  Second initial boundary problem in narrow domains of width $\epsilon\ll 1$
for linear second order differential equations with nonlinear boundary
conditions is considered in this paper. Using probabilistic methods we show
that the solution of such a problem converges as $\epsilon \downarrow 0$ to the
solution of a standard reaction-diffusion equation in a domain of reduced
dimension. This reduction allows to obtain some results concerning wave front
propagation in narrow domains. In particular, we describe conditions leading to
jumps of the wave front.
