Pseudodifferential Operators with Rough Negative Definite Symbols |
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Authors: | Alexander Potrykus |
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Institution: | 1. Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK
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Abstract: | We consider state-space dependent continuous negative definite functions and use their associated pseudodifferential operators to construct Feller semigroups. Our method works with “rough” symbols ${p(x,\xi),\,{\rm i.e.}\,\xi \mapsto p(x,\xi)}We consider state-space dependent continuous negative definite functions and use their associated pseudodifferential operators
to construct Feller semigroups. Our method works with “rough” symbols p(x,x), i.e. x? p(x,x){p(x,\xi),\,{\rm i.e.}\,\xi \mapsto p(x,\xi)} only needs to be continuous. The main part of this work concerns the development of an asymptotic expansion formula for the
composition of two pseudodifferential operators with rough negative definite symbols. This presents an improvement over other
symbolic calculi that typically require the symbols to be smooth. As an application we show how to adapt existing techniques
to construct and approximate Feller semigroups to the case of rough symbols. |
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