On the martingales which vanish on the set of Brownian zeroes

Abstract

In the study of the Hilbert transform of Brownian local times, made in Chapter 10, we encountered some Brownian martingales which vanish (≡ take the value 0) on the set H≡{(t,w):B_t(w)=0}of the zeroes of a one-dimensional BM (B_t ≥0). Such martingales were also encountered naturally in relation with the balayage formula (see, e.g., Azéma-Yor [88]). It thus seems interesting to try and describe the class

$$ {\mathcal{M}^0} \equiv \left\{ {\left( {{M_t}} \right)Brownianmartingales;{M_t}\left( \omega \right) = 0,if\left( {t,\omega } \right) \in H} \right\} $$

as well as

$$ {\mathcal{M}^{0,strict}} \equiv \left\{ {\left( {{M_t}} \right)Brownianmartingales;{M_t}\left( \omega \right) = 0iff\left( {t,\omega } \right) \in H} \right\} $$