What does Schrödinger equation reduce to in the limit of a continuous position measurement? If we measure position of a quantum particle, we force its wavefunction to collapse into a wavefunction whose probability density is given by a Dirac delta function (all the probability density of position is "squeezed" into one point in space).
Immediately after the measurement the wavefunction starts to delocalise (spread over the space), but if we measure the position very quickly again, its outcome (new position) shouldn't be far from the previous position. Moreover, by the new measurement we "localise" the wave-function again into a new Dirac delta function.
So, it looks to me that in the limit of continuous position measurements (extremely frequent measurements) the positions of the particle should form a continuous but stochastic trajectory that is probably described by a stochastic differential equation. Is it true? And, if it is the case, how does this equation look? What is its name?
ADDED
Is Balavkin equation a correct way to go? It looks like it describe a case of continuous measurement and, in particular, a case of continuous measurement of position is considered? Does this equation address some special (unrealistic) case? Is it based on some (unrealistic) assumptions?
 A: *

*The position measurement can't have infinite resolution, because then further time-evolution would be undefined. (A delta-function "wavefunction" doesn't belong to the Hilbert space.) So the resolution must be finite.


*For a single measurement event, we could approximate the effect by projecting the wavefunction into a spatial cell whose size represents the resolution of the measurement. Which cell? Whichever one we observe to be the result of the measurement. But if you really want to model a continuous position measurement (with finite resolution, of course), then this isn't a good approximation. It leads to the quantum Zeno effect, but in an artificial way: it's an artifact of the projection-approximation, which isn't how real position measurements work.


*To do better, we need to use a quantum model that includes more than just the particle of interest. We need to use a model that also includes the measurement equipment (at least). We can make that more manageable by taking a partial trace over the rest of the system, resulting in a master equation, such as a Lindblad equation, for the particle's density matrix. The density matrix evolves smoothly in time, but it doesn't remain pure: the interaction with the rest of the system causes the particle to become entangled with the rest of the system, so the entropy of the reduced density matrix increases.


*A stochastic trajectory emerges from the master equation if we occasionally apply the projection rule to account for where we actually observe the particle to be (with finite resolution), but to avoid artifacts, we should only do this occasionally, as explained above. We can do better by not taking a partial trace over the rest of the system, and letting the rest of the system include something like a physical periodically-updated digital readout of the particle's location. Then we can apply the projection rule to an observable associated with the periodic digital readout device, and you'll get a natural stochastic trajectory for the particle, without the artifacts. (Intuitively: the projection rule works great as long as you apply it far enough "downstream" from the process of interest, so that any mathematical "sharp edges" from the projection get smoothed out by the long chain of intervening physical interactions.)
For more information about point 3, this review paper looks pretty good:

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*Jacobs and Steck (2006), "A Straightforward Introduction to Continuous Quantum Measurement," Contemporary Physics 47, 279 (https://arxiv.org/abs/quant-ph/0611067)

