Electronic Journal of Probability 2014-07-12T06:08:28-07:00 Michel Ledoux (Chief Editor) Open Journal Systems The Electronic Journal of Probability applies the <a href="" target="_blank">Creative Commons Attribution License</a> (CCAL) to all articles we publish in this journal. Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles published in EJP, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. 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Any of these conditions can be waived by permission of the Corresponding Author. <p><strong> </strong></p>The <strong>Electronic Journal of Probability</strong> (EJP) publishes full-length research articles in probability theory. Short papers, those less than 12 pages, should be submitted first to its sister journal, the <a href="" target="_blank">Electronic Communications in Probability</a> (ECP). EJP and ECP share the same editorial board, but with different Editors in Chief.<p>EJP and ECP are free access official journals of the <a href="">Institute of Mathematical Statistics</a> (IMS) and the <a href=""> Bernoulli Society</a>. This web site uses the <a href="">Open Journal System</a> (OJS) free software developed by the non-profit organization <a href="">Public Knowledge Project</a> (PKP).</p><p>Please consider donating to the <a href="" target="_blank">Open Access Fund</a> of the IMS at this <a href="" target="_blank"><strong>page</strong></a> to keep the journal free.</p> Multidimensional fractional advection-dispersion equations and related stochastic processes 2014-07-12T06:08:28-07:00 Mirko D'Ovidio Roberto Garra In this paper we study multidimensional fractional advection-dispersion equations involving fractional directional derivatives both from a deterministic and a stochastic point of view. For such equations we show the connection with a class of multidimensional Lévy processes. We introduce a novel Lévy-Khinchine formula involving fractional gradients and study the corresponding infinitesimal generator of multi-dimensional random processes. We also consider more general fractional transport equations involving Frobenius-Perron operators and their stochastic solutions. Finally, some results about fractional power of second order directional derivatives and their applications are also provided. 2014-07-12T06:08:10-07:00 Markovian loop soups: permanental processes and isomorphism theorems 2014-07-12T06:08:28-07:00 Patrick J. Fitzsimmons Jay S. Rosen We construct  loop soups for general Markov processes without transition densities and show that the associated permanental process is equal in distribution to the loop soup local time. This is used to establish isomorphism theorems connecting the local time of the original process with the associated permanental process. Further properties of the loop measure are studied. 2014-07-05T05:14:13-07:00 Sensitivity analysis for stochastic chemical reaction networks with multiple time-scales 2014-07-12T06:08:28-07:00 Ankit Gupta Mustafa Khammash <!-- p, li { white-space: pre-wrap; } --> <!-- p, li { white-space: pre-wrap; } --> <p style="text-indent: 0px; margin: 0px;">Stochastic models for chemical reaction networks have become very popular in recent years. For such models, the estimation of parameter sensitivities is an important and challenging problem. Sensitivity values help in analyzing the network, understanding its robustness properties and also in identifying the key reactions for a given outcome. Most of the methods that exist in the literature for the estimation of parameter sensitivities, rely on Monte Carlo simulations using Gillespie's stochastic simulation algorithm or its variants. It is well-known that such simulation methods can be prohibitively expensive when the network contains reactions firing at different time-scales, which is a feature of many important biochemical networks. For such networks, it is often possible to exploit the time-scale separation and approximately capture the original dynamics by simulating a "reduced" model, which is obtained by eliminating the fast reactions in a certain way. The aim of this paper is to tie these model reduction techniques with sensitivity analysis. We prove that under some conditions, the sensitivity values for the reduced model can be used to approximately recover the sensitivity values for the original model. Through an example we illustrate how our result can help in sharply reducing the computational costs for the estimation of parameter sensitivities for reaction networks with multiple time-scales. To prove our result, we use coupling arguments based on the random time change representation of Kurtz. We also exploit certain connections between the distributions of the occupation times of Markov chains and multi-dimensional wave equations.</p> 2014-07-05T05:05:57-07:00 Complete localisation and exponential shape of the parabolic Anderson model with Weibull potential field 2014-07-12T06:08:28-07:00 Artiom Fiodorov Stephen Muirhead We consider the parabolic Anderson model with Weibull potential field, for all values of the Weibull parameter. We prove that the solution is eventually localised at a single site with overwhelming probability (complete localisation) and, moreover, that the solution has exponential shape around the localisation site. We determine the localisation site explicitly, and derive limit formulae for its distance, the profile of the nearby potential field and its ageing behaviour. We also prove that the localisation site is determined locally, that is, by maximising a certain time-dependent functional that depends only on: (i) the value of the potential field in a neighbourhood of fixed radius around a site; and (ii) the distance of that site to the origin. Our results extend the class of potential field distributions for which the parabolic Anderson model is known to completely localise; previously, this had only been established in the case where the potential field distribution has sub-Gaussian tail decay, corresponding to a Weibull parameter less than two. 2014-07-05T04:55:24-07:00 Vulnerability of robust preferential attachment networks 2014-07-12T06:08:28-07:00 Maren Eckhoff Peter Mörters <!-- p, li { white-space: pre-wrap; } --><!-- p, li { white-space: pre-wrap; } --> <pre style="text-indent: 0px; margin: 0px;"><span style="color: #000000;">Scale-free networks with small power law exponent are known to be robust, meaning that their qualitative topological structure </span>cannot be altered by random removal of even a large proportion of nodes. By contrast, it has been argued in the science literature that such networks are highly vulnerable to a targeted attack, and removing a small <span style="color: #000000;">number of key nodes in the network will dramatically change the topological structure. Here we </span><span style="color: #000000;">analyse</span><span style="color: #000000;"> a class of preferential attachment networks </span><span style="color: #000000;">in the robust regime and prove four main results supporting this claim: After removal of an arbitrarily small proportion </span><span style="color: #008000;">$\varepsi</span></pre><pre style="text-indent: 0px; margin: 0px;"><span style="color: #008000;">lon&gt;0$</span><span style="color: #000000;"> of the oldest </span>nodes (1) the asymptotic degree distribution has exponential instead of power law tails; (2) the largest degree in the network drops from <span style="color: #000000;">being of the order of a power of the network size </span><span style="color: #008000;">$n$</span><span style="color: #000000;"> to being just logarithmic in </span><span style="color: #008000;">$n$</span><span style="color: #000000;">; (3) the typical distances in the network increase from order </span><span style="color: #008000;">$\log\log n$</span><span style="color: #000000;"> to </span><span style="color: #000000;">order </span><span style="color: #008000;">$\log n$</span><span style="color: #000000;">; and (4) the network becomes vulnerable to random removal of nodes. Importantly, all our results explicitly quantify the dependence on the </span><span style="color: #000000;">proportion </span><span style="color: #008000;">$\varepsilon$</span><span style="color: #000000;"> of removed </span><span style="color: #000000;">vertices</span><span style="color: #000000;">. For example, we show that the critical proportion of nodes that have to be retained </span><span style="color: #000000;">for survival of the giant component undergoes a steep increase as </span><span style="color: #008000;">$\varepsilon$</span><span style="color: #000000;"> moves away from zero, and a comparison of this result with similar ones </span>for other networks reveals the existence of two different universality classes of robust network models. The key technique in our proofs is a local approximation of the network by a branching random walk with two killing boundaries, and an understanding of the particle genealogies in this process, which enters into estimates for the spectral radius of an associated operator.</pre> 2014-07-05T04:27:53-07:00