  The probabilistic representation of weak solutions to a parabolic boundary
value problem is established in the following framework. The boundary value
problem consists of a second order parabolic equation defined on a time-varying
Lipschitz domain in a Euclidean space and of a mixed boundary condition
composed of a Robin and the homogeneous Dirichlet conditions. It is assumed
that the time-varying domain is included in a fixed smooth domain and that a
certain part of the boundary of the time-varying domain is also included in the
boundary of the fixed domain, say the fixed boundary. The Robin condition is
imposed on a part of the boundary included in the fixed one and the Dirichlet
condition on the rest of the boundary. Such a parabolic boundary value problem
always has a unique weak solution for given data; however it does not possess a
classical or strong solution in general, even in the case of equations with
constant coefficients. The stochastic solution to the boundary value problem is
also considered and, by showing the equality between both the solutions, it is
obtained the probabilistic representation for the weak solution. Furthermore,
it is ensured that, for the weak solution, the stochastic solution gives a
version which is continuous up to the lateral boundary of the domain except the
border of the adjoining place imposed each of the boundary conditions. As an
application, it is shown the continuity property of a functional (cost
function) related to an optimal stopping problem motivated by an inverse
problem determining the shape of a domain.
