Beurling–Deny formula for Sobolev–Bregman forms

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