Pseudodifferential Operators with Rough Negative Definite Symbols

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.