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We present on-diagonal heat kernel estimates and quantitative homogenizationstatements for the one-dimensional Bouchaud trap model. The heat kernelestimates are obtained using standard techniques, with key inputs coming from acareful analysis of the volume growth of the invariant measure of the processunder study. As for the quantitative homogenization results, these include bothquenched and annealed Berry-Esseen-type theorems, as well as a quantitativequenched local limit theorem. Whilst the model we study here is a particularlysimple example of a random walk in a random environment, we believe the roadmapwe provide for establishing the latter result in particular will be useful forderiving quantitative local limit theorems in other, more challenging,settings.

Heat kernel fluctuations and quantitative homogenization for the one-dimensional Bouchaud trap model