  We demonstrate two examples of stochastic processes whose lifts to geometric
rough paths require a renormalisation procedure to obtain convergence in rough
path topologies. Our first example involves a physical Brownian motion subject
to a magnetic force which dominates over the friction forces in the small mass
limit. Our second example involves a lead-lag process of discretised fractional
Brownian motion with Hurst parameter $H \in (1/4,1/2)$, in which the stochastic
area captures the quadratic variation of the process. In both examples, a
renormalisation of the second iterated integral is needed to ensure convergence
of the processes, and we comment on how this procedure mimics negative
renormalisation arising in the study of singular SPDEs and regularity
structures.
