  In this paper we study the Dirichlet problem corresponding to an open bounded
set $D\subset \mathbb{R}^{d}$ and the operator \begin{equation*}
A=\sum_{i=1}^{d}a\frac{\partial ^{2}}{\partial x_{i}^{2}}
+\sum_{i=1}^{d}b_{i}\frac{\partial }{\partial x_{i}}, \end{equation*} where
$a>0$ and $b\in \mathbb{R}^{d}$. We define a mean value property and prove that
a function $u$ has such property in $D$ if and only if $Au=0$ in $D$. Using
this characterization, and a drifted Brownian motion, we define a family of
random variables that converges almost surely and the limit is used to give an
explicit representation for the solutions to the Dirichlet problem, this
immediately implies the uniqueness. On the other hand, the existence of the
solution is proved imposing a regular condition on the boundary of $D$.
