I have to solve this, but I'm not right about my result:
Having the stochastic process $\left\{W(t);t>0\right\}$ defined as $W(t)=(X_{1}+X_{2})t$. $X_1$ and $X_2$ are random variables iid. by a normal distribution with zero mean and variance one.
$\phi_{W}(t,\lambda)=\phi_{X_{1}}(t,\lambda)\phi_{X_{2}}(t,\lambda)=\text{E}\left[\exp\left(i\lambda X_{1}t\right)\right]\text{E}\left[\exp\left(i\lambda X_{2}t\right)\right]$
$\phi_{X_{1}}(t,\lambda)\phi_{X_{2}}(t,\lambda)=\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{X_{1}^{2}}{2}}e^{i\lambda tX_{1}}dX_{1}\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{X_{2}^{2}}{2}}e^{i\lambda tX_{2}}dX_{2}=e^{\frac{1}{2}\lambda^{2}t^{2}}e^{\frac{1}{2}\lambda^{2}t^{2}}=e^{(\lambda t)^{2}}$
Do you think we can solve this by $W(t)$ is a normal with zero mean and variance $2t^2$?
There is problem of sign in what you wrote. Since $X_jt\sim \mathcal N(0,t^2)$, then $$\mathbb E[e^{i\lambda X_it}]=e^{-\frac{t^2}{2}\lambda ^2},$$
and thus $$\mathbb E[e^{i\lambda W_t}]=e^{-t^2\lambda ^2},$$ what means $$W_t\sim \mathcal N(0,2t^2).$$