The Hardy space of martingales can be defined in terms of martingale differences. I'll stick to the simplest case of dyadic martingales.

**Notation** The underlying probability space can be taken to be $[0,1]$, the filtration $\{F_n\}$ has $F_n$ generated by dyadic intervals $F_n\ni I=I_+\cup I_-$ ($I_\pm\in F_{n+1}$) with $|I|=2^{-n}$ and the probability measure is Lebesgue measure. A martingale $f=\{F_n\ni f_n=\sum_0^n d_j\}$ satisfies $d_n(I_+)+d_n(I_-)=0$ if $I\in F_{n-1}$, where $d(J)=d_n(J)$ is the value of the martingale difference $d_n=f_n-f_{n-1}$ on $J\in F_n$.

The martingale $f$ is in the Hardy space $M^2$ if $$ \infty>\|f\|_2^2=\lim_{n\to\infty}{\mathbb E}f_n^2=\sum_{j=0}^\infty\sum_{|J|=2^{-j}}d_j^2(J)2^{-j}=\sum_Jd(J)^2|J|. $$ There is a vast literature on this and related martingale spaces ($M^p$'s, $BMO$, etcetera). The starting point is that $f=\lim_{n\to\infty}f_n$ exists a.e. on $[0,1]$ and $f_n={\mathbb E}[f|F_n]$.

Consider instead the *Dirichlet* martingale space $D\subset M^2$ defined by the condition
$$
\infty>\|f\|_D^2=\sum_Jd(J)^2.
$$
It is related to $M^2$ the same way the holomorphic Dirichlet space is related to the holomorphic Hardy space, or the Sobolev space $W^{1/2,2}({\mathbb R})$ ($1/2$ derivative in $L^2({\mathbb R})$) is related to $L^2({\mathbb R})$.It is easy to show that $f=\lim_{n\to\infty}f_n$ exists outside a set of null logarithmic capacity in $[0,1]$. In a sense, it is the "right" space if one wants to do calculus of variations for martingales. Also, it can be defined via martingale transforms of Hardy martingales.

The **questions** are the following. (1) Is there some literature on this space? (2) The fact that the boundary values are defined outside a set of zero capacity has some easy to understand probabilistic meaning?

[I have often used $D$ as a toy model for testing facts and proofs I wanted to verify or use dealing with holomorphic functions].