Besov measure

In mathematics — specifically, in the fields of probability theory and inverse problems — Besov 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 

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.

References 

 
 

Category:Inverse problems
Category:Measures (measure theory)
Category:Theory of probability distributions