This paper presents a concentration inequality for tree-indexed Markov chains taking values on general metric spaces. It not only generalizes known results from bounded spaces to unbounded spaces, but also captures tail behaviors that can be heavier than those of Gaussian distributions.

A Brownian motion with drift is simply a process $V_t^{\mathrm{\eta }}$ of the form $V_t^{\mathrm{\eta }}=B_t+\mathrm{\eta }t$ where $B_t$ is a standard Brownian motion and $\mathrm{\eta }>0$. In [7], the authors showed that the underlying range $R_t(V^{\mathrm{\eta }})={sup}_{0\le s\le t}{V}_t^{\mathrm{\eta }}-{inf}_{0\le s\le t}{V}_t^{\mathrm{\eta }}$ is equivalent to $\mathrm{\eta }t$ a.e in the long run, i.e

$$\frac{R_t(V^{\mathrm{\eta }})}{t}\stackrel{a.e}{\underset{t\to \mathrm{\infty }}{⟶}}\mathrm{\eta }.(0.1)$$

In this paper, we show that (0.1) follows from a deterministic property. More precisely, we show that the long run behavior of the range of a (deterministic) function is obtainable straightaway from that of the function itself.

We study the asymptotics of a two-dimensional stochastic differential system with a degenerate diffusion matrix. This system describes the dynamics of a population where individuals contribute to the degradation of their environment through two different behaviors, responding more or less intensively to their environmental perception. We exploit the almost one-dimensional form of the dynamical system to compute explicitly the Freidlin-Wentzell action functional. This allows us to give conditions under which the small noise regime of the invariant measure is concentrated around the equilibria of the dynamical system having the smallest diffusion coefficient.

We derive explicitly the adapted 2-Wasserstein distance between non-degenerate Gaussian distributions on ${\mathbb{R}}^N$ and characterize the optimal bicausal coupling(s). This leads to an adapted version of the Bures-Wasserstein distance on the space of positive definite matrices.

We examine the limiting behaviour of an order 1 Markov random walk in ${\mathbb{R}}^2$, i.e. a walk that remembers its previous step. Provided a certain symmetry of the Markovian condition, under relatively mild conditions we prove convergence to a Brownian motion and bound the rate of convergence. Therefore, an order 1 Markov condition is insufficient to alter the limiting behaviour of a random walk.

In this paper, under a weak and general condition, we establish explicit McDiarmid type concentration inequalities for general functionals of Wasserstein contractive Markov chains. Our approach is different from those commonly used to establish McDiarmid’s inequalities. In particular cases, our result recovers some existing concentration inequalities.

Within the context of rough path analysis via fractional calculus, we show how variability can be used to prove the existence of integrals with respect to Hölder continuous multiplicative functionals in the case of Lipschitz coefficients with first order partial derivatives of bounded variation. We discuss applications to certain Gaussian processes, in particular, fractional Brownian motions with Hurst index $\frac{1}{3}<H\le \frac{1}{2}$.

We propose a new random process to construct the eigenvectors of some random operators which gives a short and clean connection with the resolvent and could be used to simplify previous proofs about Anderson localization. In this process the center of localization has to be chosen randomly.