eurandom
11.40 - 12.30 h.

Name and Affiliation                                                                                       Title  

ABSTRACTS

2012

Florin Ciucu (T-Labs / TU Berlin)

Randomness and Network Capacity

This talk presents a recent methodology to compute stochastic bounds on the distribution of per-flow capacity and delay in a wireless network with a general random topology covering uniform, Poisson, heavy-tailed, etc. distributions for nodes' densities and number of hops. The capacity results are obtained in both asymptotic and non-asymptotic regimes, in time scale and network size.  The delay results are obtained for the class of Markov arrival processes. 
Interestingly, the presence of randomness in topology is shown to be theoretically advantageous to the per-flow capacity in the sense that it can yield larger scaling laws than in corresponding non-random settings, by a factor as large as O(n).  In turn, the presence of spatial correlations along an e2e path is shown to be theoretically detrimental to the per-flow capacity by a logarithmic term.

Clement Dombry (Univ. Poitiers, France)

Regular conditional distributions of max-stable processes

We present in this talk recent results on the prediction problem in extreme value theory. Our main result is an explicit expression of the regular conditional distribution of a max-stable (or max-infinitely divisible) process $\{\eta(t)\}_{t\in T}$ given observations $\{\eta(t_i)=y_i,\ 1\leq i\leq k\}$. The starting point is the point process representation of max-infinitely divisible processes by Gin\'e, Hahn and Vatan (1990). We carefully analyze the structure of the underlying point process and introduce the  notions of extremal function, sub-extremal functions and hitting scenario associated to the constraints.  This allows us to explicit the conditional distribution as a mixture over all hitting scenarios compatible with the conditioning constraints. This extends a result by Wang and Stoev (2011) dealing with the case of spectrally discrete max-stable random fields. We believe this work offers new tools and perspective for prediction in extreme value theory together with numerous potential applications.

Remco van der Hofstad, (TU/e)

Hypercube percolation

Consider bond percolation on the hypercube {0,1}^n at the critical edge probability p_c defined such that the expected cluster size equals 2^{n/3}, where 2^{n/3} acts as the cube root of the number of vertices of the n-dimensional hypercube. Percolation on the hypercube was proposed by Erdos and Spencer (1979), and has proved to be substantially harder than percolation on the complete graph due to the non-trivial geometry of the hypercube. In this talk, I will describe the phase transition for percolation on the hypercube, and show that it shares many features with that on the complete graph. In previous work, we have identified the subcritical and critical regimes of percolation on the hypercube. In particular, we know that for p=p_c(1+O(2^{-n/3})), the largest connected component is of size roughly 2^{2n/3} and that this quantity is non-concentrated. So far, we were missing an analysis of the behavior of the largest connected component just above the critical value. In this work, we identify the supercritical behavior of percolation on the hypercube by showing that, for any sequence \varepsilon_n tending to zero, but \varepsilon_n being much larger than 2^{-n/3}, percolation on the hypercube with edge probability p=p_c(1+\varepsilon_n) has, with high probability, a unique giant component of size (2+o(1))\varepsilon_n 2^n. A main tool is the use of non-backtracking random walk, which we use to show that long percolation paths have endpoints that are almost uniform on the hypercube. This is joint work with Asaf Nachmias, building on previous work with Markus Heydenreich, Gordon Slade, Christian Borgs, Jennifer Chayes and Joel Spencer.

Phil Whiting (Alcatel-Lucent Bell Labs)

Backlog-Based Random Access in Wireless Networks: Fluid Limits and Instability Issues

Backlog-based wireless access schemes are simple and inherently distributed, yet provide a striking capability to match the optimal throughput performance of centralized scheduling mechanisms in a wide range of scenarios. Unfortunately, the type of activation rules for which throughput optimality has been established, may result in excessive backlogs and delays. The use of more aggressive/persistent access schemes than these can improve the delay performance, but does not offer any universal maximum-stability guarantees.
Motivated by the above issues, we use fluid limits to explore the (in)stability properties of backlog-based random-access algorithms. Such fluid limits have varying qualitative properties, dependent on the specific scenario, ranging from ones with smooth deterministic features,
to others which exhibit random oscillatory characteristics. It turns out that more aggressive access schemes continue to provide maximum stability in some networks, e.g. complete interference graphs. As we show however, in other topologies such schemes can drive the system into inefficient states and thus cause instability. Simulation experiments are presented to illustrate and validate the analytical results.

Mikhail Prokopenko (CSIRO ICT Centre, Sydney)

Information Dynamics at the Edge of Chaos

Many evolutionary and self-organization pressures can be characterized information-theoretically not only because it's an approximation useful in designing biologically-inspired systems, but also because numerous optimal structures evolve/self-organize in nature when information transfer within certain channels is maximized. Specifically, distributed computation can be described in terms of three fundamental operations: information storage, transfer, and modification. The talk will focus on information dynamics of computation within spatio-temporal systems, quantifying these operations on a local scale in space and time. The approach will be exemplified in different contexts, including cellular automata, computational neuroscience, swarm dynamics, and random Boolean networks (RBNs). In addition, we shall/may discuss a relation between Fisher information and phase transitions / order parameters, drawing from both thermodynamics and statistical estimation theory.

Dr Mikhail Prokopenko is a Principal Research Scientist in CSIRO ICT Centre. He has a strong international reputation in the areas of complex self-organizing systems, with more than 110 publications and patents, including an edited book ("Advances in Applied Self-organizing Systems", Springer, 2008). He received a PhD in Computer Science (2002, Australia), and MSc in Applied Mathematics (1988, USSR). Dr Prokopenko has co-organized and co-chaired the series of International Workshops on Guided Self-organization (GSO); was a keynote speaker at 6th International Workshop on Agent-Based Simulation (2005) and 3rd International Workshop on GSO (2010). Most recently, Dr Prokopenko served as an editor of special issues on Complex Networks (Artificial Life), and Guided Self-organization (HFSP, Theory in Biosciences, Advances in Complex Systems), and a section editor for Encyclopaedia of Machine Learning (Evolutionary Computation). He is an Honorary Associate at University of Sydney.

Radmila Erkocevic-Pribic (Thales, Delft)

Stochastics with Compressive Sensing in Radar

I will start with showing the Compressive Sensing potential in sensor/radar by giving an overview of Compressive Sensing and Radar with(out) Compressive Sensing. After that, I will point out where in the processing chain Compressive Sensing brings crucial (fundamental) changes. These changes lead to several fundamental mathematical problems that need to be investigated and where industry-academia cooperation is required. I will conclude by emphasizing the stochastic nature of these changes.

Alessandro Di Bucchianico (TU/e)

Applications in simulation methods and statistics

I will start by briefly reviewing the spectral representation of a reversible transition matrix and try to put this into a somewhat broader context. Next I will discuss a bound on the total variation distance of the chain to its stationary distribution. I will also discuss in detail a specific  use of Markov chains in simulation known as Markov Chain Monte Carlo. This technique is widely used in Bayesian statistics. I will show the Markov chain setting behind the two main forms (the Gibbs sampler and the Metropolis-Hastings sampler) and prove convergence in probability results of Section 12.6 that are important in statistics for deciding how long to simulate. If time permits, I will discuss a contraction bound on the relaxation time, which is  the start of Chapter 13 .

Gideon Weiss (University of Haifa)

Sojourn Time Estimation in an M/G/Infinity Queue with Partial Information

Prof. Gideon Weiss will discuss the estimation of the sojourn time in an M/G/1 system when one
only observes arrivals and departures, but not their identities.

This is a joint work with Nafna Nelgabats and Yuval Nov.

Istvan Kolossvary & Julia Komjathy

First passage percolation on Inhomogeneous Random Graphs

This talk discusses first passage percolation (FPP) on inhomogeneous random graphs (IHRG). The random graph model G(n;\kappa) we first study is the so-called finite type case of the general model introduced by Bollobas, Janson and Riordan. Each edge of G(n;\kappa) is given by an independent exponential edge weight with rate 1. Our main assumption is that the average number of neighbors \lambda_n +1 of each vertex is independent of its type. We consider the cases where the limit as n goes to infinity gives a finite or infinite \lambda. Afterwards the general model is also considered.
This can be considered a generalization of Bhamidi, van der Hofstad and Hooghiemstra, where FPP is explored on the Erdos-Renyi random graphs, a special case where all the vertices are of the same type. We find analogous results for the minimal weight of the path between uniformly chosen vertices in the giant component and for the hopcount, i.e. the number of edges on this minimal weight path. The proofs make use of a direct relation between FPP on the IHRG and thinned continuous-time multi-type branching processes.

Peter van Elsacker (TU Eindhoven)

Small Sample Size Quantile Estimation for the Negative Binomial Distribution

In this talk I will present the results of my bachelor's thesis research project.
The Negative Binomial distribution is a suitable distribution to describe count processes in which overdispersion is involved. Such processes are common in quality control settings in the pharmaceutical industries. Timely detection of trends in such processes requires that quantiles of the Negative Binomial Distribution have to be estimated from very small samples. Estimation of these quantiles can be divided into two problems. First, we consider three methods to estimate the parameters delta and eta of the Negative Binomial Distribution and evaluate them by observing the bias and the standard deviation.
After that, each parameter estimation method is applied to a quantile estimation method. In total, we investigated four different quantile estimation methods for each of the parameter estimation methods. It turns out that three of these methods consistently yield too low values, except for cases where both delta and eta are small. The other method produces consistently yields too high values.

Estate Khmaladze (University of Wellington)

Contiguity theory for statistical problems with set as a parameter of interest

To give a flavor of practical questions we have in mind in this talk consider this: is the pollution site the given hypothetical set K or is it rather one of its small perturbations K_\epsilon? Or the same question with an ore deposit site: is it a given K of one of K_\epsilon? Given n observation how small perturbations can we detect? The proper answer requires development of quite a bit of a theory.
As we know, contiguity theory for statistical problems with a finite-dimensional parameter, or with an infinite-dimensional parameter such as a function, for example -- a density or a regression function, is a central part of the statistical theory of testing. This is the theory which answers the question: if the sample size increases, how small deviations from the hypothesis can one hope to detect with some non-trivial power?
The more observations we have, the smaller deviations from the hypothesis we should be able to detect. Then, how small? and with what power?
In this talk we ask similar question for the case, when the parameter, still infinite-dimensional, is of a different nature, namely, it is a set. The problem becomes much more complex, because we did not have until recently an appropriate theory of small (infinitesimal) changes in sets. For this reason the literature on statistical estimation of sets is vast, but on testing hypothesis about sets there existed practically nothing.
The aim of this talk is to present a relatively general theory of differentiation of set-valued functions. In particular, we are now able to define a derivative of a set-valued function in the neighborhood of any closed set, equal to closure of its interior. As a result we show a limit theorem for the likelihood process as a process, which eventually lives on the class of derivatives of set-valued functions involved.

Mark Peletier (TU Eindhoven)

Understanding the origins of the Wasserstein gradient flows

Many evolutionary systems described by parabolic partial differential equations can be written as a gradient flow of some energy with respect to some metric. When present, this gradient-flow structure provides both high-level insight into the behaviour of the system, and low-level, practical tools for the analysis of the system and its solutions. Since the pioneering work of Jordan, Kinderlehrer, and Otto (1998) an impressive collection of evolutionary PDEs has been formulated as a gradient flow of some energy with respect to the Wasserstein metric.
In this talk I will briefly introduce the Wasserstein metric and the concept of a Wasserstein gradient flow, and then turn to the main topic of the talk. This is the question: how can we understand _why_ the Wasserstein metric appears in so many systems?
I will focus on the simplest of all examples, the linear diffusion equation, which is the Wasserstein gradient flow of the entropy. I will show how the gradient-flow structure is intimately connected to the large-deviation behaviour of a system of independent Brownian particles. This connection gives us an explanation for the prevalence of Wasserstein gradient flows, and suggests directions for the derivation of a large number of related systems.
(joint work with Stefan Adams, Nicolas Dirr, and Johannes Zimmer)

Peter Grünwald (CWI - University Leiden)

Safe Learning: How to modify Bayesian inference when the model is wrong

Standard Bayesian inference can behave suboptimally if the model under consideration is wrong: in some simple settings, the posterior may fail to concentrate even in the limit of infinite sample size. We introduce a test that can tell from the data whether we are in such a situation. If we are, we can adjust the learning rate (equivalently: make the prior lighter-tailed) in a data-dependent way. The resulting “safe” estimator continues to achieve good rates with wrong models. When applied to classification problems, the safe estimator achieves the optimal rates for the Tsybakov exponent of the underlying distribution, thereby establishing a connection between Bayesian inference and statistical learning theory.

Arnoud den Boer (CWI)

Dynamic Pricing and Learning

'Revenue management and dynamic pricing' (RM&DP) is an umbrella term for practices where the selling price of a product or service is not a fixed quantity, but can easily be adjusted over time and adapted to changing circumstances. Classical examples are found in the airline and hotel industry, where prices are controlled by opening or closing 'fare classes', but nowadays many more applications can be found, e.g. in restaurants, concert halls, theaters, and amusement parks.
The availability of digital sales data enables firms to continuously learn about consumer behavior, and optimize pricing decisions accordingly. This has inspired a stream of literature on RM&DP problems where estimation and optimization takes place simultaneously. The
decision maker then faces the task of not only optimizing profit, but also optimizing the 'learning process'. A key question in these type of
problems is whether a myopic or learning-by-doing approach - always choosing the optimal price w.r.t. current estimates - has a good
performance, or whether the decision maker should actively experiment in order to improve his/her knowledge on consumer behavior.
We show that in the classical RM&DP model (where finite inventory is sold during finite, consecutive selling seasons), learning-by-doing
performs well. The regret, which quantifies the costs for learning, is O(log(T)2) in the number of selling seasons T. In comparison, a setting
with no inventory restrictions has regret that grows with rate at least T^(1/2). We offer an explanation why in these two models, the
cost-for-learning behaves differently.

Florian Simatos (CWI-Eurandom)

Fluid and heavy traffic regime of a stochastic network with mobility

After briefly introducing myself and my research interests, I will focus in this talk on a model of stochastic network with mobile users. I will present scaling results in the fluid and heavy traffic regimes, which provide an important step towards analyzing the performance of bandwidth allocation policy of wireless networks. The study of the heavy traffic regime provides an example of a general result that allows to reduce the convergence of regenerative processes to the convergence of their excursions.

Hao Peng (TU/e, Department of Industrial Engineering) CANCELLED

Component Reliability Criticality or Importance Measures for Systems with Degrading Components

This paper proposes two new importance measures: one new importance measure for systems with s-independent degrading components, and another one for systems with s-correlated degrading components. Importance measures in previous research are inadequate for systems with degrading components because they are only applicable to steady-state cases and problems with discrete states without considering the continuously changing status of the degrading components. Our new importance measures are proposed as functions of time that can provide timely feedback on the critical components prior to failure based on the measured or observed degradation. Furthermore, the correlation between components is considered for developing these importance measures through a multivariate distribution. To evaluate the criticality of components, we analyzed reliability models for multi-component systems with degrading components, which can also be utilized for studying maintenance models. Numerical examples show that the proposed importance measures can be used as an effective tool to assess component criticality for systems with degrading components.

Jean-Bernard Martens (TU/e, Department of Industrial Design)

Statistics from an HCI Perspective – Introducing Illmo (Interactive Log Likelihood Modeling)

While statistics is recognized as an important topic (or necessary evil) in the area of human-computer interaction (HCI) and other scientific fields, it is also a topic that spurns continuous debate. The current practice is that, despite extensive developments in the area of statistics in the last decades, most practitioners stick to the most simple parametric methods. Increasingly, we see the argument arise that scientists will have to resort to more advanced methods, which are unfortunately only available in advanced statistical packages that require a specific (often, command-line) syntax and a substantial understanding of the underlying statistical principles. This talk introduces and discusses a new program, called Illmo, for performing statistics. In order to operate the program successfully, the user only needs to understand a single statistical principle, i.e., the likelihood as a goodness-of-fit measure between the actual data and the proposed statistical model. Illmo is unique in the sense that it not only provides extensive graphical renderings of the data analysis results, but provides an advanced visual interface for navigating between different data analysis methods. A one-week course on statistics with non-mathematically trained students (i.e., design students) confirmed that the program is relatively easy to operate and understand, which allowed students to come up with original ideas for how to apply the program within their own design practice.

Marko Boon (TU/e)

Waiting times in queueing networks with a single shared server

We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel servers, and many others.

The present research develops a novel unifying framework to find the waiting time distribution, which can be applied to a wide variety of models which lacked an analysis of the waiting time distribution until now. That is, we derive the waiting time distributions for stable systems as well as various asymptotic results (heavy traffic, light traffic, and infinite switch-over times) for systems with general renewal arrival processes. By interpolating between these asymptotic regimes, we develop simple closed-form approximations for the waiting time distribution for arbitrary loads
 

Oliver Amft (TU/e, Department of Electrical Engineering)

Activity pattern modelling and recognition in ubiquitous systems

Behaviour analysis using ubiquitous systems could facilitate many areas, from assisting in daily life situations of independently living chronic patients to saving energy in public buildings. Detecting nuances in daily behaviour routines from sensor data, such as in social interaction or dietary activities, may allow physicians to understand trends in cognitive impairments, e.g., related to the effect of a therapy, and status of a patient. Likewise, occupancy and activity of users in public buildings affects actual energy needs far beyond the dynamics that are currently considered in building operations. However, on-body and ambient data from ubiquitous sensors is noisy and activity patterns vary or drift over time.
Starting from application examples of ubiquitous systems, this talk will illustrate statistical pattern recognition techniques considered in
activity and context recognition. A specific focus is given to challenges pertained to dynamic adaptivity in pattern modelling, unsupervised
learning, and distributed recognition systems.

Jef Teugels (KU Leuven, Eurandom)

Insurance and Reinsurance

INSURANCE MATHEMATICS
Our intention is to cover the essentials of reinsurance treaties. Since reinsurance is insurance in itself, we first discuss some classical problems from insurance mathematics. We assume that we are looking at an homogeneous portfolio where the number of claims over the time period till t is denoted by N (t) while the claim sizes are considered to form a sample from a distribution F . We assume that claim times and claim sizes are independent. We will give an overview of results dealing with the total claim amount and with ruin problems, both in infinite and in finite time.
We will pay special attention to the differences that occur when the distribution F has either an exponentially bounded tail or when it is sub-exponential. This first case treats the situation where the claims are considered to be light-tailed while the second covers instances where claims are heavy-tailed.

REINSURANCE
When the underlying distribution is heavy-tailed, the insurance company might opt for reinsurance. Traditionally, reinsurance has been used as one possibility to share insurance risks between partners. In practice, reinsurance treaties have either a proportional or a non-proportional character. Nevertheless, the usual motive for reinsurance is the possible advent of large, even catastrophic claims.
After treating these classical reinsurance treaties, it seems worthwhile to add another form of treaty, genuinely based on large claims and related to Extreme Value Analysis. The most well- known such treaty is ECOMOR, a reinsurance form introduced in 1950 by the French actuary Thépaut. Despite this long-standing existence, large claims reinsurance treaties have never enjoyed great popularity. We try to understand the reasons behind this lack of recognition. In particular, we derive some mathematical results about ECOMOR. Based on these results, we formulate some pro’s and con’s of this and other reinsurance treaties.

PRESENTATION 1 ; PRESENTATION 2

Debjit Roy (RSM)

Stochastic Modeling of Automated Storage Systems using Semi-open Queuing Networks

Semi-open queuing network (SOQN) realistically models the synchronization process between external customer arrivals and fixed system resources. Hence, SOQNs provide better estimates of customer waiting times and resource utilization. In this research, stochastic models are developed for multi-tier automated storage systems where vehicles (resources) are self-powered to travel in horizontal directions (x- and y- axes), but use conveyors for vertical motion (z-axis). The SOQN for a single tier forms the building block for the multi-tier queuing network model. The conveyor system is modeled as a GI/G/1 queue. By using a combination of parametric decomposition and embedded Markov chain analysis, the queuing network is solved, and the effect of conveyors on system throughput capacity and transaction cycle times is determined.

Nikhil Bansal (TU/e)

Finding needles in exponential haystacks using Brownian Motion

Let S_1,...,S_n be arbitrary sets on the elements 1,...n. The celebrated "six standard deviations suffice" result of Spencer says that there always exists a red-blue coloring of the elements that colors each set almost evenly (i.e. up to 6 n(1/2) ).
Spencer's result is non-constructive and it had been long conjectured that no polynomial time algorithm to find such a coloring would exist. In this talk I will describe such an algorithm.
Our algorithm finds the desired coloring by performing a discrete brownian motion in the n-dimensional cube. This motion is defined using several semi-definite programs (SDPs) over time, where the choice of these SDPs themselves is guided by non-constructive methods.

Ton Dieker (Georgia Tech)
(joint work with Jinwoo Shin,ARC, Georgia Tech)

Global stability of stochastic processing networks through local quadratic Lyapunov functions

The size of many of today's stochastic networks prohibits optimal scheduling due to high computational demands, even when optimality merely requires that the network be stable. This motivates the search for simple scheduling algorithms for which throughput guarantees can be given. We present such a scheduling algorithm.
Our starting point is a set of 'local' scheduling policies, for instance at each station, which can be shown to achieve a given throughput using a quadratic Lyapunov function. We devise a simple policy for the whole network and show that it achieves the same throughput. Informally one can think of our result saying that 'local' stability yields 'global' stability under our policy. The proof relies on the construction of a novel quadratic Lyapunov function for the whole network using the local quadratic Lyapunov functions. Our methodology is applicable to multiclass networks, parallel server networks, networks of switches, and special wireless networks.
 

Tim Hulshof (TU/e)

Geometric properties of critical percolation clusters

Percolation is a paradigm model of statistical mechanics because it is one of the simplest models to undergo a phase transition. What happens at the phase transition is not well understood. I will talk about some new results for the geometry of critical percolation clusters in high-dimensions. In particular, I will highlight a few remarkable links between critical percolation and random walk.

Yifan Xu (Binghamton University)

First crossing time of compound Poisson processes(CPP) with linear boundaries

The question on the first passage time arises in topics such as, Wald's sequential probability ratio test, the time to ruin in classic risk model, the length of a busy period in M/G/1 queues, and certain inventory management problems. The exact distribution of the stopping time is found through the defective pdf of the level of CPP at any fixed time t, restricted on the set {processes that have not crossed any boundaries at t}. The said defective pdf itself is interesting since it describes the state of CPP within certain linear boundaries. A method of numerical approximation of the result is also discussed.

Britt Mathijssen (TU/e)

Distributed Control of Light Networks

Wireless networks are integrated into many of today's technologies. A new wireless communication system, is Visible Light Communication, which uses visible light to transmit data. In this talk, we investigate the performance of an application, currently researched by Philips, of this communication system in grid mesh networks. An introduction to wireless networking protocols and mesh networks is given, explaining the main concepts of the field. We consider two of these protocols, ALOHA and CSMA/CA, more extensively. Using these protocols, we analyze and optimize linear mesh networks. The main subject of the talk is the application of a protocol similar to CSMA/CA to mesh networks positioned on a grid. Our focus lies on finding the optimal value of the sensing range used by the protocol, minimizing hitting time and nodes that are not reached. Using a specially built simulation application, we derive some qualitative results on these grid networks.

Kiamars Vafayi (University of Leiden)

Brownian Momentum Process and Symmetric Inclusion Process, duality properties and weak coupling limits

I will talk about Brownian Momentum Process (BMP), a model of heat conduction with stochastic diffusion of energy. This model is related to and is dual (in the probabilistic sense) to an interacting particle system, the Symmetric Inclusion Process (SIP) that we have introduced, which is a Bosonic counterpart to the Fermionic symmetric exclusion process. SIP helps obtaining probabilistic properties of BMP and for example it describes the evolution of all the correlation functions. 
I will consider BMP in close to equilibrium conditions, i.e. when it is weakly coupled to the boundary heat baths or if the two heat baths are at almost same temperatures. We obtain the temperature profile in the system and talk about the form of the correlation functions and the behavior of the measure as the coupling to heat baths vanishes. At the end I will explain more about the duality property and its relation to the non-abelian symmetries of the generator of Markov process which is valid also in more general settings.
References:
1) Frank Redig and Kiamars Vafayi, Weak coupling limits in a stochastic model of heat conduction, J. Math. Phys. 52, 093303 (2011)
http://arxiv.org/abs/1101.2877
2) Cristian Giardina, Jorge Kurchan, Frank Redig, Kiamars Vafayi, Duality and hidden symmetries in
interacting particle systems, J. Stat. Phys. 135, 25 (2009)
http://arxiv.org/abs/0810.1202

Last updated 14-05-12
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