  We consider linear parabolic equations on a random non-cylindrical domain.
Utilizing the domain mapping method, we write the problem as a partial
differential equation with random coefficients on a cylindrical deterministic
domain. Exploiting the deterministic results concerning equations on
non-cylindrical domains, we state the necessary assumptions about the velocity
filed and in addition, about the flow transformation that this field generates.
In this paper we consider both cases, the uniformly bounded with respect to the
sample and log-normal type transformation. In addition, we give an explicit
example of a log-normal type transformation and prove that it does not satisfy
the uniformly bounded condition. We define a general framework for considering
linear parabolic problems on random non-cylindrical domains. As the first
example, we consider the heat equation on a random tube domain and prove its
well-posedness. Moreover, as the other example we consider the parabolic Stokes
equation which illustrates the case when it is not enough just to study the
plain-back transformation of the function, but instead to consider for example
the Piola type transformation, in order to keep the divergence free property.
