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The problems considered in the present paper have their roots in twodifferent cultures. The 'true' (or myopic) self-avoiding walk model (TSAW) wasintroduced in the physics literature by Amit, Parisi and Peliti. This is anearest neighbor non-Markovian random walk in Z^d which prefers to jump tothose neighbors which were less visited in the past. The self-repellingBrownian polymer model (SRBP), initiated in the probabilistic literature byDurrett and Rogers (independently of the physics community), is the continuousspace-time counterpart: a diffusion in R^d pushed by the negative gradient ofthe (mollified) occupation time measure of the process. In both cases, similarlong memory effects are caused by a pathwise self-repellency of thetrajectories due to a push by the negative gradient of (softened) local time.We investigate the asymptotic behaviour of TSAW and SRBP in the non-recurrentdimensions. First, we identify a natural stationary (in time) and ergodicdistribution of the environment (the local time profile) as seen from themoving particle. The main results are diffusive limits. In the case of TSAW,for a wide class of self-interaction functions, we establish diffusive lowerand upper bounds for the displacement and for a particular, more restrictedclass of interactions, we prove full CLT for the finite dimensionaldistributions of the displacement. In the case of SRBP, we prove full CLTwithout restrictions on the interaction functions. These results settle part ofthe conjectures, based on non-rigorous renormalization group arguments (equally'valid' for the TSAW and SRBP cases). The proof of the CLT follows thenon-reversible version of Kipnis-Varadhan theory. On the way to the proof, weslightly weaken the so-called graded sector condition.

Diffusive limits for "true" (or myopic) self-avoiding random walks and self-repellent Brownian polymers in d >= 3