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We prove a version of the strong half-space theorem between the classes ofrecurrent minimal surfaces and complete minimal surfaces with bounded curvatureof $\mathbb{R}^{3}_{\raisepunct{.}}$ We also show that any minimal hypersurfaceimmersed with bounded curvature in $M\times \R_+$ equals some $M\times \{s\}$provided $M$ is a complete, recurrent $n$-dimensional Riemannian manifold with$\text{Ric}_M \geq 0$ and whose sectional curvatures are bounded from above.For $H$-surfaces we prove that a stochastically complete surface $M$ can not bein the mean convex side of a $H$-surface $N$ embedded in $\R^3$ with boundedcurvature if $\sup \vert H_{_M}\vert < H$, or ${\rm dist}(M,N)=0$ when $\sup\vert H_{_M}\vert = H$. Finally, a maximum principle at infinity is shownassuming $M$ has non-empty boundary.

Stochastic half-space theorems for minimal surfaces and $H$-surfaces of $\mathbb{R}^{3}$