Mosco convergence of nonlocal to local quadratic forms

The Wiener criterion for nonlocal Dirichlet problems

“…Remark 2.2. (i) The notation V in the definition (2.3) is adopted from [14,16,19]. Note that some authors denote the space in (2.3) by W s,p (Ω|R n ) or W s,p (Ω).…”

A Remake on the Bourgain-Brezis-Mironescu Characterization of Sobolev Spaces

“…Secondly, from a probabilistic point of view, one obtains a family of pure jumps Lévy processes (X ε ) ε each associated with the measure ν ε (h)dh from a Lévy process X associated with ν(h)dh. In fact, the family of stochastic processes (X ε ) ε converges in finite dimensional distributional sense (see [FKV19]) to a Brownian motion provided that one in addition assumes that ν is radial. Proposition 2.4 (ii) below shows that the generator of L ε of the X ε converges to − 1 2d ∆ the generator of a Brownian motion.…”

“…These convergences are involved in a more significant context. For example in [FKV19] the convergence in Mosco sense of the Dirichlet forms associated with process in play is established.…”

“…We also examine the situation where Ω is not an extension domain. In fact, we want to generalize [FKV19,Theorem 3.4] where the case p = 2 is already considered. We will see that the boundedness condition on Ω can be lifted.…”

Nonlocal complement value problem for a global in time parabolic equation

“…The operator L in (1.2) arises naturally in the study of pure Lévy stochastic processes with the jump interaction measure ν(h)dh. Analogous to the classical case, attempts have been made recently (see, e.g., [19]), where elliptic and parabolic problems related to this type of nonlocal operators are studied, see also [12,13,14,15] where various other typical studies related to this of nonlocal operator are considered.…”