  We present a well-posedness and stability result for a class of nondegenerate
linear parabolic equations driven by rough paths. More precisely, we introduce
a notion of weak solution that satisfies an intrinsic formulation of the
equation in a suitable Sobolev space of negative order. Weak solutions are then
shown to satisfy the corresponding energy estimates which are deduced directly
from the equation. Existence is obtained by showing compactness of a suitable
sequence of approximate solutions whereas uniqueness relies on a doubling of
variables argument and a careful analysis of the passage to the diagonal. Our
result is optimal in the sense that the assumptions on the deterministic part
of the equation as well as the initial condition are the same as in the
classical PDEs theory.
