## A one-dimensional drift-jump particle process.

In the *Hammersley process* (Aldous-Diaconis 1995), at each time \( t>0\)
there are a (two-sided) infinite number of particles at positions
\( 0 < \ldots < X_{-1}(t) < X_0(t) < X_1(t) < \ldots < \infty\)
and the dynamics are
Events occur as a rate-1 space-time Poisson point process on \( (0,\infty) \times (0,\infty) \).
When an event occurs at \( (x_0,t_0) \), the closest particle to the right of \(x_0\) is moved to \(x_0\).

This arose as one of many aspects of the "longest increasing subsequence of a uniform random permutations" problem,
and the relation is most clearly seen using the *patience sorting*
representation (Aldous-Diaconis 1999).
But there is also a different limit process which arises by considering, instead of a scaling limit, only the first
\(k\) piles in patience sorting, corresponding in a sense to considering only the start of the subsequence.
In this process there are \(k\) particles at positions
\( 0 < X_1(t) < X_2(t) < \ldots < X_k(t) \)
and the dynamics are
Events occur as a rate-1 space-time Poisson point process on \( (0,\infty) \times (0,\infty) \).
When an event occurs at \( (x_0,t_0) \), the closest particle (if any) to the right of \(x_0\) is moved to \(x_0\).
Also the particles move to the right at speed linearly increasing in space:
\( dX_i(t)/dt = X_i(t) \).

In fact these distributions are consistent as \(k\) varies, and so define an infinite particle process.
This process is mentioned briefly in section 5.2 of Aldous-Diaconis 1999.
By soft arguments this process has a unique stationary distribution.
**Problem.**
Give some (fairly explicit) description of the stationary distribution.

**History.**
Some analysis is attempted in section 4.1 of these unfinished unpublished 1993 notes
from which the two published papers above were mined and refined.
The problem was often mentioned in subsequent conversations. Posted here February 2018.

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