As far as I understand, many Gaussian Processes can be either described by their corresponding mean and kernel functions or by a stochastic differential equation (SDE). For my purposes it is sufficient that the predictors $x$ just include time, thus, I will use $t$ instead of $x$. The stochastic differential equation $dX_t = a X_t + bW_t$ where $a, b$ are scalars and $W_t$ the Wiener process leads to the exponential kernel function $k(t,t')= \sigma^2 \exp(\frac{|t-t'|}{l})$ where $l>0$, $\sigma>0$. There is a one to one relationship between the parameters $a,b$ and the parameters $l,\sigma$. By solving the SDE one can even obtain an analytic expression for $X_t$.
My question is: Is there also a SDE interpretation of the squared exponential kernel $$\sigma^2\exp\left(\frac{(t-t')^2}{l}\right)$$ with $l,\sigma>0$, assuming that $t,t'$ are representatives of time, that is $t,t' \in \mathbb{R}_0^+$. If this is the case what is the solution of the SDE, that is, the analytic expression for $X_t?$
