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A new class of stochastic processes, termed Brownian semistationary processes (BSS), is introduced and discussed. This class has similarities to that of Brownian semimartingales (BSM), but is mainly directed towards the study of stationary processes, and BSS processes are not in general of the semimartingale type. We focus on semimartingale-nonsemimartingale issues and on inference problems concerning the underlying volatility/intermittency process, in the nonsemimartingale case and based on normalised realised quadratic variation. The concept of BSS processes has arisen out of an ongoing study of turbulent velocity fields and is the purely temporal version of the general tempo-spatial framework of ambit processes. The latter, which may have applications also to the finance of energy markets, is briefly considered at the end of the paper, again with reference to the question of inference on the volatility …