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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 $Z^{i+1}_t - Z^{i}_T \leq 2$.

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A small variation of the problem is to add one more constraint: $|B^1_t -B^n_t|\leq 1$. This should not change the result much but adds additional symmetry. –  Piotr Miłoś Jan 10 '12 at 18:05
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3 Answers 3

You can split. I'll do it for $B_1,B_2$. Let's go in small time steps to avoid talking about stochastic differential equations and other stuff I don't really know. Note that the increments $\xi$ and $\eta$ of $B_2-B_1$ and $B_1$ at each step are correlated Gaussians with certain covariance. Now write $\eta=a\xi+\gamma$ where $\gamma$ is a Gaussian orthogonal to and, thereby, independent of $\xi$. Note that then you have free Brownian motion controlled by $\gamma$ that takes care of the overall drift combined with the bounded motion controlled by $((1+a)\xi,a\xi)$, which is conditioned to stay in some domain. The same can be done with any number of $B$'s. Note that the orthogonal projection of $(1,0,\dots,0)$ to the orthogonal complement of the plane $\sum_j x_j=0$ is of length $n^{-1/2}$ confirming your first conjecture. The second conjecture then says (after a linear transformation) that the standard $n-1$-dimensional BM conditioned on staying in a certain parallelepiped stays fairly concentrated. I do not see it immediately but that may be well-known to probabilists.

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I am not sure if I understand well. Is the $n-1$-dimensional BM (in the second part): $(B_2 -B_1, B_3 - B_2,\ldots, B_n - B_{n-1})$? If, yes, it is not a standard BM, i.e. its components are not independent. –  Piotr Miłoś Jan 10 '12 at 8:54
    
It is rather the motion with coordinates $B_j-\frac 1n\sum_j B_j$, which is the orthogonal projection of the full motion to the plane $\sum x_j=0$. Now, the orthogonal projection of a standard BM is a standard BM again, just in lower dimension. The real catch is the shape of the restriction domain P, which is here a parallelepiped vs. the shape of the desired concentration domain C. You basically want to say that a standard BM conditioned on staying in P stays in C most of the time. Such natural problem must have been studied. I just don't know the literature. –  fedja Jan 10 '12 at 11:33
    
Thx, now I agree. –  Piotr Miłoś Jan 10 '12 at 15:42
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In the case of $n=3$ (by the procedure outlined by fedja) the problem boils down to study two dimensional BM in the yellow domain. Let now $G$ be a group generated by the reflections in the blue lines, then the transition density of the BM killed on the hitting boundary, $h_t(x,y)$, is

$h_t(x,y) := \sum_{g\in G} (-1)^{r(g)}p_t(x,g(x)),$

where $p_t(x,y) = (2\pi t)^{-1} e^{-|x-y|^2/(2t)}$ is the transition density of the BM in $\mathbb{R}^2$ and $r$ is "the rank" of $g$ (to be explained in a moment).

I know almost nothing about the group theory but it seems to me that $G$ is what is called "a reflection group" or a special case of a Coxeter group. $r(g)$ as far as I understand is the length of the shortest way in which $g$ can be represented using the generator only.

So, the questions are:

  1. Is there a way of presenting the above sum in by a closed formula.

  2. What is asymptotic behaviour of $h_t(x,y)/\int h_t(x,y)$?

Domain

P.S. It is probably not very common to answer own questions but I think it is more then a comment and I do not know how to paste images into comments.

P.P.S. Following Omer's comment, it should be not hard to prove that

$l(y):=\lim_{t\rightarrow +\infty} h_t(0,y)/h_t(0,0)$

exists and $l$ is the leading eigenvalue of the Laplacian in the considered domain.

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The first claim is correct. As Fedja and Piotr suggest, the key is a change of basis. The centre of mass $Z_t$ has law $W_t/n$ where $W$ is B.M. independent of the differences. Since the condition only affects the differences, the claim follows.

To estimate fluctuations around $Z_t$, you need to estimate the leading eigenfunction for the Laplacian on the domain given by the constraints $|B^i-B^{i+1}|<1$ -- a parallelogram. In the case $n=3$ this is Piotr's figure (with the additional constraint $|B^1-B^3|<1$).

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Thanks for the answer. You are right. One way of calculating the eigenfunction would be to use the formula in my previous post (as explained in P.P.S.). But, as for a moment we do not know how to handle the formula, may be there are methods of proving some concentration properties of the eigenfunction without finding it explicitly. –  Piotr Miłoś Jan 24 '12 at 10:10
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