Variable t times a Wiener Process W(1/t)

Question

If $W(t)$ is a Wiener process and $V(t) = t\cdot W(1/t)$ is it possible to say that

Since $W(1/t)\space \sim N(0,1/t)$

that $V(t) \sim t\cdot N(0,1/t)$?

And if so then is $t\cdot N(0,1/t) = N(0,t^2\cdot 1/t) = N(0,t)$?

Thanks.

Answer 1

Yes. In fact $t\mapsto t\cdot W(1/t)$ is itself a Wiener process, strange as it may seem.