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algebras analysis analytic associated assume bm(q Brezis and J L C+(t canonical transformations change of variables chapter Cj(t College de France compute consider coordinates defined Definition denote dualization ek(q equations of motion Evolution Semigroups exp q exp t G exponential Feinsilver's Feynman-Kac formula follows Fourier transforms France Seminar functions f(q,p gauge transformation given group of canonical gt(q,ik H Brezis Hamilton-Jacobi Equation Hamilton-Jacobi function Hamiltonian flows Hamiltonian H(q,p Hamiltonian system hand side Hilbert space identity infinitesimal inverse J L Lions Lemma Let F(q,P mapping Martingale Math Nonlinear partial differential notations operational calculus p(ik partial differential equations phase space Pm(q Poisson bracket polynomial sequences Proof Proposition Ptf(q quadratic Hamiltonians quantum mechanics reader to verify respect result satisfies semigroup Pt sinh solution solve test functions tH(P theorem theory and applications transformation q,p trivially integrable systems TS(t U(ik vacuum vector Volume