| Abstract: |
| In this talk we will consider fractional powers of parabolic operators, $(\partial_t +L)^s$, where L is an elliptic operator in divergence form in a bounded domain. They arise, for example, in semipermeable membrane problems. We will see that they are also examples of master equations, that are fundamental in continuous time random walks. Using the Gamma function, the Cauchy integral theorem and the method of semigroups we are able to obtain the pointwise nonlocal formula. Then using a parabolic extension theorem we prove parabolic Harnack inequalities, both interior and boundary. This is joint work with Marta de Le\`on-Contreras (Universidad of Reading, UK) and Pablo Ra\`ul Stinga (Iowa State University). |
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