Elliptic Boundary Value Problems and Construction of by Benedict Baur

By Benedict Baur

Benedict Baur provides glossy sensible analytic equipment for building and research of Feller procedures usually and diffusion tactics specifically. themes lined are: building of Lp-strong Feller strategies utilizing Dirichlet shape equipment, regularity for ideas of elliptic boundary worth difficulties, development of elliptic diffusions with singular float and mirrored image, Skorokhod decomposition and purposes to Mathematical Physics like finite particle platforms with singular interplay. Emphasize is put on the dealing with of singular go with the flow coefficients, in addition to at the dialogue of element clever and course clever houses of the built techniques instead of simply the quasi-everywhere houses regularly recognized from the overall Dirichlet shape theory.

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Extra resources for Elliptic Boundary Value Problems and Construction of Lp-Strong Feller Processes with Singular Drift and Reflection

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Sturm provides an entirely different approach for constructing diffusions on general locally compact metric spaces with given reference measure, see [Stu98a] and [Stu98b]. Instead of a gradient Dirichlet form, he considers a family of Dirichlet forms consisting of difference operators. Then convergence of these forms to a limit Dirichlet form is proven. Under an additional property, namely the measure contraction property, a Feller process associated with the limiting Dirichlet form is constructed.

Now the statement follows using the Markov property of (Xt )t≥0 and that (PtΔ )t≥0 is the transition semigroup of (Xt )t≥0 . 4 Some Examples We close this section with examples. These examples are based on [AKR03], [BGS13] and [Sti10]. We show that the results presented there can be deduced from our construction scheme together with the elliptic regularity results which are used in the cited works. First we consider the distorted Brownian motion on Rd , d ∈ N. This is done in [AKR03]. Let : Rd → R+ 0 be Borel measurable.

In the sequel we apply this lemma for M1 , M2 open subsets of Rd and M1 ⊂ Ω an open set, M2 = Rd ∩ {xd > 0}. In the latter case ψ is the diffeomorphism flattening the boundary of Ω locally. Using these lemmata we prove the regularity result. We start with the interior point case. We provide a local representation for weak solutions in terms of potentials. This representation has been already derived in [Sha06, Lem. 2], but we include it here to keep our exposition self-contained. 1, interior point case.

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