Let’s distinguish two versions of this question. First version: why does a generic non-Born ensemble of Bohmian worlds tend to become Born-like? I think the technical answer is to be found in footnote 9 and the discussion around equation 20. But ultimately I think it will come back to a Liouville theorem in the space of distributions. There is some natural metric under which the Born-like distributions are the majority. (Or perhaps it is that non-Born regions are traversed relatively quickly.)
Second version: why does an individual Bohmian world contain a Born distribution of outcomes? This follows from the first part. An individual Bohmian world consists of a universal wavefunction and a quasiclassical trajectory. If you pick just a few of the classical variables, you can construct a corresponding reduced density matrix in the usual fashion, and a reduced Bohmian equation of motion in which the evolution of those variables depends on that density matrix and on influences coming from all the degrees of freedom that were traced over. So when you look at all the instances, within a single Bohmian history, of a particular physical process, you are looking at an ensemble of noisy Bohmian microhistories. The argument above suggests that even if this starts as a non-Born ensemble, it will evolve into a Born-like ensemble. The only complication is the noise factor. But it is at least plausible that in the majority of Bohmian worlds, this nonlocal noise is just noise and does not introduce an anti-Born tendency.
From an all-worlds-exist perspective, which we both favor, I would summarize as follows: (1) the Born distribution is the natural measure on the subset of worlds consisting of the Bohmian worlds (2) most Bohmian worlds will exhibit an internal Born distribution of physical outcomes. At present these are conjectures rather than theorems, but I would consider them plausible conjectures in the light of Valentini’s work.
“What about decoherence?”
As we’ve just discussed, Bohmian dynamics both preserves exact Born distributions and evolves non-Born distributions towards Born-like distributions (and this is true for subsystems of a Bohmian world as well as for the whole). So the sub-ensembles in the decohered branches will preserve or evolve towards Born.
“part 2 adds more complexity and computational burden without any apparent benefit”
This is a complicated matter to discuss, not least because there is an interpretation of Bohmian mechanics, the nomological interpretation, according to which the “wavefunction” is a law of motion and not a thing. In nomological Bohmian mechanics, the configuration is all that exists, evolving according to a nonlocal potential.
“Why do they converge to the Born distribution?”
Let’s distinguish two versions of this question. First version: why does a generic non-Born ensemble of Bohmian worlds tend to become Born-like? I think the technical answer is to be found in footnote 9 and the discussion around equation 20. But ultimately I think it will come back to a Liouville theorem in the space of distributions. There is some natural metric under which the Born-like distributions are the majority. (Or perhaps it is that non-Born regions are traversed relatively quickly.)
Second version: why does an individual Bohmian world contain a Born distribution of outcomes? This follows from the first part. An individual Bohmian world consists of a universal wavefunction and a quasiclassical trajectory. If you pick just a few of the classical variables, you can construct a corresponding reduced density matrix in the usual fashion, and a reduced Bohmian equation of motion in which the evolution of those variables depends on that density matrix and on influences coming from all the degrees of freedom that were traced over. So when you look at all the instances, within a single Bohmian history, of a particular physical process, you are looking at an ensemble of noisy Bohmian microhistories. The argument above suggests that even if this starts as a non-Born ensemble, it will evolve into a Born-like ensemble. The only complication is the noise factor. But it is at least plausible that in the majority of Bohmian worlds, this nonlocal noise is just noise and does not introduce an anti-Born tendency.
From an all-worlds-exist perspective, which we both favor, I would summarize as follows: (1) the Born distribution is the natural measure on the subset of worlds consisting of the Bohmian worlds (2) most Bohmian worlds will exhibit an internal Born distribution of physical outcomes. At present these are conjectures rather than theorems, but I would consider them plausible conjectures in the light of Valentini’s work.
“What about decoherence?”
As we’ve just discussed, Bohmian dynamics both preserves exact Born distributions and evolves non-Born distributions towards Born-like distributions (and this is true for subsystems of a Bohmian world as well as for the whole). So the sub-ensembles in the decohered branches will preserve or evolve towards Born.
“part 2 adds more complexity and computational burden without any apparent benefit”
This is a complicated matter to discuss, not least because there is an interpretation of Bohmian mechanics, the nomological interpretation, according to which the “wavefunction” is a law of motion and not a thing. In nomological Bohmian mechanics, the configuration is all that exists, evolving according to a nonlocal potential.