Beurling-Deny formula for Sobolev-Bregman forms

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