  A five-dimensional scenario with a non compact extra dimension of infinite
extent is studied, in which a single three-brane is affected by small Gaussian
fluctuations in the extra dimension. The average magnitude of the fluctuations
is of order of the electro-weak length scale ($\sigma\sim m_{EW}^{-1}$). The
model provides an stochastic approach to gravity that accounts for an
alternative resolution of the mass hierarchy problem. The cosmological constant
problem can be suitably treated as well. Surprisingly the Mach's principle
finds a place in the model. It is argued that the Mach's principle, the mass
hierarchy and the cosmological constant problem, are different aspects of a
same property of gravity in this model: its stochastic character. Thin-brane
scenarios are recovered in the "no-fluctuations" limit ($\sigma\to 0$).
