2. Linear Integro-differential Equations

We define it with a minus sign in front for future convenience.

General operators of the form (2.1.10) are called pseudodifferential operators (\(\psi \)do); we refer to [139] or [129] for a nice introduction.

When \(A>0\), the corresponding operators are of second order with lower-order nonlocal terms. We refer to [118] for a study of such operators.

Recall from the definition of Lévy process, Definition 2.1.1, that we do not necessarily have continuous paths, so in the first hitting time it is not necessarily true that \(x+X_{\tau _x}\in \partial \Omega \). Still, we are working with càdlàg modifications of the underlying processes, and hence, \(\tau _x\) is a stopping time.

This ensures that the matrix A does not degenerate in any direction.

A measure \(\mu \) is rotationally invariant if \(\mu (B) = \mu (O B)\) for any orthogonal transformation \(O\in \mathcal {O}(n)\) and for any Borel set \(B\subset \mathbb {R}^n\).

This can be seen either through the properties of the Fourier transform or by means of the equivalent ellipticity conditions (2.1.25)–(2.1.26).

Observe that, in this case, given \(\rho \in \mathbb {R}\setminus \{0\}\), we have \(K(\rho \, dy) = |\rho |{ }^n K(\rho y)\, dy\).

Notice, though, that there are purely singular measures for which \([K]_\alpha \) is finite.

Observe that we can use it, since \(\mathcal {L} v \in L^1(\mathbb {R}^n)\cap L^\infty (\mathbb {R}^n)\) by Lemma 2.2.11, and hence \(\mathcal {L} v\in L^2(\mathbb {R}^n)\).

We note that the first term in (2.2.26) is the analogue of the Dirichlet energy, \(\frac 12\int |\nabla u|{ }^2\), in the local case \(s = 1\) with \(\mathcal {L} = -\Delta \). The constant \(1/4\) appears here because the integration by parts in Lemma 2.2.16 holds for the scalar product (2.2.15), which has a constant \(1/2\) in front.

When \(K > 0\) in \(\mathbb {R}^n\), such argument already yields a strong maximum principle, since one has \(\mathcal {L} u(x_\circ ) > 0\) unless u is constant. However, for \(\mathcal {L} \in {\mathfrak {G}_s(\lambda , \Lambda )}\), the strong maximum principle turns out to be more delicate; see Sect. 2.4.6.

We are using here that if \(u_r(x) := u(rx)\), then \(\|u_r\|{ }_{L^\infty _{2s-\varepsilon }(\mathbb {R}^n)}\le \|u\|{ }_{L^\infty _{2s-\varepsilon }(\mathbb {R}^n)}\) for \(r < 1\), and that if \(\mathcal {L} u = f\), then there exists some \(\mathcal {L}_r\in {\mathfrak {G}_s(\lambda , \Lambda )}\) such that \(\mathcal {L}_r u_r (x) = r^{2s} f(rx)\) (see Remark 2.1.19), so that the same estimate applies.

This also happens for operators in divergence form \(\mathrm {div}(A(x) \nabla u)\) in the local case \(s = 1\).

See [199] for the case of non-homogeneous kernels.

See also [165, 166] for regularity results in \(L^p\) spaces.