Title: An energy method for rough partial differential equations,
Abstract: 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 en- ergy estimates which are deduced
directly from the equation. Existence is obtained by showing compactness of a
suitable sequence of approximate solutions whereas unique- ness 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.