We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpi´nski carpet that is invariant with respect to the local symmetries of the carpet. Consequently, for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

We consider the one-sided exit problem – also called one-sided barrier problem – for ($\alpha$-fractionally) integrated random walks and Lévy processes. Our main result is that there exists a positive, non-increasing function $\alpha \mapsto \theta \left(\alpha \right)$ such that the probability that any $\alpha$-fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time $T$ behaves as $T^{-\theta \left(\alpha \right)+\mathrm{o}\left(1\right)}$ for large $T$. We also investigate when the fixed level can be replaced by a different barrier...