We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non existence of moments for first-passage and last-exit times. In our proofs we also make use of estimates for hitting probabilities…

Suppose that $\{X_{jk}, j=1,\cdots,p_1; k=1,\cdots,n\}$ are independent and identically distributed (i.i.d) real random variables with $EX_{11}=0$ and $EX_{11}^{2}=1$, and that $\{Y_{jk}, j=1,\cdots,p_2; k=1,\cdots,n\}$ are i.i.d real random variables with $EY_{11}=0$ and $EY_{11}^{2}=1$, and that $\{X_{jk}, j=1,\cdots,p_1; k=1,\cdots,n\}$ are independent of $\{Y_{jk}, j=1,\cdots,p_2;…

Consider a Markov process $\omega_t$ at stationarity and some event $\mathcal{C}$ (a subset of the state-space of the process). A natural measure of correlations in the process is the pairwise correlation $\mathbb{P}[\omega_0,\omega_t \in \mathcal{C}] - \mathbb{P}[\omega_0 \in \mathcal{C}]^2$. A second…

Let $\mathcal{T}$ be a rooted supercritical multi-type Galton-Watson (MGW) tree with types coming from a finite alphabet, conditioned to non-extinction. The $\lambda$-biased random walk $(X_t)_{t\ge0}$ on $\mathcal{T}$ is the nearest-neighbor random walk which, when at a vertex $v$ with $d_v$ offspring,…

Consider random symmetric Toeplitz matrices $T_{n}=(a_{i-j})_{i,j=1}^{n}$ with matrix entries $a_{j}, j=0,1,2,\cdots,$ being independent real  random variables such that $$ \mathbb{E}[a_{j}]=0, \ \ \mathbb{E} [|a_{j}|^{2}]=1 \ \mathrm{for}\,\ \ j=0,1,2,\cdots,$$ (homogeneity of 4-th moments) $$\kappa=\mathbb{E} [|a_{j}|^{4}],$$ and further (uniform boundedness) $$\sup\limits_{j\geq 0} \mathbb{E} [|a_{j}|^{k}]=C_{k}<\infty\…

We consider population models in which the individuals reproduce, die and also migrate in space. The population size scales according to some parameter $N$, which can have different interpretations depending on the context. Each individual is assigned a mass of $1/N$ and…

Random interlacements (at level $u$) is a one parameter family of random subsets of $\mathbb{Z}^d$ introduced by Sznitman. The vacant set at level $u$ is the complement of the random interlacement at level $u$. While the random interlacement induces a…

We study percolation in the hierarchical lattice of order N where the probability of connection between two points separated by distance k is of the form ck/Nk(1+δ), δ>-1. Since the distance is an ultrametric, there are significant differences with percolation…

We consider the one-dimensional diffusion $X$ that satisfies the stochastic differential equation$$dX_t = b(X_t) \, dt + \sigma (X_t) \, dW_t$$in the interior $int(I) = \mbox{} ]\alpha, \beta[$ of a given interval $I \subseteq [-\infty, \infty]$, where $b, \sigma: int(I)\rightarrow \mathbb{R}$…

We consider an equilibrium birth and death type process for a particle system in infinite volume, the latter is described by the space of all locally finite point configurations on $\mathbb{R}^d$. These Glauber type dynamics are Markov processes constructed for…