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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Infinite dimensional forward-backward stochastic differential equations and the KPZ equation http://ejp.ejpecp.org/article/view/2709 Kardar-Parisi-Zhang (KPZ) equation is a quasilinear stochastic partial differential equation (SPDE) driven by a space-time white noise. In recent years there have been several works directed towards giving a rigorous meaning to a solution of this equation. Bertini, Cancrini and Giacomin have proposed a notion of a solution through a limiting procedure and a certain renormalization of the nonlinearity. In this work we study connections between the KPZ equation and certain infinite dimensional forward-backward stochastic differential equations. Forward-backward equations with a finite dimensional noise have been studied extensively, mainly motivated by problems in mathematical finance. Equations considered here differ from the classical works in that, in addition to having an infinite dimensional driving noise, the associated SPDE involves a non-Lipschitz (specifically, a quadratic) function of the gradient. Existence and uniqueness of solutions of such infinite dimensional forward-backward equations is established and the terminal values of the solutions are then used to give a new probabilistic representation for the solution of the KPZ equation. Sergio Almada Monter Amarjit Budhiraja 2014-04-04 2014-04-04 19 Thermodynamic formalism and large deviations for multiplication-invariant potentials on lattice spin systems http://ejp.ejpecp.org/article/view/3189 We introduce the multiplicative Ising model and prove basic properties of its thermodynamic formalism such as existence of pressure and entropies. We generalize to one-dimensional "layer-unique'' Gibbs measures for which the same results can be obtained. For more general models associated to a $d$-dimensional multiplicative invariant potential, we prove a large deviation theorem in the uniqueness regime for averages of multiplicative shifts of general local functions. This thermodynamic formalism is motivated by the statistical properties of multiple ergodic averages. Jean-René Chazottes Frank Redig 2014-04-01 2014-04-01 19 Variance-Gamma approximation via Stein's method http://ejp.ejpecp.org/article/view/3020 Variance-Gamma distributions are widely used in financial modelling and contain as special cases the normal, Gamma and Laplace distributions. In this paper we extend Stein's method to this class of distributions. In particular, we obtain a Stein equation and smoothness estimates for its solution. This Stein equation has the attractive property of reducing to the known normal and Gamma Stein equations for certain parameter values. We apply these results and local couplings to bound the distance between sums of the form $\sum_{i,j,k=1}^{m,n,r}X_{ik}Y_{jk}$, where the $X_{ik}$ and $Y_{jk}$ are independent and identically distributed random variables with zero mean, by their limiting Variance-Gamma distribution. Through the use of novel symmetry arguments, we obtain a bound on the distance that is of order $m^{-1}+n^{-1}$ for smooth test functions. We end with a simple application to binary sequence comparison. Robert Edward Gaunt 2014-03-29 2014-03-29 19 On the expectation of normalized Brownian functionals up to first hitting times http://ejp.ejpecp.org/article/view/3049 Let $B$ be a Brownian motion and $T_1$ its first hitting time of the level $1$. For $U$ a uniform random variable independent of $B$, we study in depth the distribution of $B_{UT_1}/\sqrt{T_1}$, that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered. Romuald Elie Mathieu Rosenbaum Marc Yor 2014-03-29 2014-03-29 19 Müntz linear transforms of Brownian motion http://ejp.ejpecp.org/article/view/2424 We consider a class of Volterra linear transforms of Brownian motion associated to a sequence of Müntz Gaussian spaces and determine explicitly their kernels; the kernels take a simple form when expressed in terms of Müntz-Legendre polynomials. These are new explicit examples of progressive Gaussian enlargement of a Brownian filtration. We give a necessary and sufficient condition for the existence of kernels of infinite order associated to an infinite dimensional Müntz Gaussian space; we also examine when the transformed Brownian motion remains a semimartingale in the filtration of the original process. This completes some already obtained partial answers to the aforementioned problems in the infinite dimensional case. Larbi Alili Ching-Tang Wu 2014-03-22 2014-03-22 19