Coupled Quantum Harmonic Oscillators and Feynman–Kac path integrals for Linear Diffusive Particles

We propose a new solvable class of multidimensional quantum harmonic oscillators for linear diffusive particle and quadratic energy absorbing well associated with semi-definite positive matrix force. Under natural easily checked controllability conditions, non necessarily reversible models possibly transient free diffusions, the ground state zero-point are explicitly computed in terms unique fixed point continuous time algebraic Riccati equation. also present an explicit solution normalised dependent Feynman–Kac measures on path spaces varying dynamical system coupled differential A refined asymptotic analysis stability these is developed based recently Floquet-type representation exponential semigroups matrices. provide estimates decays to equilibrium Wasserstein distances or Boltzmann-relative entropy. For we develop series functional inequalities including de Bruijn identity, Fisher’s information decays, log-Sobolev inequalities, entropy contraction estimates. In this context, complete description all spectrum excited states Hamiltonian, yielding what seems be first result type models. illustrate formulae traditional oscillator real Brownian particles Mehler’s formula. The article can extended solve Schrodinger equations equipped diffusions potential functions.