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Consider $w(t)$ as Guassian random process, with $w(t)$ being $\mathcal{N}(\mu,\sigma)$ and i.i.d for all t.

I consider applying a (stochastic)derivative operation to the random process. What is the distribution of resulting random process? The derivative operation is w.r.t t. Based on this operation I what to get insight on the following.

(a) When is the derivative operation defined?

(b) With derivative being a linear operation is the resulting random process also Gaussian.

(c) Can we relate the mean and variance of the resulting r.v of the random process to the mean ($\mu$) and variance ($\sigma^2$) of $w(t)$?

I hope the question is clear but I will be happy to correct if it is not.

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Stochastic processes with i.i.d. values in continuous time are not nice. Such a process $X$ with nondegenerate marginals (e.g. Gaussian with nonzero variance) satisfies $\limsup_{t\to t_0} X(t)\ne\liminf_{t\to t_0}X(t)$ for any time $t_0$, so the trajectories cannot even be continuous, leave alone differentiable in a reasonable sense.

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