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Smooth measures and capacities associated with nonlocal parabolic operators

  • Tomasz KlimsiakEmail author
  • Andrzej Rozkosz
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Abstract

We consider a family \(\{L_t,\, t\in [0,T]\}\) of closed operators generated by a family of regular (non-symmetric) Dirichlet forms \(\{(B^{(t)},V),t\in [0,T]\}\) on \(L^2(E;m)\). We show that a bounded (signed) measure \(\mu \) on \((0,T)\times E\) is smooth, i.e. charges no set of zero parabolic capacity associated with \(\frac{\partial }{\partial t}+L_t\), if and only if \(\mu \) is of the form \(\mu =f\cdot m_1+g_1+\partial _tg_2\) with \(f\in L^1((0,T)\times E;\mathrm{d}t\otimes m)\), \(g_1\in L^2(0,T;V')\), \(g_2\in L^2(0,T;V)\). We apply this decomposition to the study of the structure of additive functionals in the Revuz correspondence with smooth measures. As a by-product, we also give some existence and uniqueness results for solutions of semilinear equations involving the operator \(\frac{\partial }{\partial t}+L_t\) and a functional from the dual \({{\mathcal {W}}}'\) of the space \({{\mathcal {W}}}=\{u\in L^2(0,T;V):\partial _t u\in L^2(0,T;V')\}\) on the right-hand side of the equation.

Keywords

Dirichlet form Parabolic capacity Smooth measure Hunt process Additive functional 

Mathematics Subject Classification

Primary 31C25 Secondary 35K58 31C15 60J45 

Notes

Acknowledgements

This work was supported by Polish National Science Centre (Grant No. 2016/23/B/ST1/01543).

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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityToruńPoland

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