Publication

Regularity Structures on Manifolds and Vector Bundles

2023

Abstract

We develop a generalisation of the original theory of regularity structures, [Hai14], which is able to treat SPDEs on manifolds with values in vector bundles. Assume M is a Riemannian manifold and E→M and Fi→M are vector bundles (with a metric and connection), this theory allows to solve subcritical equations of the form ∂tu+u=∑i=0mGi(u,∇u,…,∇nu)ξi , where u is a (generalised) section of E,  is a uniformly elliptic operator on E of order strictly greater than n, the ξi are Fi-valued random distributions (e.g. Fi-valued white noises), and the Gi:E×TM∗⊗E×…×(TM∗)⊗n⊗E→L(Fi,E) are local functions. We apply our framework to three example equations which illustrate that when  is a Laplacian it is possible in most cases to renormalise such equations by adding spatially homogeneous counterterms and we discuss in which cases more sophisticated renormalisation procedures (involving the curvature of the underlying manifold) are required.

Official source

https://infoscience.epfl.ch/entities/publication/64788c22-8732-419a-9560-9f5f54dd8c8a

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2023

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/64788c22-8732-419a-9560-9f5f54dd8c8a

About this result

Related concepts (32)

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

Kähler manifold

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.

Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.

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