| Abstract: |
| For an arbitrary regular Dirichlet form $\mathcal{E}$ and the associated symmetric Markovian semigroup $T_t$,
we consider the corresponding \emph{Sobolev--Bregman form}
$
\mathcal{E}_p [u]
:= \lim_{t\to0^+} \tfrac{1}{t} \langle u-T_tu,u|u|^{p-2} \rangle
$,
where $p\in(1,\infty)$.
The Sobolev--Bregman form describes the rate of decrease of the $L^p$ norm of $T_t u$ with respect to time: we have
$
\mathcal{E}_p [u] = -\tfrac{1}{p}
\tfrac{d}{dt} \| T_t u \|_p^p \big\vert_{t = 0}
$.
When $p=2$, $\mathcal{E}_p$ coincides with the original Dirichlet form $\mathcal{E}$. Therefore, the Sobolev--Bregman form can be treated as a $L^p$-extension of the Dirichlet form.
The celebrated Beurling--Deny formula provides a decomposition of a regular Dirichlet form $\mathcal{E}$ into the strongly local term $\mathcal{E}^c$, the purely nonlocal part given in terms of the \emph{jumping kernel} $J$, and the killing term described by the \emph{killing measure} $k$.
It is well known that there is a one-to-one correspondence between the class of regular Dirichlet forms $\mathcal{E}$ and the class of symmetric Hunt processes (strong Markovian, quasi-left continuous with c\`adl\`ag paths).
The three parts of the Beurling--Deny decomposition describe the local, jumping, and killing behaviour of the process.
The aim of the talk will be to provide a similar decomposition of the corresponding Sobolev--Bregman form.
The talk is based on joint work with Mateusz Kwa{\`s}nicki (Wroc{\l }aw University of Science and Technology). |
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