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We characterize the Besov-Lipschitz spaces with zero boundary conditions on bounded smooth domains. We prove that the appropriate first and second difference norms are equivalent to the norm given in terms of the transition kernel of the Brownian motion killed upon exit from the domain.

It is commonly believed that the nature of the generating structure of a stochastic process can be determined by examining the distributional form of the stachastic process in time. For example, it is believed that correlations between time intervals can be used to distinguish between contagion (after-effect) and heterogeneity (stratification). These beliefs are incorrect, and no amount of statistical information internal to the sampled process, is sufficient for such a distinction.