Besov Priors: Wavelet-Based Non-Gaussian Models

• Besov priors are non-Gaussian function-space distributions defined via wavelet expansions with scale weights reflecting Besov regularity.
• They promote sparsity and edge preservation by employing heavy-tailed coefficient laws, effectively modeling spatial inhomogeneity and sharp features.
• Applications span Bayesian inverse problems and imaging, ensuring discretization invariance and convergence to continuum variational formulations.

Besov priors are function-space prior distributions whose construction is tied to Besov regularity and, typically, to a wavelet expansion of the unknown. In the Bayesian literature they serve as non-Gaussian alternatives to Gaussian or Sobolev priors when the target object is expected to exhibit multiscale sparsity, edges, jumps, or spatial inhomogeneity rather than globally homogeneous smoothness. The modern formulation originates in the wavelet-based constructions of Lassas–Saksman–Siltanen and their extension to nonlinear inverse problems, and has since developed into a broad framework spanning Bayesian inverse problems, classification, density and intensity estimation, variational inference, and structured priors with tree, spatiotemporal, or variable-exponent geometry (Dashti et al., 2011).

1. Wavelet construction and function-space definition

A Besov prior is typically defined by randomizing the coefficients of a wavelet expansion with scale weights matched to a Besov norm. In a standard construction on the torus $T^d$, one writes

$$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$

where $\{\psi_l\}$ is an orthonormal basis, $s>0$, $1\le q<\infty$, $\kappa>0$, and the coefficients $\xi_l$ are i.i.d. with density

$$\pi_\xi(x)\propto \exp\!\left(-\frac12 |x|^q\right).$$

Formally, this induces a Gibbs-type law proportional to

$$\exp\!\left(-\frac{\kappa}{2}\|u\|_{X^{s,q}}^q\right),$$

with

$$\|f\|_{X^{s,q}} = \left( \sum_{l=1}^\infty l^{(sq/d+q/2-1)}|f_l|^q \right)^{1/q}.$$

When the basis is an $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$0-regular wavelet basis with $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$1, $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$2 coincides with the Besov space $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$3, so the prior becomes a $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$4 measure (Dashti et al., 2011).

The most prominent edge-preserving case is $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$5, where the prior is Laplace in wavelet coordinates. In the periodic $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$6 construction one uses

$$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$7

The Besov norm is then

$$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$8

so the prior has the formal density

$$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$9

This is the canonical Besov–Laplace prior in inverse-problem applications (Agapiou et al., 2017).

Equivalent coefficient characterizations appear throughout the literature. In one dimension,

$\{\psi_l\}$0

and for $\{\psi_l\}$1,

$\{\psi_l\}$2

Discrete implementations often write this as $\{\psi_l\}$3, with $\{\psi_l\}$4 a discrete wavelet transform and $\{\psi_l\}$5 a scale-weight matrix (Wang et al., 2016).

2. Sparsity, edge preservation, and regularity structure

The characteristic feature of Besov priors is that they encode regularity through weighted $\{\psi_l\}$6-summability of wavelet coefficients rather than through a quadratic Hilbert norm. For $\{\psi_l\}$7, and especially for $\{\psi_l\}$8, this yields heavier-tailed coefficient laws than the Gaussian case $\{\psi_l\}$9, with an associated preference for many small coefficients and comparatively few large ones. Because wavelets are localized in both position and scale, this coefficient geometry is naturally aligned with piecewise smooth signals, sharp interfaces, localized spikes, and blocky structures (Agapiou et al., 2017).

This contrast with Gaussian priors is central. Gaussian wavelet priors correspond to $s>0$0, hence to Sobolev-type quadratic penalties, whereas Laplace or more general $s>0$1-exponential Besov priors replace the $s>0$2-type geometry by an $s>0$3- or $s>0$4-type geometry in wavelet space. In applications to imaging and permeability reconstruction, Haar-based Besov priors are particularly prominent because Haar wavelets are spatially localized and represent jumps and sharp interfaces directly, whereas Fourier or trigonometric bases are global and tend to smear discontinuities or introduce Gibbs-type oscillations near jumps (Wacker et al., 2017).

The connection to spatial inhomogeneity is explicit in the nonparametric Bayes literature. The scale $s>0$5 is repeatedly identified as the canonical Besov class for spatially inhomogeneous functions, and one has

$s>0$6

This is the basis for viewing Besov–Laplace priors as Bayesian analogues of $s>0$7- or total-variation-like regularization, but realized as genuine infinite-dimensional probability measures rather than only as formal penalties (Giordano, 2022).

Besov priors also come with precise sample-path regularity statements. For the generic $s>0$8 construction, the following are equivalent: $s>0$9 almost surely, $1\le q<\infty$0 for $1\le q<\infty$1, and $1\le q<\infty$2. In the wavelet case this identifies the support of the prior in function space (Dashti et al., 2011). For the $1\le q<\infty$3-Laplace prior one has

$1\le q<\infty$4

for $1\le q<\infty$5 (Agapiou et al., 2017). A Fernique-like theorem also holds in Hölder scales: if $1\le q<\infty$6 and $1\le q<\infty$7, then $1\le q<\infty$8 for sufficiently small $1\le q<\infty$9 (Dashti et al., 2011).

These facts clarify a common misconception. Besov priors are not defined by edge preservation alone; they are probability measures whose edge-preserving behavior follows from a specific wavelet-domain geometry, regularity scale, and coefficient law.

3. Bayesian inverse problems, posterior well-posedness, and MAP estimators

The canonical inverse-problem model is

$\kappa>0$0

with unknown $\kappa>0$1 in an infinite-dimensional Banach space, finite-dimensional data $\kappa>0$2, forward map $\kappa>0$3, and Gaussian noise $\kappa>0$4. The posterior is defined by

$\kappa>0$5

with Gaussian data misfit

$\kappa>0$6

in the finite-dimensional noise setting (Agapiou et al., 2017). In the more general abstract formulation,

$\kappa>0$7

and posterior well-definedness and Hellinger stability follow under lower-bound, local boundedness, and local Lipschitz assumptions on $\kappa>0$8, combined with the Fernique-like integrability of the Besov prior (Dashti et al., 2011).

A major structural result concerns maximum a posteriori estimators. For the $\kappa>0$9-Besov prior, the generalized Onsager–Machlup functional of the posterior is

$\xi_l$0

and the small-ball ratio satisfies

$\xi_l$1

Hence minimizing $\xi_l$2 is equivalent to maximizing asymptotic posterior small-ball probabilities. In this setting weak MAP and strong MAP estimates coincide, and both are exactly the minimizers of $\xi_l$3 (Agapiou et al., 2017).

This result gives a precise infinite-dimensional meaning to the common computational practice of solving a Tikhonov-type problem with an $\xi_l$4-Besov penalty. It also explains why Besov MAP estimators are interpreted as sparsity-promoting and edge-preserving reconstructions: the penalty term is exactly the Besov norm appearing in the prior law, not an ad hoc discretized surrogate.

The same theory yields consistency results. With repeated observations

$\xi_l$5

the corresponding MAP functional is

$\xi_l$6

If $\xi_l$7 minimizes $\xi_l$8, then under a local Lipschitz assumption on $\xi_l$9 there exists a subsequence converging in $$\pi_\xi(x)\propto \exp\!\left(-\frac12 |x|^q\right).$$0 for any $$\pi_\xi(x)\propto \exp\!\left(-\frac12 |x|^q\right).$$1 to some $$\pi_\xi(x)\propto \exp\!\left(-\frac12 |x|^q\right).$$2 satisfying $$\pi_\xi(x)\propto \exp\!\left(-\frac12 |x|^q\right).$$3; injectivity of $$\pi_\xi(x)\propto \exp\!\left(-\frac12 |x|^q\right).$$4 yields $$\pi_\xi(x)\propto \exp\!\left(-\frac12 |x|^q\right).$$5 (Agapiou et al., 2017).

A related and practically important point is discretization invariance. The function-space Bayesian literature treats Besov priors as priors whose finite-dimensional approximations remain consistent under refinement, and for Besov–Laplace priors this underpins the claim that increasingly fine discretized MAP problems converge to the continuum variational problem rather than representing purely numerical artifacts (Dashti et al., 2011).

4. Computational methods and algorithmic consequences

The non-Gaussian and often non-Hilbertian structure of Besov priors makes posterior computation substantially different from Gaussian-prior sampling and optimization. Three computational themes recur.

First, wavelet parameterizations can accelerate MAP optimization. In the elliptic inverse problem for permeability reconstruction, the misfit gradient can be extracted in a Haar basis by solving one forward PDE, one adjoint PDE, and then taking a wavelet decomposition of the field $$\pi_\xi(x)\propto \exp\!\left(-\frac12 |x|^q\right).$$6. This replaces a prohibitively large number of directional PDE solves by a localized basis calculation and yields markedly faster optimization than trigonometric parameterizations in interface-rich problems (Wacker et al., 2017).

Second, several works transform Besov or more general $$\pi_\xi(x)\propto \exp\!\left(-\frac12 |x|^q\right).$$7-type priors into Gaussian reference variables to enable optimization-based sampling. For discrete $$\pi_\xi(x)\propto \exp\!\left(-\frac12 |x|^q\right).$$8 priors of the form

$$\pi_\xi(x)\propto \exp\!\left(-\frac12 |x|^q\right).$$9

with $$\exp\!\left(-\frac{\kappa}{2}\|u\|_{X^{s,q}}^q\right),$$0, a componentwise Laplace-to-Gaussian map $$\exp\!\left(-\frac{\kappa}{2}\|u\|_{X^{s,q}}^q\right),$$1 is used to write

$$\exp\!\left(-\frac{\kappa}{2}\|u\|_{X^{s,q}}^q\right),$$2

where $$\exp\!\left(-\frac{\kappa}{2}\|u\|_{X^{s,q}}^q\right),$$3. In transformed coordinates the posterior becomes Gaussian-prior-like and can be sampled by Metropolized randomize-then-optimize (RTO) (Wang et al., 2016). A later generalization treats Besov priors with generalized Gaussian coefficient laws, using a CDF-based coefficient transform, MAP computation in the transformed variable, and an RTO-MH sampler. Numerical studies there report discretization-invariant posterior means in deconvolution and effective uncertainty quantification in inpainting and sparse-angle CT (Horst et al., 20 Jun 2025).

Third, structured Besov priors require specialized MCMC. For Besov random tree priors, the parameter space includes both wavelet coefficients and a Galton–Watson tree, hence is Polish but not a normed linear space. Posterior sampling is handled by independence Metropolis–Hastings with proposal equal to the prior, while multilevel estimators use cutoffs to compensate for non-integrable contributions caused by the loss of uniform ellipticity in the forward PDE (Stein et al., 2023).

These algorithmic developments support a broader methodological point. Besov priors are not merely regularizers rewritten in probabilistic notation; they have generated dedicated optimization, sampling, and multilevel strategies precisely because their geometry differs materially from the Gaussian case.

5. Contraction theory, minimax rates, and adaptation

A large later literature studies Besov priors from a frequentist asymptotic viewpoint. In nonlinear inverse problems with Gaussian white noise, Laplace wavelet priors matched to $$\exp\!\left(-\frac{\kappa}{2}\|u\|_{X^{s,q}}^q\right),$$4 truth yield minimax-optimal contraction for spatially inhomogeneous signals, whereas Gaussian priors are proved to be limited by the slower linear minimax rate in the direct-observation case. For forward maps with smoothing order $$\exp\!\left(-\frac{\kappa}{2}\|u\|_{X^{s,q}}^q\right),$$5, the contraction rate for the forward image is

$$\exp\!\left(-\frac{\kappa}{2}\|u\|_{X^{s,q}}^q\right),$$6

and the Gaussian lower bound is polynomially slower over Besov classes with $$\exp\!\left(-\frac{\kappa}{2}\|u\|_{X^{s,q}}^q\right),$$7 (Agapiou et al., 2021).

In parallel, Besov–Laplace priors have been analyzed in non-inverse settings. In i.i.d. density estimation, carefully rescaled under-smoothing Laplace priors attain the optimal total-variation contraction rate

$$\exp\!\left(-\frac{\kappa}{2}\|u\|_{X^{s,q}}^q\right),$$8

for $$\exp\!\left(-\frac{\kappa}{2}\|u\|_{X^{s,q}}^q\right),$$9-smooth, spatially inhomogeneous densities, and hierarchical hyper-priors on the regularity parameter achieve adaptation over all $$\|f\|_{X^{s,q}} = \left( \sum_{l=1}^\infty l^{(sq/d+q/2-1)}|f_l|^q \right)^{1/q}.$$0 (Giordano, 2022). In the white noise model, $$\|f\|_{X^{s,q}} = \left( \sum_{l=1}^\infty l^{(sq/d+q/2-1)}|f_l|^q \right)^{1/q}.$$1-exponential priors with empirical-Bayes or hierarchical-Bayes tuning attain minimax or near-minimax rates over both homogeneous and inhomogeneous Besov spaces; the distinction is explicit: Gaussian priors are optimal in homogeneous regimes, while Laplace-type priors remain effective when $$\|f\|_{X^{s,q}} = \left( \sum_{l=1}^\infty l^{(sq/d+q/2-1)}|f_l|^q \right)^{1/q}.$$2 (Agapiou et al., 2022).

Recent papers extend the same pattern to classification and covariate-driven point-process intensity estimation. In binary classification with logistic link, rescaled Besov–Laplace priors lead to the contraction rate

$$\|f\|_{X^{s,q}} = \left( \sum_{l=1}^\infty l^{(sq/d+q/2-1)}|f_l|^q \right)^{1/q}.$$3

over $$\|f\|_{X^{s,q}} = \left( \sum_{l=1}^\infty l^{(sq/d+q/2-1)}|f_l|^q \right)^{1/q}.$$4-type truth, and a hierarchical hyper-prior on $$\|f\|_{X^{s,q}} = \left( \sum_{l=1}^\infty l^{(sq/d+q/2-1)}|f_l|^q \right)^{1/q}.$$5 yields adaptation to unknown regularity at the same optimal rate (Giordano, 9 Sep 2025). An analogous hierarchical Besov–Laplace construction for binary classification achieves

$$\|f\|_{X^{s,q}} = \left( \sum_{l=1}^\infty l^{(sq/d+q/2-1)}|f_l|^q \right)^{1/q}.$$6

in $$\|f\|_{X^{s,q}} = \left( \sum_{l=1}^\infty l^{(sq/d+q/2-1)}|f_l|^q \right)^{1/q}.$$7, again adaptively over $$\|f\|_{X^{s,q}} = \left( \sum_{l=1}^\infty l^{(sq/d+q/2-1)}|f_l|^q \right)^{1/q}.$$8, and is implemented by a dimension-robust wpCN-within-Gibbs sampler in whitened coordinates (Dolmeta et al., 26 Nov 2025). In increasing-domain asymptotics for covariate-based point-process intensity estimation, Besov–Laplace priors yield

$$\|f\|_{X^{s,q}} = \left( \sum_{l=1}^\infty l^{(sq/d+q/2-1)}|f_l|^q \right)^{1/q}.$$9

with the matched choice $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$00 giving the minimax-optimal $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$01 rate for $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$02-smooth intensities (Dolmeta et al., 11 May 2026).

Variational inference has also been incorporated into the same theory. For nonlinear inverse problems with PDE forward maps and Besov priors on $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$03, variational posteriors over Besov-type or mean-field families achieve the same statistical rate as the exact posterior up to logarithmic factors, and the resulting rates are minimax-optimal over $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$04 classes (Zu et al., 8 Aug 2025).

Setting	Prior regime	Representative rate/result
Nonlinear inverse problems	Laplace/Besov over $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$05	$$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$06 (Agapiou et al., 2021)
Density estimation	Rescaled under-smoothing Besov–Laplace	$$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$07 (Giordano, 2022)
Binary classification	Rescaled or hierarchical Besov–Laplace	$$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$08 (Giordano, 9 Sep 2025)
Covariate-based intensity estimation	Besov–Laplace with link function	$$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$09 (Dolmeta et al., 11 May 2026)

One consequence is that Besov priors are no longer confined to inverse problems in the narrow sense. They have become a general nonparametric Bayesian mechanism for modeling spatial inhomogeneity, with theory that now covers exact Bayes, empirical and hierarchical Bayes, and variational approximations.

6. Structured variants and generalizations

The basic wavelet-product construction has been generalized in several directions.

A first extension is the random tree Besov prior, where only a random subset of wavelet coefficients is active. The active set is generated by a Galton–Watson tree, while the retained coefficients remain $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$10-exponential. In one formulation,

$$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$11

This model introduces exact zeros, cross-scale support structure, and random fractal geometry. In the earlier fractal-imaging formulation, singularities are confined to a random fractal set $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$12, and on the event $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$13 its Hausdorff dimension is $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$14; in the Galton–Watson parameterization used later, when $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$15 the active set can survive indefinitely and has Hausdorff dimension $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$16 (Kekkonen et al., 2021). These priors motivate multilevel MCMC/FEM algorithms and, more recently, data-driven selection of the scale-dependent wavelet-density parameter $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$17 through hierarchical models and bottom-up tree-pruning MAP recursions (Stein et al., 2023).

A second extension is variable-index Besov priors. Here the regularity and integrability indices become functions of position, producing priors on spaces $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$18 defined by wavelet coefficients with generalized $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$19-exponential laws. The resulting non-Gaussian prior admits exponential-integrability and sample-regularity results analogous to the constant-index case, and supports posterior well-posedness for inverse problems including integer and fractional backward diffusion (Jia et al., 2015).

A third direction is spatiotemporal Besov modeling. The spatiotemporal Besov process replaces scalar wavelet coefficients by stochastic time functions: $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$20 with $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$21-exponential temporal processes controlling temporal correlation. This construction preserves the spatial edge-preserving features of Besov priors while incorporating temporal dependence, and includes the spatiotemporal Gaussian process as the special case $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$22 (Lan et al., 2023).

A fourth line of work uses Besov priors in decomposition models. For linear inverse problems where the unknown is $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$23 with one piecewise constant and one smooth component, one may place a Haar-wavelet Besov prior on $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$24 and a smooth-wavelet Besov prior on $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$25, or combine a smooth Besov prior with a hierarchical Gaussian prior on the gradient of the jump component. This exposes an important modeling issue: two simultaneous wavelet-based Besov components can suffer from identifiability or coherence problems, so hybrid constructions may be preferable when distinct structural components are to be separated (Horst et al., 23 Jun 2025).

Variant	Construction	Representative papers
Random tree Besov	$$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$26-exponential coefficients on a Galton–Watson-supported wavelet tree	(Kekkonen et al., 2021, Stein et al., 2023)
Variable-index Besov	Wavelet series in $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$27 with generalized $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$28-exponential coefficients	(Jia et al., 2015)
Spatiotemporal Besov process	Wavelet series with temporally correlated $$u(x) = \sum_{l=1}^\infty l^{-\left(\frac{s}{d}+ \frac{1}{2} -\frac{1}{q}\right)}\left(\frac{1}{\kappa}\right)^{1/q}\xi_l\,\psi_l(x),$$29-exponential coefficient processes	(Lan et al., 2023)
Decomposition priors	Multiple Besov components or Besov-plus-gradient hierarchies	(Horst et al., 23 Jun 2025)

These structured variants underscore the breadth of the framework. The core Besov prior remains a wavelet-scaled non-Gaussian law reflecting Besov geometry; the variants alter support structure, local regularity, temporal dependence, or multicomponent composition without abandoning that underlying principle.