Related papers: Invariance principle for non-homogeneous random wa…

Consider a random walk on $\mathbb{Z}^d$ in a translation-invariant and ergodic random environment and starting from the origin. In this short note, assuming that a quenched invariance principle for the opportunely-rescaled walks holds, we…

Commonly, normal diffusive behavior is characterized by a linear dependence of the second central moment on time, $< x^2(t) >\propto t$, while anomalous behavior is expected to show a different time dependence, $ < x^2(t) > \propto…

We introduce and study branching interval partition diffusions in their natural generality. We let interval widths evolve independently according to a general real-valued diffusion subject only to conditions that ensure finite lifetimes of…

This is a preprint of Chapter 2 in the following work: Marta Lewicka, A Course on Tug-of-War Games with Random Noise, 2020, Springer, reproduced with permission of Springer Nature Switzerland AG. We present the basic relation between the…

We consider a recurrent random walk (RW) in random environment (RE) on a strip. We prove that if the RE is i. i. d. and its distribution is not supported by an algebraic subsurface in the space of parameters defining the RE then the RW…

A random flight on a plane with non-isotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the…

We establish a notion of universality for the parabolic Anderson model via an invariance principle for a wide family of parabolic stochastic partial differential equations. We then use this invariance principle in order to provide an…

A random walk on a countable group $G$ acting on a metric space $X$ gives a characteristic called the drift which depends only on the transition probability measure $\mu$ of the random walk. The drift is the `translation distance' of the…

We consider simple random walks on random graphs embedded in $\mathbb{R}^d$ and generated by point processes such as Delaunay triangulations, Gabriel graphs and the creek-crossing graphs. Under suitable assumptions on the point process, we…

We consider a locally uniformly strictly elliptic second order partial differential operator in $\mathbb{R}^d$, $d\ge 2$, with low regularity assumptions on its coefficients, as well as an associated Hunt process and semigroup. The Hunt…

Traditionally, time-development of the mean square displacement has been employed to determine the diffusion coefficient from the trajectories of single particles. However, this approach is sensitive to the noise and the motion blur upon…

We obtain non-Gaussian limit laws for one-dimensional random walk in a random environment assuming that the environment is a function of a stationary Markov process. This is an extension of the work of Kesten, M. Kozlov and Spitzer for…

We consider a branching random walk initiated by a single particle at location 0 in which particles alternately reproduce according to the law of a Galton-Watson process and disperse according to the law of a driftless random walk on the…

Let $G$ be a connected simple real Lie group, $\Lambda_{0}\subseteq G$ a lattice and $\Lambda \unlhd \Lambda_{0}$ a normal subgroup such that $\Lambda_{0}/\Lambda\simeq \mathbb{Z}^d$. We study the drift of a random walk on the…

We consider random walks on the line given by a sequence of independent identically distributed jumps belonging to the strict domain of attraction of a stable distribution, and first determine the almost sure exponential divergence rate, as…

We study a symmetric random walk (RW) in one spatial dimension in environment, formed by several zones of finite width, where the probability of transition between two neighboring points and corresponding diffusion coefficient are…

Two fundamental theorems by Spitzer/Erickson and Kesten/Maller on the fluctuation type (positive divergence, negative divergence or oscillation) of a real-valued random walk $(S_{n})_{n\ge 0}$ with iid increments $X_{1},X_{2},\ldots$ and…

Let X_0=0, X_1, X_2, ..., be an aperiodic random walk generated by a sequence xi_1, xi_2, ..., of i.i.d. integer-valued random variables with common distribution p(.) having zero mean and finite variance. For an N-step trajectory…

We prove that the dimension drop phenomenon holds for the harmonic measure associated to a transient random walk in a random environment (as defined by R. Lyons and R. Pemantle in 1992 and generalized by G. Faraud in 2011) on an infinite…

We derive diffusion constants and martingales for senile random walks with the help of a time-change. We provide direct computations of the diffusion constants for the time-changed walks. Alternatively, the values of these constants can be…