Welcome to Harprit Singh's Website

I have mostly been working on Stochastic Partial Differential Equations (SPDEs). More specifically, the following are my articles/preprints:

• Rough geometric integration: We introduce a notion of distributional k-forms on d-dimensional manifolds which can be integrated against suitably regular k-submanifolds. The approach combines ideas from Whitney's geometric integration with those of sewing approaches to rough integration. Joint work with A. Chandra, arxiv.

• Periodic space-time homogenisation of the phi^4_2 equation: We consider the homogenisation problem for the phi^4_2 equation and establish joint continuity of the solution map in the regularisation and homogenisation lenth-scale. Furthermore we investigate the required renormalisation counterterms for several regularisation schemes. Joint work with  M. Hairer, arxiv.

• Canonical solutions to non-tranlsattion invariant singular SPDEs: We exhibit a canonical, finite dimensional solution familly for certain singular SPDEs and show that the diverging renormalisation functions are proportional to local functions of the differential operator. We also establish continuity of the solution map with respect to the differential operator for these equations, arxiv. 

• Regularity Structures on Manifolds and Vector bundles: We develop a generalisation of the original Theory of Regularity Structures which is able to treat SPDEs on manifolds with values in vector bundles. This is done in full generality. Joint work with M. Hairer, arxiv.

• Singular SPDEs on homogeneous Lie Groups: We show that Regularity Structures, without too much change, work on general homogeneous Lie groups. In particular we obtain a solution theory for SPDEs where the differential operator stems from a large class of hypo-elliptic operators. Joint work with A. Mayorcas, arxiv.

• Strong convergence of parabolic rate 1 of discretisations of stochastic Allen-Cahn-type equations: We show that the optimal convergence rate of a fully discrete space-time explicit finite difference approximation to a class of stochastic Allen-Cahn-type equations driven by space time white noise in 1+1 dimensions can be made arbitrarily close to (but not exceeding) 1, and not just 1/2, if studied in an appropriate Besov norm.  Joint work with M. Gerencser, Transactions of the AMS, arxiv version.

• An elementary proof of the reconstruction theorem: We provide a proof of the reconstruction theorem as the unique extension of the explicitly given reconstruction operator on smooth models. This relies on new mollification procedures for models as well as the introduction of a universal dense subspace of modelled distributions. All results are formulated in a generic (p,q) Besov setting. Joint work with J. Teichmann, arxiv.

Some related activities during my PhD: 

• Participation at the Oberwolfach Arbeitsgemeinschaft: Quantitative Stochastic Homogenization (2022)

• Teaching exercise classes on "Progress on Yang Mills",  XXV Brazilian School of Probability (2022)

• Teaching exercise classes on "Brownian Motion and Stochastic Calculus", ETH Zürich (2021)