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Any of these conditions can be waived by permission of the Corresponding Author. ecp@iam.uni-bonn.de (Anton Bovier (Chief Editor)) ejpecp@chafai.net (Djalil Chafaï) Thu, 09 Jan 2014 00:05:12 -0800 OJS http://blogs.law.harvard.edu/tech/rss 60 On harmonic functions of killed random walks in convex cones http://ecp.ejpecp.org/article/view/3219 We prove the existence of uncountably many nonnegative harmonic functions for random walks in the euclidean space with non-zero drift, killed when leaving general convex cones with vertex in 0. We also make the natural conjecture about the Martin boundary for lattice random walks in general convex cones in two dimensions. Proving that the set of harmonic functions found is the full Martin boundary for these processes is an open problem. Jetlir Duraj http://ecp.ejpecp.org/article/view/3219 Mon, 17 Nov 2014 02:37:32 -0800 Recurrence for the frog model with drift on $\mathbb{Z}^d$ http://ecp.ejpecp.org/article/view/3740 In this paper we present a recurrence criterion for the frog model on $\mathbb{Z}^d$ with an i.i.d. initial configuration of sleeping frogs and such that the underlying random walk has a drift to the right. Christian Döbler, Lorenz Pfeifroth http://ecp.ejpecp.org/article/view/3740 Fri, 14 Nov 2014 22:26:29 -0800 Wald for non-stopping times: the rewards of impatient prophets http://ecp.ejpecp.org/article/view/3609 Let $X_1,X_2,\ldots$ be independent identically distributed nonnegative random variables. Wald's identity states that the random sum $S_T:=X_1+\cdots+X_T$ has expectation $\mathbb{E} T \cdot \mathbb{E} X_1$ provided $T$ is a stopping time. We prove here that for any $1&lt;\alpha\leq 2$, if $T$ is an arbitrary nonnegative random variable, then $S_T$ has finite expectation provided that $X_1$ has finite $\alpha$-moment and $T$ has finite $1/(\alpha-1)$-moment. We also prove a variant in which $T$ is assumed to have a finite exponential moment. These moment conditions are sharp in the sense that for any i.i.d. sequence $X_i$ violating them, there is a $T$ satisfying the given condition for which $S_T$ (and, in fact, $X_T$) has infinite expectation.An interpretation of this is given in terms of a prophet being more rewarded than a gambler when a certain impatience restriction is imposed. Alexander E Holroyd, Yuval Peres, Jeffrey E Steif http://ecp.ejpecp.org/article/view/3609 Wed, 12 Nov 2014 21:21:38 -0800 Some limit results for Markov chains indexed by trees http://ecp.ejpecp.org/article/view/3601 We consider a sequence of Markov chains $(\mathcal X^n)_{n=1,2,...}$ with $\mathcal X^n = (X^n_\sigma)_{\sigma\in\mathcal T}$, indexed by the full binary tree $\mathcal T = \mathcal T_0 \cup \mathcal T_1 \cup ...$, where $\mathcal T_k$ is the $k$th generation of $\mathcal T$. In addition, let $(\Sigma_k)_{k=0,1,2,...}$ be a random walk on $\mathcal T$ with $\Sigma_k \in \mathcal T_k$ and $\widetilde{\mathcal R}^n = (\widetilde R_t^n)_{t\geq 0}$ with $\widetilde R_t^n := X_{\Sigma_{[tn]}}$, arising by observing the Markov chain $\mathcal X^n$ along the random walk. We present a law of large numbers concerning the empirical measure process $\widetilde{\mathcal Z}^n = (\widetilde Z_t^n)_{t\geq 0}$ where $\widetilde{Z}_t^n = \sum_{\sigma\in\mathcal T_{[tn]}} \delta_{X_\sigma^n}$ as $n\to\infty$. Precisely, we show that if $\widetilde{\mathcal R}^n \Rightarrow{n\to\infty} \mathcal R$ for some Feller process $\mathcal R = (R_t)_{t\geq 0}$ with deterministic initial condition, then $\widetilde{\mathcal Z}^n \Rightarrow{n\to\infty} \mathcal Z$ with $Z_t = \delta_{\mathcal L(R_t)}$. Peter Czuppon, Peter Pfaffelhuber http://ecp.ejpecp.org/article/view/3601 Tue, 11 Nov 2014 09:12:35 -0800 Monotone interaction of walk and graph: recurrence versus transience http://ecp.ejpecp.org/article/view/3607 We consider recurrence versus transience for models of random walks on growing in time, connected subsets $\mathbb{G}_t$ of some fixed locally finite, connected graph, in which monotone interaction enforces such growth as a result of visits by the walk (or probes it sent), to the neighborhood of the boundary of $\mathbb{G}_t$. Amir Dembo, Ruojun Huang, Vladas Sidoravicius http://ecp.ejpecp.org/article/view/3607 Thu, 06 Nov 2014 23:27:32 -0800