Besov measure

Let ${\displaystyle H}$ be a separable Hilbert space of functions defined on a domain ${\displaystyle D\subseteq \mathbb {R} ^{d}}$, and let ${\displaystyle \{e_{n}\mid n\in \mathbb {N} \}}$ be a complete orthonormal basis for ${\displaystyle H}$. Let ${\displaystyle s\in \mathbb {R} }$ and ${\displaystyle 1\leq p<\infty }$. For ${\displaystyle u=\sum _{n\in \mathbb {N} }u_{n}e_{n}\in H}$, define

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

This defines a norm on the subspace of ${\displaystyle H}$ for which it is finite, and we let ${\displaystyle X^{s,p}}$ denote the completion of this subspace with respect to this new norm. The motivation for these definitions arises from the fact that ${\displaystyle \|u\|_{X^{s,p}}}$ 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 the variance) 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 Lebesgue probability density function proportional to ${\displaystyle \exp(-{\tfrac {1}{2}}|\xi _{n}|^{p})}$. Informally, ${\displaystyle u}$ can be said to have a probability density function proportional to ${\displaystyle \exp(-{\tfrac {\kappa }{2}}\|u\|_{X^{s,p}}^{p})}$ 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, when t ≤ s, the X^t,p norm is finite whenever the X^s,p norm is. Therefore, the spaces X^s,p and X^t,p are nested:

${\displaystyle X^{s,p}\subseteq X^{t,p}{\mbox{ when }}t\leq s.}$

This is consistent with the usual nesting of smoothness classes of functions f: D → R: for example, the Sobolev space H²(D) is a subspace of H¹(D) and in turn of the Lebesgue space L²(D) = H⁰(D); the Hölder space C¹(D) of continuously differentiable functions is a subspace of the space C⁰(D) of continuous functions.

It can be shown that the series defining u converges in X^t,p almost surely for any t < s − d / p, and therefore gives a well-defined X^t,p-valued random variable. Note that X^t,p is a larger space than X^s,p, and in fact thee random variable u is almost surely not in the smaller space X^s,p. The space X^s,p is rather the Cameron-Martin space of this probability measure in the Gaussian case p = 2. The random variable u is said to be Besov distributed with parameters (κ, s, p), and the induced probability measure is called a Besov measure.

• 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.