Joint moments of Brownian motion

Question

My approach to this SE question uses the following joint moments of Brownian motion. For $n=1,2$ they are obvious and well-known, the others are not terribly hard to work out. Is there a reference where these formulas are given, or/and is there a pattern to the coefficients?

Fix $t_1\leq t_2\leq t_3\leq\cdots \leq t_n$. For odd values of $n$ we have $\mathbb{E}[W(t_1)\ W(t_2) \cdots W(t_n)]=0$ while for even values of $n$ we get

\begin{eqnarray*} \mathbb{E}[W (t_1)\ W(t_2)]&=& t_1 \cr \mathbb{E}[W (t_1)\ W(t_2)\ W(t_3)\ W(t_4)]&=& 2t_1 t_2+t_1t_3 \cr \mathbb{E}[W (t_1)\ W(t_2)\ W(t_3)\ W(t_4)\ W(t_5)\ W(t_6)]&=& 2t_1t_2t_5+t_1 t_3 t_5 +4 t_1 t_2 t_4 +2 t_1 t_3 t_4 +6 t_1 t_2 t_3 \end{eqnarray*}

I suppose everything about Brownian motion has been worked out, but I can't find this in any of my books. It's not very important, but I'm just curious!

Answer 1

Another formula for this joint moment can be found as Lemma 4.5 in a paper by J. Rosen and M.B. Marcus [Annals of Probability, vol. 20, no. 4 (1992) pp. 1603-1684]; the authors refer to it as "well-known". The formula is this (for even $n$): The joint moment $E[W(t_1)W(t_2)\cdots W(t_n)]$ is the sum over all pairings $\{\{a(1),b(1)\},\{a(2),b(2)\},\ldots,\{a(n/2),b(n/2)\}\}$ of $\{1,2,\ldots,n\}$ of $$ \prod_{i=1}^{n/2}E[W(t_{a(i)},t_{b(i)})]. $$ A pairing is simply a partition of $\{1,2,\ldots,n\}$ into $n/2$ doubletons.

Answer 2

By considering $t_1=t_2=...=t_{2n}$, the sum of the coefficients is the $2n$th moment of $W(1)$, $(2n-1)!!$, which is the number of paths of length $2n$ in Young's lattice from the empty partition to itself.

I haven't yet proven the following, but I think induction should work.

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Conjecture:

The indices of the terms with positive coefficients correspond to Dyck words, so the number of terms in $E[W(t_1)W(t_2)...W(t_{2n})]$ is the $n$th Catalan number. $t_1 t_2 t_4$ corresponds to the Dyck word $++-+--$ with pluses in positions $(1, 2, 4)$.

The coefficient of the term corresponding to a Dyck word is the number of paths of length $2n$ in Young's lattice from the empty partition to itself such that the partition at each step has size equal to the height of the Dyck path. The coefficient of $t_{i_1}t_{i_2}...t_{i_n}$ is $\prod_{k=1}^n (2k-i_k)$. For example, the coefficient of $t_1 t_2 ... t_n$ equals $\prod_{k=1}^n (2k-k) = n!$ and paths in Young's lattice which ascend $n$ times and then descend $n$ times correspond to pairs of standard tableaux of the same shape, which are in bijection with permutations of $\{1,2,...,n\}$.

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