Singular paths spaces and applications

Motivated by recent applications in rough volatility and regularity structures, notably the notion of singular modelled distribution, we study paths, paths related objects with a quantified singularity at zero. In pure path setting this allows us to leverage on existing SLE Besov estimates see that traces takes values H\"older space, which quantifies well-known boundary effect regime $\kappa < 1$. We then consider integration theory against some extensions thereof. This gives method reconcile, from structure point view, different kernels used construct (fractional) models an effective reduction stationary case is crucial apply general renormalisation methods.