| Abstract: |
| Let $D\subset \mathbf R^d$, be an open $d$-set, and $L$ a nonlocal operator of the form $L= -\phi(-\Delta)$, where $\phi:(0,\infty)\to (0,\infty)$ is a complete Bernstein function. The operator $L$ is a generator for the subordinate Brownian motion, with the Laplace exponent $\phi$ of the subordinator, and admits the following singular integral form
\[
Lu(x)=\textrm{P.V.}\int_{\mathbf R^d} (u(y)-u(x))j(|y-x|)dy,\ \ u\in C_c^2(\mathbf R^d),
\]
where $j:(0,\infty)\to(0,\infty)$ is the corresponding radially decreasing L\` evy density. In this talk, we revisit the regional subordinate Laplacian on $D$, i.e. the nonlocal operator of the form
\[
L_D u(x)=\textrm{P.V.}\int_{D} (u(y)-u(x))j(|y-x|)dy,\ \ u\in C_c^2(D),
\]
and two associated strong Markov processes on $\overline{D}$ -- the censored and resurrected subordinate Brownian motions on $D$. Specifically, we discuss the pathwise behaviour of these processes and give conditions under which they are equivalent. |
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