Maximum principles for integro-differential parabolic operators. (English) Zbl 0822.45005

The parabolic integro-differential operator in a cylinder \(Q_ T= \Omega\times (0, T)\), where \(\Omega\) is either a bounded or an unbounded domain in \(\mathbb{R}^ d\), \(T\leq +\infty\), \[ {\mathcal A}(x, t, \partial_ x, \partial_ t) u(x, t)= \partial_ t u(x, t)- a_{ij} \partial_{ij} u(x, t)+ a_ i \partial_ i u(x, t)+ a_ 0 u(x, t)- Iu(x, t) \] is considered. It is supposed that \(I\) is an integro-differential operator related with a diffusion process with jumps, and \(a_{ij}(x, t) \xi_ i \xi_ j\geq \mu| \xi|^ 2\) \(\forall(x, t)\in \overline Q_ T\), \(\xi\in \mathbb{R}^ d\), \(\mu> 0\), \(a_{ij}, a_ i, a_ 0\in L^ \infty(Q_ T)\). Several versions of the classical maximum principle, which are valid for parabolic differential problems, are proved to hold for second-order integro-differential problems.


45K05 Integro-partial differential equations