Find parameters $ a,b,c $ for process $ aW_t^2+cW_{bt^2+a} $ to make it gaussian.
The process is gaussian if
$$ E \left [ \exp \left ( i \sum_{l=1}^{k} s_l Y_{t_l} \right ) \right ] = \exp \left ( -\frac{1}{2} \sum_{j,l} \sigma_{j,l}s_j s_l + i \sum_{l=1}^{k} \mu_l s_l \right ) $$
$ \mu = E[aW_t^2+cW_{bt^2+a}] = at $
$ \sigma = \operatorname{Var}[aW_t^2+cW_{bt^2+a}] = 2ac\,\operatorname{cov}(W_t^2, cW_{bt^2+a}) + 3a^2t^2 + c^2(bt^2+a)$, if calculated correctly
What should I do with $2ac\,\operatorname{cov}(W_t^2, cW_{bt^2+a})$?
And how to get rid of mean.
We are first looking for conditions on $(a,b,c)$ that guarantee that fixed $t$, $aW_t^2+cW_{bt^2+a}$ has a Gaussian distribution. Such a random variable has the same distribution as $aX^2+Y$, where $(X,Y)$ is a Gaussian vector. Substracting and adding $KX$ for a constant $K$ such that $Y-KX$ is independent of $X$, we are in the situation where $aU^2+KU+V$ has a Gaussian distribution, where $(U,V)$ is a vector of independent normal random variables.
If $Z'$, $Z+Z'$ have a Gaussian distribution and $Z$ is independent of $Z'$, then $Z$ has a Gaussian distribution (look at the characteristic functions of the involved random variable).
If $a\neq 0$, then $aU^2+KU$ cannot have a Gaussian distribution because of properties of the range of a second order polynomial.
Therefore, we need $a=0$. Now I think that checking that $cW_{bt^2}$ is Gaussian should not be a hard task.