There are two notions of sparsity associated to a frame Ψ = (ψ i ) i∈I : Analysis sparsity of f means that the analysis coefficients ( f, ψ i ) i∈I are sparse, while synthesis sparsity means that we can write f = i∈I c i ψ i with sparse synthesis coefficients (c i ) i∈I . Here, sparsity of a sequence c = (c i ) i∈I means c ∈ ℓ p (I) for a given p < 2. We show that both notions of sparsity coincide if Ψ = SH (ϕ, ψ; δ) = (ψ i ) i∈I is a discrete (cone-adapted) shearlet frame with sufficiently nice generators ϕ, ψ and sufficiently small sampling density δ > 0. The required 'niceness' of ϕ, ψ is explicitly quantified in terms of Fourier-decay and vanishing moment conditions. In addition to ℓ p -sparsity, we even allow weighted ℓ p -spaces ℓ p w s as a sparsity measure, with weights of the form w s = 2 js (j,ℓ,δ,k) where j encodes the scale of the corresponding shearlet elements.More precisely, we show that the shearlet smoothness spaces S p,q s R 2 introduced by Labate et al. simultaneously characterize analysis and synthesis sparsity with respect to a shearlet frame, in the sense that-for suitable ϕ, ψ, δ-the following are equivalent:As an application, we prove that shearlets yield (almost) optimal approximation rates for the class of cartoonlike functions:, where f N is a linear combination of N shearlets. This might appear to be a well-known statement, but an inspection of the existing proofs reveals that these only establish analysis sparsity of cartoon-like functions, which implies f, where g N is a linear combination of N elements of the dual frame Ψ to the shearlet frame Ψ. This is not completely satisfying, since only limited knowledge about the structure and properties of Ψ is available.In addition to classical shearlets, we also consider more general α-shearlet systems. For these, the parabolic scaling is replaced by α-parabolic scaling. The resulting systems range from ridgelet-like systems (for α = 0) over classical shearlets (α = 1 2 ) to wavelet-like systems (α = 1). In this more general case, the shearlet smoothness spaces S p,q s R 2 have to be replaced by the α-shearlet smoothness spaces S p,q α,s R 2 . We completely characterize the existence of embeddings between these spaces for different values of α. This allows us to decide whether sparsity with respect to α 1 -shearlets implies sparsity with respect to α 2 -shearlets, even for α 1 = α 2 .
Estimates for sample paths of fast–slow stochastic ordinary differential equations have become a key mathematical tool relevant for theory and applications. In particular, there have been breakthroughs by Berglund and Gentz to prove sharp exponential error estimates. In this paper, we take the first steps in order to generalise this theory to fast–slow stochastic partial differential equations. In a simplified setting with a natural decomposition into low- and high-frequency modes, we demonstrate that for a short-time period the probability for the corresponding sample path to leave a neighbourhood around the stable slow manifold of the system is exponentially small as well.
We prove the existence of a global random attractor for a certain class of stochastic partly dissipative systems. These systems consist of a partial (PDE) and an ordinary differential equation (ODE), where both equations are coupled and perturbed by additive white noise. The deterministic counterpart of such systems and their long-time behaviour have already been considered but there is no theory that deals with the stochastic version of partly dissipative systems in their full generality. We also provide several examples for the application of the theory.
We prove the existence of a global random attractor for a certain class of stochastic partly dissipative systems. These systems consist of a partial and an ordinary differential equation, where both equations are coupled and perturbed by additive white noise. The deterministic counterpart of such systems and their long-time behaviour have already been considered but there is no theory that deals with the stochastic version of partly dissipative systems in their full generality. We also provide several examples for the application of the theory.
The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton-Watson tree conditioned to survive. We prove that the solution at time t is concentrated at a single site with high probability and at two sites almost surely as t → ∞. Moreover, we identify asymptotics for the localisation sites and the total mass, and show that the solution u(t, v) at a vertex v can be well-approximated by a certain functional of v. The main difference with earlier results on Z d is that we have to incorporate the effect of variable vertex degrees within the tree, and make the role of the degrees precise.
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