Feynman's formula in a phase space for a certain class of systems of pseudodifferential equations (Q1100360)

The author considers the Cauchy problem: \(D_ tu=H(x,D_ x)u\), \(u(0,x)=u_ 0(x)\), where \(u_ 0: {\mathbb{R}}^ d\to {\mathbb{C}}^ r\) is a regular functional. The solution u is a function of \(t\in [0,T]\) with its values in the Schwartz space \({\mathcal S}'({\mathbb{R}}^ d,{\mathbb{C}}^ r)\). Assuming that \(| \partial_ p^{\alpha}H(q,p)| \leq c(1+| p|)^{m_{\alpha}}\) and that \[ H(q,p)=\int \exp (i(q,h)+i(p,k))\hat H(dh dk), \] where \(\hat H\) is a bounded Borel measure, the author constructs the solution as a Feynman integral in the phase space.