Schauder estimates for drifted fractional operators in the supercritical case

Abstract

We consider a non-local operator Lα which is the sum of a fractional Laplacian △ α/2 , α ∈ (0, 1), plus a first order term which is measurable in the time variable and locally β-Hölder continuous in the space variables. Importantly, the fractional Laplacian ∆ α/2 does not dominate the first order term. We show that global parabolic Schauder estimates hold even in this case under the natural condition α + β > 1. Thus, the constant appearing in the Schauder estimates is in fact independent of the L ∞ -norm of the first order term. In our approach we do not use the so-called extension property and we can replace △ α/2 with other operators of α-stable type which are somehow close, including the relativistic α-stable operator. Moreover, when α ∈ (1/2, 1), we can prove Schauder estimates for more general α-stable type operators like the singular cylindrical one, i.e., when △ α/2 is replaced by a sum of one dimensional fractional Laplacians d k=1 (∂ 2x k x k ) α/2 .

Weak well-posedness of multidimensional stable driven SDEs in the critical case

“…Following the proof of Lemma 4.3 in [10] (see also Section 4.2 in [4]) we can as well show the following result (the sketch of the proof is postponed to Appendix).…”

“…If the drift b is itself Hölder continuous and bounded, stronger assumption than (C), then the well-posedness of the martingale problem for L can be established following [17] (see also Section 3 in [18]). For unbounded Hölder drifts this property follows from the Schauder estimates established in [4]. All these results are based on perturbative techniques which exploit that the singularities of the corresponding heat-kernel serving as a proxy appearing in the analysis can be absorbed thanks to the Hölder continuity.…”

Well-posedness of some non-linear stable driven SDEs

“…In our previously mentionned Hölder setting, the main idea to complete our fixed point approach, consists in exploiting some parabolic Schauder estimates but under some rough regularity conditions on the final condition, whose regularity will be somehow related to the spatial one of the coefficients. Such estimates are obtained through a forward parametrix perturbation argument in the same spirit as those established by Chaudru de Raynal et al [7], [8] in the Kolmogorov degenerate diffusive setting or in the α-stable supercritical case.…”

“…We importantly point out that the former condition appearing in (B H ) is the natural constraint arising in [8] to derive Schauder estimates for a general, and potentially singular, non-degenerate spherical measure ω in the sense of (12) in the super-critical case. We can as well assume without loss of generality (see e.g.…”

“…Note carefully that we denote by δU t, x, µ, ν := U t, x, µ − U t, x, ν the difference of two time-space dependent functions which also involve a measure argument. Such a notation is not to be confused with the flat derivative δ δm U associated with a real-valued, measure dependent function U as defined in (8). To avoid ambiguity, we specify that, in what follows, flat derivatives are always denoted with an explicit dependence w.r.t.…”