Beurling-Deny formula for Sobolev-Bregman forms
Michał Gutowski, Mateusz Kwaśnicki
TL;DR
The paper develops a full Beurling--Deny-type decomposition for Sobolev--Bregman forms $\mathscr{E}_p$ associated with a regular Dirichlet form $\mathscr{E}$ and its symmetric Markov semigroup $T_t$. It establishes a domain characterization $\mathscr{D}(\mathscr{E}_p)=\{u\in L^p(E): u^{p/2}\in \mathscr{D}(\mathscr{E})\}$ and a precise Beurling--Deny expansion of $\mathscr{E}_p$ into local, jump, and killing parts with explicit constants, extending known cases without translation-invariance or strong regularity assumptions. The approach hinges on elementary bounds for the heat kernel approximation, a detailed $L^2$-to-$L^p$ comparison, and a limit argument that recovers the energy contributions from $\mathscr{E}^{\mathrm{c}}$, $J$, and $k$; a Hardy--Stein identity follows as a natural corollary. The Euclidean-space example illustrates how the local gradient term and nonlocal jump term combine in the $p$-form, connecting energy measures to Bregman divergences and broadening applicability to nonlinear PDEs and stochastic analysis for general symmetric Markov processes.
Abstract
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.