A one-shot isotropic point-like actuator can trigger a super-tiny spherical shock front, which is a solution of a homogeneous second order partial differential equation. It integrates into the Green's function of the affected field. The excitation quickly fades away but in the meantime, the field deforms a tiny bit. The phenomenon adds a persistent bit of "volume" to the field. Thus, it temporarily deforms and it permanently expands the field.
The effect is so small that it cannot be observed in isolation. However, a recurrently regenerated dense and coherent swarm of such actuators, such that the shock fronts overlap, can produce a significant and persistent deformation.
A point-like object that hops around in a stochastic hopping path such that each hop landing triggers a spherical shock front that owns a private stochastic process that generates the landing locations and that on its turn owns a characteristic function that ensures the coherence of the generated hop landing location swarm because it equals the Fourier transform of the location density distribution of the swarm can do the job.
The deformation of the field can be described by the convolution of the location density distribution of the swarm and the Green's function of the field.
Does this story not elucidate how super-tiny spherical shock fronts offer mass to particles for which the location density distribution of the swarm equals the square of the modulus of the wavefunction of the particle?