Consider the linear nonhomogeneous fixed-point equation R=D∑i=1NCiRi+Q, where (Q,N,C1,C2,…) is a random vector with N∈{0,1,2,3,…}∪{∞},Ci≥0 for all i∈N, P(|Q|0)0, and {Ri}i∈N is a sequence of i.i.d. random variables independent of (Q,N,C1,C2,…) having the same distribution as...

Consider a probability measure μ supported by a regular geodesic ball in a manifold. For any p≥1 we define a stochastic algorithm which converges almost surely to the p-mean ep of μ. Assuming furthermore that the functional to minimize is regular around ep, we prove that a natural...

We study a fractional stochastic perturbation of a first-order hyperbolic equation of nonlinear type. The existence and uniqueness of the solution are investigated via a Lax–Oleĭnik formula. To construct the invariant measure we use two main ingredients. The first one is the notion of a...

The Bernoulli sieve is the infinite “balls-in-boxes” occupancy scheme with random frequencies Pk=W1⋯Wk−1(1−Wk), where (Wk)k∈N are independent copies of a random variable W taking values in (0,1). Assuming that the number of balls equals n, let Ln denote the number of empty boxes...

We consider the random conductance model where the underlying graph is an infinite supercritical Galton–Watson tree, and the conductances are independent but their distribution may depend on the degree of the incident vertices. We prove that if the mean conductance is finite, there is a...

In this paper, Hunt’s hypothesis (H) and Getoor’s conjecture for Lévy processes are revisited. Let X be a Lévy process on Rn with Lévy–Khintchine exponent (a,A,μ). First, we show that if A is non-degenerate then X satisfies (H). Second, under the assumption that μ(Rn∖ARn)∞, we...

Let η=(η(t))t∈T be a sample continuous max-infinitely random field on a locally compact metric space T. For a closed subset S⊂T, we denote by ηS the restriction of η to S. We consider β(S1,S2), the absolute regularity coefficient between ηS1 and ηS2, where S1,S2 are two disjoint...

We prove a law of large numbers for random walks in certain kinds of i.i.d. random environments in Zd that is an extension of a result of Bolthausen et al. (2003) [4]. We use this result, along with the lace expansion for self-interacting random walks, to prove a monotonicity result for the...

We study the long time behavior of a Brownian particle moving in an anomalously diffusing field, the evolution of which depends on the particle position. We prove that the process describing the asymptotic behavior of the Brownian particle has bounded (in time) variance when the particle...

We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a...