  The problems considered in the present paper have their roots in two
different cultures. The 'true' (or myopic) self-avoiding walk model (TSAW) was
introduced in the physics literature by Amit, Parisi and Peliti. This is a
nearest neighbor non-Markovian random walk in Z^d which prefers to jump to
those neighbors which were less visited in the past. The self-repelling
Brownian polymer model (SRBP), initiated in the probabilistic literature by
Durrett and Rogers (independently of the physics community), is the continuous
space-time counterpart: a diffusion in R^d pushed by the negative gradient of
the (mollified) occupation time measure of the process. In both cases, similar
long memory effects are caused by a pathwise self-repellency of the
trajectories due to a push by the negative gradient of (softened) local time.
We investigate the asymptotic behaviour of TSAW and SRBP in the non-recurrent
dimensions. First, we identify a natural stationary (in time) and ergodic
distribution of the environment (the local time profile) as seen from the
moving particle. The main results are diffusive limits. In the case of TSAW,
for a wide class of self-interaction functions, we establish diffusive lower
and upper bounds for the displacement and for a particular, more restricted
class of interactions, we prove full CLT for the finite dimensional
distributions of the displacement. In the case of SRBP, we prove full CLT
without restrictions on the interaction functions. These results settle part of
the conjectures, based on non-rigorous renormalization group arguments (equally
'valid' for the TSAW and SRBP cases). The proof of the CLT follows the
non-reversible version of Kipnis-Varadhan theory. On the way to the proof, we
slightly weaken the so-called graded sector condition.
