Differential operators $D_x$, $D_y$, and $D_z$ are formed using the action of the $3$-dimensional discrete Heisenberg group $G$ on a set $S$, and the operators will act on functions on $S$. The Laplacian operator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator with variable differences which can be associated to a unitary representation of $G$ on the Hilbert space $L^2(S)$. Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable.

We give upper and lower Gaussian estimates for the diffusion kernel of a divergence and nondivergence form elliptic operator in a Lipschitz domain. On donne des estimations Gaussiennes pour le noyau d'une diffusion, r\'eversible ou pas, dans un domaine Lipschitzien.

We prove comparison theorems for the probability of life in a Lipschitz domain between Brownian motion and random walks. On donne des th\'eor\`emes de comparaison pour la probabilit\'e de vie dans un domain Lipschitzien entre le Brownien et de marches al\'eatoires.