In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
— specifically, in the fields of
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
inverse problem
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...
s — Besov measures and associated Besov-distributed random variables are generalisations of the notions of
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named ...
s and
random variables
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
,
Laplace distribution
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponen ...
s, and other classical distributions. They are particularly useful in the study of
inverse problems
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...
on
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s for which a Gaussian
Bayesian prior
In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into ...
is an inappropriate model. The construction of a Besov measure is similar to the construction of a
Besov space In mathematics, the Besov space (named after Oleg Vladimirovich Besov) B^s_(\mathbf) is a Complete metric space, complete quasinormed space which is a Banach space when . These spaces, as well as the similarly defined Triebel–Lizorkin spaces, ser ...
, hence the nomenclature.
Definitions
Let
be a
separable Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
of functions defined on a domain
, and let
be a
complete orthonormal basis for
. Let
and
. For
, define
:
This defines a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
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
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
) 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
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
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
(although this Is not quite true — see below).
Properties
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:
:
This is consistent with the usual nesting of smoothness classes of functions ''f'': ''D'' → R:
for example, the
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
''H''
2(''D'') is a subspace of ''H''
1(''D'') and in turn of the
Lebesgue space ''L''
2(''D'') = ''H''
0(''D''); the
Hölder space Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
* Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modu ...
''C''
1(''D'') of continuously differentiable functions is a subspace of the space ''C''
0(''D'') of continuous functions.
It can be shown that the series defining ''u'' converges in ''X''
''t'',''p'' almost surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
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
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
''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
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...
is called a Besov measure.
See also
*
*
*
*
*
References
*
*
{{Measure theory
Inverse problems
Measures (measure theory)
Theory of probability distributions