Beurling-Deny formula for Sobolev-Bregman forms

For an arbitrary regular Dirichlet form $\mathscr{E}$ and the associated symmetric Markovian semigroup $T_t$, we consider the corresponding Sobolev-Bregman form $\mathscr{E}_p(u) = -\tfrac{1}{p} \frac{d}{d t}\bigr\vert_{t = 0} \|T_t u\|_p^p$, where $p \in (1, \infty)$. We prove a variant of the Beurling-Deny formula for $\mathscr{E}_p$. As an application, we prove the corresponding Hardy-Stein identity. Our results extend previous works in this area, which either required that $\mathscr{E}$ is translation-invariant, or that $u$ is sufficiently regular.