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Inmathematics — specifically, in the fields ofprobability theory andinverse problems —Besov measures and associatedBesov-distributed random variables are generalisations of the notions ofGaussian measures andrandom variables,Laplace distributions, and other classical distributions. They are particularly useful in the study ofinverse problems onfunction spaces for which a GaussianBayesian prior is an inappropriate model. The construction of a Besov measure is similar to the construction of aBesov space, hence the nomenclature.

Let${\displaystyle H}$ be aseparableHilbert space of functions defined on a domain${\displaystyle D\subseteq {\mathbb{R}}^d}$, and let${\displaystyle \{e_n\mid n\in \mathbb{N}\}}$ be acomplete orthonormal basis for${\displaystyle H}$. Let${\displaystyle s\in \mathbb{R}}$ and. For${\displaystyle u=\sum _{n\in \mathbb{N}}{u}_n{e}_n\in H}$, define

${\displaystyle \Vert u{\Vert }_{X^{s,p}}={\Vert \sum _{n\in \mathbb{N}}{u}_n{e}_n\Vert }_{X^{s,p}}:={\left(\sum _{n=1}^{\mathrm{\infty }}{n}^{(\frac{ps}{d}+\frac{p}{2}-1)}|u_n{|}^p\right)}^{1/p}.}$

This defines anorm on the subspace of${\displaystyle H}$ for which it is finite, and we let${\displaystyle X^{s,p}}$ denote thecompletion of this subspace with respect to this new norm. The motivation for these definitions arises from the fact that${\displaystyle \Vert u{\Vert }_{X^{s,p}}}$ is equivalent to the norm of${\displaystyle u}$ in the Besov space${\displaystyle B_{pp}^s(D)}$.

Let${\displaystyle \kappa >0}$ be a scale parameter, similar to the precision (the reciprocal of thevariance) of a Gaussian measure. We now define a${\displaystyle X^{s,p}}$-valued random variable${\displaystyle u}$ by

${\displaystyle u:=\sum _{n\in \mathbb{N}}{n}^{-(\frac{s}{d}+\frac{1}{2}-\frac{1}{p})}{\kappa }^{-\frac{1}{p}}{\xi }_n{e}_n,}$

where${\displaystyle {\xi }_1,{\xi }_2,\dots }$ are sampled independently and identically from the generalized Gaussian measure on${\displaystyle \mathbb{R}}$ with Lebesgueprobability density function proportional to${\displaystyle \exp (-\frac{1}{2}|{\xi }_n{|}^p)}$. Informally,${\displaystyle u}$ can be said to have a probability density function proportional to${\displaystyle \exp (-\frac{\kappa }{2}\Vert u{\Vert }_{X^{s,p}}^p)}$ with respect to infinite-dimensional Lebesgue measure (which does not make rigorous sense), and is therefore a natural candidate for a "typical" element of${\displaystyle X^{s,p}}$ (although this Is not quite true — see below).

It is easy to show that, whent ≤ s, theX^t,p norm is finite whenever theX^s,p norm is. Therefore, the spacesX^s,p andX^t,p are nested:

${\displaystyle X^{s,p}\subseteq X^{t,p}\text{ when }t\le s.}$

This is consistent with the usual nesting of smoothness classes of functionsf: D → R:for example, theSobolev spaceH²(D) is a subspace ofH¹(D) and in turn of theLebesgue spaceL²(D) =H⁰(D); theHölder spaceC¹(D) of continuously differentiable functions is a subspace of the spaceC⁰(D) of continuous functions.

It can be shown that the series definingu converges inX^t,palmost surely for anyt < s − d / p, and therefore gives a well-definedX^t,p-valued random variable. Note thatX^t,p is a larger space thanX^s,p, and in fact thee random variableu isalmost surelynot in the smaller spaceX^s,p. The spaceX^s,p is rather the Cameron-Martin space of this probability measure in the Gaussian casep = 2. The random variableu is said to beBesov distributed with parameters (κ,s,p), and the inducedprobability measure is called aBesov measure.

• Abstract Wiener space – Mathematical construction relating to infinite-dimensional spaces
• Cameron–Martin theorem – Theorem defining translation of Gaussian measures (Wiener measures) on Hilbert spaces.
• Feldman–Hájek theorem – Theory in probability theory
• Structure theorem for Gaussian measures – Mathematical theorem
• There is no infinite-dimensional Lebesgue measure – Mathematical folklorePages displaying short descriptions of redirect targets

• Dashti, Masoumeh; Harris, Stephen;Stuart, Andrew M. (2012). "Besov priors for Bayesian inverse problems".Inverse Problems & Imaging.6 (2):183–200.arXiv:1105.0889.doi:10.3934/ipi.2012.6.183.ISSN 1930-8337.MR 2942737.S2CID 88518742.
• Lassas, Matti; Saksman, Eero; Siltanen, Samuli (2009). "Discretization-invariant Bayesian inversion and Besov space priors".Inverse Problems & Imaging.3 (1):87–122.arXiv:0901.4220.doi:10.3934/ipi.2009.3.87.ISSN 1930-8337.MR 2558305.S2CID 14122432.

• Inverse problems
• Measures (measure theory)
• Theory of probability distributions

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