Consider a system of n-sdes in $\mathbb{R}$ ( the formula is not important). The corresponding particle system $X(t)=(X_{1}(t),X_{2}(t),...,X_{n}(t))$ lives in $\mathbb{R}^{n}$ and assume that for this system we know that its center of mass is a Brownian Motion. ie $C(t) =\frac{1}{n} \sum \limits_{i} X_{i}(t) = $ BrMotion $ \sim N(0,t)$

The paper that I am studying says (quote) : " the direction (1,1,...,1) is quite singular for the particle system. Indeed, the center of mass being a Brownian Motion prevents the law of $X(t)$ from converging as $t\longrightarrow\infty$ "

I can't understand why this property of the center of mass prevents convergence. My idea/interpetation is that $law(X(t))(1_{\mathbb{R}^{n}})\leq law(C(t))(1) \propto \frac{1}{\sqrt{t}} \exp(-\frac{1}{2t})$ where the last quantity goes to 0 as $t\longrightarrow\infty$

So we get that $law(X(t))(k1_{\mathbb{R}^{n}}) \longrightarrow 0$ as $t\longrightarrow\infty $ for all $k \in \mathbb{R}$ .

Can we deduce from this that there is no limit for $law(X(t))$ ?

What does "singular direcion" mean in this context ?


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