Electronic Journal of Probability http://ejp.ejpecp.org/ <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="http://ecp.ejpecp.org/" 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="http://www.imstat.org/">Institute of Mathematical Statistics</a> (IMS) and the <a href="http://isi.cbs.nl/BS/bshome.htm"> Bernoulli Society</a>. 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Applying this standard license to your work will ensure your right to make your work freely and openly available.<br /><br /><strong>Summary of the Creative Commons Attribution License</strong><br /><br />You are free<br /><ul><li> to copy, distribute, display, and perform the work</li><li> to make derivative works</li><li> to make commercial use of the work</li></ul>under the following condition of Attribution: others must attribute the work if displayed on the web or stored in any electronic archive by making a link back to the website of EJP via its Digital Object Identifier (DOI), or if published in other media by acknowledging prior publication in this Journal with a precise citation including the DOI. For any further reuse or distribution, the same terms apply. Any of these conditions can be waived by permission of the Corresponding Author. An Itô type formula for the fractional Brownian motion in Brownian time http://ejp.ejpecp.org/article/view/3184 Let $X$ be a (two-sided) fractional Brownian motion of Hurst parameter $H\in (0,1)$ and let $Y$ be a standard Brownian motion independent of $X$. Fractional Brownian motion in Brownian motion time (of index $H$), recently studied, is by definition the process $Z=X\circ Y$. It is a continuous, non-Gaussian process with stationary increments, which is selfsimilar of index $H/2$. The main result of the present paper is an Itô's type formula for $f(Z_t)$, when $f:\mathbb{R}\to\mathbb{R}$ is smooth and $H\in [1/6,1)$. When $H&gt;1/6$, the change-of-variable formula we obtain is similar to that of the classical calculus. In the critical case $H=1/6$, our change-of-variable formula is in law and involves the third derivative of $f$ as well as an extra Brownian motion independent of the pair $(X,Y)$. We also discuss briefly the case $H&lt;1/6$. Ivan Nourdin Raghid Zeineddine 2014-10-24 2014-10-24 19 The harmonic measure of balls in critical Galton-Watson trees with infinite variance offspring distribution http://ejp.ejpecp.org/article/view/3498 We study properties of the harmonic measure of balls in large critical Galton-Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index $\alpha\in (1,2]$. Here the harmonic measure refers to the hitting distribution of height $n$ by simple random walk on the critical Galton-Watson tree conditioned on non-extinction at generation $n$. For a ball of radius $n$ centered at the root, we prove that, although the size of the boundary is roughly of order $n^{\frac{1}{\alpha-1}}$, most of the harmonic measure is supported on a boundary subset of size approximately equal to $n^{\beta_{\alpha}}$, where the constant $\beta_{\alpha}\in (0,\frac{1}{\alpha-1})$ depends only on the index $\alpha$. Using an explicit expression of $\beta_{\alpha}$, we are able to show the uniform boundedness of $(\beta_{\alpha}, 1&lt;\alpha\leq 2)$. These are generalizations of results in a recent paper of Curien and Le Gall. Shen Lin 2014-10-20 2014-10-20 19 A compact containment result for nonlinear historical superprocess approximations for population models with trait-dependence http://ejp.ejpecp.org/article/view/3506 We consider an approximating sequence of interacting population models with branching, mutation and competition. Each individual is characterized by its trait and the traits of its ancestors. Birth- and death-events happen at exponential times. Traits are hereditarily transmitted unless mutation occurs. The present model is an extension of the model used in [Méléard and Tran, EJP, 2012], where for large populations with small individual biomasses and under additional assumptions, the diffusive limit is shown to converge to a nonlinear historical superprocess. The main goal of the present article is to verify a compact containment condition in the more general setup of Polish trait-spaces and general mutation kernels that allow for a dependence on the parent's trait. As a by-product, a result on the paths of individuals is obtained. An application to evolving genealogies on marked metric measure spaces is mentioned where genealogical distance, counted in terms of the number of births without mutation, can be regarded as a trait. Because of the use of exponential times in the modeling of birth- and death-events the analysis of the modulus of continuity of the trait-history of a particle plays a major role in obtaining appropriate bounds. Sandra Kliem 2014-10-18 2014-10-18 19 On the heat kernel and the Dirichlet form of Liouville Brownian motion http://ejp.ejpecp.org/article/view/2950 In a previous work, a Feller process called Liouville Brownian motion on $\mathbb{R}^2$ has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field $e^{\gamma\, X}$ and is the right diffusion process to consider regarding $2d$-Liouville quantum gravity.  In this note, we discuss the construction of the associated  Dirichlet form, following essentially Fukushima, Oshima, and Takeda, and the techniques introduced in our previous work. Then we carry  out the analysis of the Liouville resolvent. In particular, we prove that it is strong Feller, thus obtaining the existence of  the Liouville heat kernel via a non-trivial theorem of Fukushima and al. One of the motivations which led to introduce the Liouville Brownian motion in our previous work was to investigate the puzzling Liouville metric through the eyes of this new stochastic process. In particular,  the theory developed for example in Stollmann and Sturm, whose aim is to capture the "geometry" of the underlying space out of the Dirichlet form of a process living on that space, suggests a notion of distance associated to a Dirichlet form. More precisely, under some mild hypothesis on the regularity of the Dirichlet form, they provide a distance in the wide sense, called intrinsic metric, which is interpreted as an extension of Riemannian  geometry applicable to non differential structures. We prove  that the needed mild hypotheses are satisfied but that the associated intrinsic metric unfortunately vanishes, thus showing that renormalization theory remains out of reach of  the metric aspect  of Dirichlet forms. Rémi Rhodes Christophe Garban Vincent Vargas 2014-10-16 2014-10-16 19 Martingale inequalities and deterministic counterparts http://ejp.ejpecp.org/article/view/3270 We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the martingale inequality is determined by a fixed point of a simple nonlinear operator involving a concave envelope. Our results yield an explanation for certain inequalities that arise in mathematical finance in the context of robust hedging. Mathias Beiglböck Marcel Nutz 2014-10-16 2014-10-16 19