  We study a semilinear elliptic equation with a pure power nonlinearity with
exponent $p>1$, and provide sufficient conditions for the existence of positive
solutions. These conditions involve expected exit times from the domain, $D$,
where a solution is defined, and expected occupation times in suitable
subdomains of $D$. They provide an alternative new approach to the geometric or
topological sufficient conditions given in the literature for exponents close
to the critical Sobolev exponent. Moreover, unlike standard results, in our
probabilistic approach no \emph{a priori} upper bound restriction is imposed on
$p$, which might be supercritical. The proof is based on a fixed point argument
using a probabilistic representation formula. We also prove a multiplicity
result and discuss possible extensions to the existence of sign changing
solutions. Finally, we conjecture that necessary conditions for the existence
of solutions might be obtained using a similar probabilistic approach. This
motivates a series of natural questions related to the characterisation of
topological and geometrical properties of a domain in probabilistic terms.
