Decorated Trees for singular (stochastic) PDEs

We will introduce the notion of decorated trees and their associated multi-pre-Lie/Hopf algebraic structures. We will see how this structure can be used in numerous applications ranging from singular SPDEs up to a numerical scheme for dispersive PDEs. This structure is also well adapted for handling renormalisation and symmetries.

The key references are:

[1] I. Bailleul, Y. Bruned, "Renormalised singular stochastic PDEs", 16 pages, arxiv:2101.11949 .

[2] Y. Bruned, F. Gabriel, M. Hairer, L. Zambotti, "Geometric stochastic heat equations", Journal of the American Mathematical Society, 2022, Volume 35, Issue 1, pp 1-80. arxiv:1902.02884 . doi:10.1090/jams/977 .

[3] Y. Bruned, D. Manchon, "Algebraic deformation for (S)PDEs", 43 pages, submitted, arxiv:2011.05907 .

[4] Y. Bruned, K. Schratz, "Resonance based schemes for dispersive equations via decorated trees", Forum of Mathematics, Pi, 2022, Volume 10, e2, pp 1 76. arxiv:2005.01649 . doi:10.1017/fmp.2021.13.