Mean field dynamics of a quantum tracer particle interacting with a boson gas

We consider the dynamics of a heavy quantum tracer particle coupled to a non-relativistic boson field in ${\mathbb{R}}^3$. The pair interactions of the bosons are of mean-field type, with coupling strength proportional to $\frac{1}{N}$ where $N$ is the expected particle number. Assuming that the mass of the tracer particle is proportional to $N$, we derive generalized Hartree equations in the limit $N\to \mathrm{\infty }$. Moreover, we prove the global well-posedness of the associated Cauchy problem for sufficiently weak interaction potentials.