Let N be a simply connected nilpotent Lie group and let $S=N⋊{\left(ℝ⁺\right)}^d$ be a semidirect product, ${\left(ℝ⁺\right)}^d$ acting on N by diagonal automorphisms. Let (Qₙ,Mₙ) be a sequence of i.i.d. random variables with values in S. Under natural conditions, including contractivity in the mean, there is a unique stationary measure ν on N for the Markov process Xₙ = MₙXn-1 + Qₙ. We prove that for an appropriate homogeneous norm on N there is χ₀ such that $li{m}_{t\to \infty }{t}^{\chi ₀}\nu x:\left|x\right|>t=C>0$. In particular, this applies to classical Poisson kernels on symmetric spaces,...

We consider a semidynamical system $(X,ℬ,\Phi ,w)$. We introduce the cone $𝔸$ of continuous additive functionals defined on $X$ and the cone $𝒫$ of regular potentials. We define an order relation “$\le$” on $𝔸$ and a specific order “$\prec$” on $𝒫$. We will investigate the properties of $𝔸$ and $𝒫$ and we will establish the relationship between the two cones.

Let $W_t:0\le t\le 1$ be a linear Brownian motion, starting from 0, defined on the canonical probability space (Ω,ℱ,P). Consider a process $u_t:0\le t\le 1$ belonging to the space ${ℒ}^{2,1}$ (see Definition II.2). The Skorokhod integral $U_t={ʃ}_0^{t}u\delta W$ is then well defined, for every t ∈ [0,1]. In this paper, we study the Besov regularity of the Skorokhod integral process $t\mapsto U_t$. More precisely, we prove the following THEOREM III.1. (1)If 0 < α < 1/2 and $u\in {ℒ}^{p,1}$ with 1/α < p < ∞, then a.s. $t\mapsto U_t\in {ℬ}_{p,q}^{\alpha }$ for all q ∈ [1,∞], and $t\to U_t\in {ℬ}_{p,\infty }^{\alpha ,0}$. (2) For every even integer p ≥...

Let $(B_t,t\in [0,1])$ be a linear Brownian motion starting from 0 and denote by $(L_t\left(x\right),t\ge 0,x\in ℝ)$ its local time. We prove that the spatial trajectories of the Brownian local time have the same Besov-Orlicz regularity as the Brownian motion itself (i.e. for all t>0, a.s. the function $x\to L_t\left(x\right)$ belongs to the Besov-Orlicz space $B_{M_2,\infty }^{1/2}$ with $M_2\left(x\right)=e^{{\left|x\right|}^2}-1$). Our result is optimal.