2 I add gr.group-theory tag, as now, the questions seems to be related.
1

# Many Brownian motions moving together

Let $(B^i),\:{{i=1,\ldots,n}}$ be a set of independent Brownian motions. By $(X^i)$ we denote $(B^i)$ conditioned on the event

$|B^i_t-B_t^{i+1}|\leq 1,\quad \forall_{1\leq i\leq n-1}, \forall_{t\geq 0}.$

(this is a $0$-measure event but one can make the definition correct by conditioning on a finite time horizon and them sending it to $\infty$).

Is such process known? What is its behaviour?

My conjecture is that:

1. The mass centre of the process, i.e. $Z_t := \frac{1}{n} \sum_{i=1}^{n} X^i_t$ behaves as $n^{-1/2} W_t$ ($W_t$ is a BM again).
2. The process of fluctionations around $Z_t$, i.e. $\hat{X}^i_t := X^i_t - Z_t$ is well concentrated (e.g. $\sup_{i} \hat{X}^i_t \sim \sqrt{\log{n}}$).

(I am much less sure of 2 then 1).

These predictions come from considering a very crude version of the model as follows. We let the Brownian motions to move unconstrained for time $[0,1]$ then we calculate their mean $z_1 := n^{-1} \sum_{i=1}^n B^i_1$ and set all process to start from this position, i.e. $B^i_{1+}:= z_1$ . We repeat this procedure on each interval $[n,n+1]$.

The further questions would be:

1. Can this process be described as a diffusion. A standard way is to perform a $h$-transform but one needs to find a harmonic function first. I tried this but beyond $n=2$ calculations become messy.

2. Does this process have connections to the random matrices theory? E.g. by defining $Z^i_t:= Z^i_t+i$ one can regard this process as the Dyson Brownian motion with additional conditions $Z^{i+1}_t - Z^{i}_T \leq 2$.