Quantum mechanics, in its standard formulations, is well-known to suffer from conceptual and technical problems such as the quantum measurement problem and what might be called the 'problem of ontology' (i.e., the problem that it is unclear what the ontology of the physical world is according to standard quantum mechanics). In the 20th century, one of the most promising approaches developed to address these problems was “stochastic mechanics”, originally proposed by Imres Fenyes in 1952, and again by Edward Nelson in 1966. In stochastic mechanics, the central goal is the reconstruction of quantum mechanics from a more fundamental theory of point particles interacting with a classical-like ether medium, where the interaction causes the particles to undergo a diffusion process on configuration space that conserves the total energy of the particles on the average. In this way, the measurement problem can easily be solved and there is no problem of ontology.
On a formal level, stochastic mechanics succeeds in this central goal. However, as pointed out in 1989 by Wallstrom, on a technical level, stochastic mechanics runs into the technical problem that it cannot recover the Schroedinger equation of quantum mechanics unless an ad hoc supplementary condition is assumed; namely, that the circulation of the 'current velocity' of the diffusing particles is quantized in integer multiples of Planck's constant (basically the Bohr-Sommerfeld quantization condition of the Old Quantum Theory). This observation was one of the primary reasons that research interest in stochastic mechanics diminished significantly in the '90's and onward.
In this thesis, I reformulate stochastic mechanics so that the aforementioned quantization condition arises as a natural consequence of the classical point particles interacting with the classical-like ether. This is done by combining stochastic mechanics with a proposal by Louis de Broglie and David Bohm, in which an elementary particle in its rest frame is viewed as a localized periodic phenomenon of fixed frequency, from which it follows that the phase of the periodic phenomenon in the lab frame satisfies a relation that's equivalent to the quantization condition. In doing so, it is argued that the stochastic mechanics approach can once again be regarded as a viable approach to reconstructing quantum mechanics.
In addition to reformulating stochastic mechanics, I show that stochastic mechanics yields straightforward and empirically viable models of semiclassical Newtonian gravity and electrodynamics, simply by incorporating classical Newtonian gravitational and electrostatic interactions between the particles. I also show how the Schroedinger-Newton equation and the Schroedinger-Coulomb equation arise as mean-field approximations from within these stochastic mechanical models of semiclassical Newtonian gravity and electrodynamics, and how classical Newtonian gravity can be recovered from a center-of-mass description of many interacting stochastic mechanical particles. Thus I argue that stochastic mechanics has novel implications for the problem of combining quantum mechanics with gravity in a semiclassical way (at least at the Newtonian level), and that it has certain advantages over other approaches to combining quantum mechanics and gravity semiclassically (e.g., approaches based on standard quantum mechanics or other alternative quantum theories).