  We present a fundamentally new proof of the dimensionless Lp boundedness of
the Bakry Riesz vector on manifolds with bounded geometry. Our proof has the
significant advantage that it allows for a much stronger conclusion than
previous arguments, namely that of some new dimensionless weighted estimates
with optimal exponent. Part of the importance of this task lies in the novelty
of the techniques: we develop the self similarity argument known as sparse
domination in the setting of uniformly integrable cadlag Hilbert space valued
martingales and extend the domination to a process with infinite memory. We
provide a range of optimal weighted estimates and weak type estimates for these
stochastic processes. Previous geometric Riesz transform estimates relied on
Bellman functions and did not provide this range of weighted estimates. The
development of sparse domination in this probabilistic setting and its use for
high dimensional problems is new.
