Let $B^{1}$ and $B^{2}$ be independent brownian motions and $\alpha, \beta \in \mathbb R$. What conditions are needed for $B:=\alpha B^{1}+\beta B^{2}$ to be a brownian motion
My ideas: $B_{0}=0$ and $t\to B_{t}$ continuous a.s. are clear
In terms of the distributions I claim that $\alpha+\beta=1 $ in order for $B$ to be a brownian motion, since:
$B_{t}=\alpha B^{1}_{t}+\beta B^{2}_{t}$~$\mathcal{N}(0,\alpha t)+\mathcal{N}(0,\beta t)=\mathcal{N}(0,(\alpha+ \beta)t)$
What do I need for the independence of increments to hold?
Because for $u > t > s$, we know that
$B_{u}-B_{t}=\alpha (B_{u}^{1}-B_{t}^{1})+\beta(B_{u}^{2}-B_{t}^{2})$
and
$B_{t}-B_{s}=\alpha (B_{t}^{1}-B_{s}^{1})+\beta(B_{t}^{2}-B_{s}^{2})$
It is clear that $\alpha (B_{u}^{1}-B_{t}^{1})$ and $\alpha (B_{t}^{1}-B_{s}^{1})$ are independent and likewise that
$\beta(B_{u}^{2}-B_{t}^{2})$ and $\beta(B_{t}^{2}-B_{s}^{2})$ are also independent
But how can I be sure that $\alpha (B_{t}^{1}-B_{s}^{1})$ is independent of $\beta(B_{u}^{2}-B_{t}^{2})$ and also independent of $\beta(B_{t}^{2}-B_{s}^{2})$ since all we have is independence of $B^{1}$ to $B^{2}$ and note of their increments
$\alpha + \beta = 1$ is not correct. You have the right idea, but made a common mistake. If $B_t^1 \sim N(0,t)$, then $\alpha B_t^1 \sim N(0, \alpha^2 t)$.
It helps to recall what exactly independence means for two stochastic processes. The two processes $B^1$ and $B^2$ are be independent, so for every choice of times $t_1,t_2,\ldots,t_n$ we have the random vector $(B_{t_1}^1,\ldots,B_{t_n}^1)$ is independent of the random vector $(B_{t_1}^2,\ldots,B_{t_n}^2))$. Does that help you convince yourself why $\alpha( B_t^1 - B_s^1)$ is independent of $\beta (B_u^2 - B_t^2)$?
There's one last thing to show after independence of increments. You also need to show the increments are stationary: $B_t - B_s$ has the same distribution as $B_{t-s}$.
To answer your more general question of what conditions are necessary: continuity, zero at the origin, independent increments, and stationary increments. If you process satisfies those, then your process must look like some scaling of Brownian motion, e.g., $a B_t$.
The other major criteria you often use to show something Brownian motion is Lévy's martingale characterization of Brownian Motion.