Is Brownian motion $B_t$ the only Gaussian $\mathcal N (0,t)$ process that has a.s continuous paths?

Question

Is it possible for a stochastic process $X_t$ to have law $\mathcal N(0,t)$, continuous paths a.s and not be a Brownian motion ? If it is not possible, how to prove it ?

Answer 1

It is possible, like pointed out in the comments. In fact, we can characterize Brownian Motion as a stochastic process $X_{t}$ with a.s. continuous sample paths such that

1. For any $0\le t_{1}<...<t_{n}<\infty$, $(X_{t_{1}},...,X_{t_{n}})$ follows a multivariate normal distribution

2. For any $t\ge 0$, we have $E[X_{t}]=0$

3. For any $s,t \ge 0$, we have $Cov(X_{t},X_{s})=\min(s,t)$