Regularity Structures on Manifolds and Vector Bundles

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 $E\to M$ and $F^i\to M$ are vector bundles (with a metric and connection), this theory allows to solve subcritical equations of the form $$ \partial_t u + \mathcal{L}u = \sum_{i=0}^m G_i(u, \nabla u,\ldots, \nabla^n u)\xi_i\ , $$ where $u$ is a (generalised) section of $E$, $\mathcal{L}$ is a uniformly elliptic operator on $E$ of order strictly greater than $n$, the $\xi_i$ are $F^i$-valued random distributions (e.g. $F^i$-valued white noises), and the $G_i:E\times TM^*\otimes E \times\ldots\times (TM^*)^{\otimes n} \otimes E \to L(F^i ,E)$ are local functions. We apply our framework to three example equations which illustrate that when $\mathcal{L}$ 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.