Publication

Spectral gap for projective processes of linear SPDEs

2023

Abstract

This work studies the angular component πt=ut/‖ut‖ associated to the solution u of a vector-valued linear hyperviscous SPDE on a d-dimensional torus duα=−να(−Δ)auαdt+(u⋅dW)α,α∈{1,…,m} for u:𝕋d→ℝm, a⩾1 and a sufficiently smooth and non-degenerate noise W. We provide conditions for existence, as well as uniqueness and spectral gaps (if a>d/2) of invariant measures for π in the projective space. Our proof relies on the introduction of a novel Lyapunov functional for πt, based on the study of dynamics of the ``energy median'': the energy level M at which projections of u onto frequencies with energies less or more than M have about equal L2 norm. This technique is applied to obtain -- in an infinite-dimensional setting without order preservation -- lower bounds on top Lyapunov exponents of the equation, and their uniqueness via Furstenberg-Khasminskii formulas.

Official source

https://infoscience.epfl.ch/entities/publication/2bf60c3b-7ce2-408e-b371-e00993927f5d

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2023

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/2bf60c3b-7ce2-408e-b371-e00993927f5d

About this result

Related concepts (32)

Duality (projective geometry)

In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration.

Projective space

In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs.

Complex projective space

In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines through the origin of a complex Euclidean space (see below for an intuitive account). Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space.