Besov measure

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In mathematics — specifically, in the fields of probability theory and inverse problemsBesov measures and associated Besov-distributed random variables are generalisations of the notions of Gaussian measures and random variables, Laplace distributions, and other classical distributions. They are particularly useful in the study of inverse problems on function spaces for which a Gaussian Bayesian prior is an inappropriate model. The construction of a Besov measure is similar to the construction of a Besov space, hence the nomenclature.

Definitions[edit]

Let be a separable Hilbert space of functions defined on a domain , and let be a complete orthonormal basis for . Let and . For , define

This defines a norm on the subspace of for which it is finite, and we let denote the completion of this subspace with respect to this new norm. The motivation for these definitions arises from the fact that is equivalent to the norm of in the Besov space .

Let be a scale parameter, similar to the precision (the reciprocal of the variance) of a Gaussian measure. We now define a -valued random variable by

where are sampled independently and identically from the generalized Gaussian measure on with Lebesgue probability density function proportional to . Informally, can be said to have a probability density function proportional to with respect to infinite-dimensional Lebesgue measure (which does not make rigorous sense), and is therefore a natural candidate for a “typical” element of . It can be shown that the series defining converges in almost surely for any , and therefore gives a well-defined -valued random variable. Note that in particular this random variable is almost surely not in . The space is rather the Cameron-Martin space of this probability measure in the Gaussian case . The random variable is said to be Besov distributed with parameters , and the induced probability measure on is called a Besov measure.

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