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We introduce a new notion of irregularity of paths, in terms of control ofgrowth of the size of small balls by means of the occupation measure of thepath. This notion ensures Besov regularity of the occupation measure and thusextends the analysis of Catellier and Gubinelli (2016) to general Besov spaces.On stochastic processes this notion is granted by suitable properties of localnon-determinism.

Yet another notion of irregularity through small ball estimates