Questions about smoothness and determinacy of stochastic processes Recently I've been dealing with a problem involving stochastic processes. However, I found myself not so familiar with this topic. I have the following two questions regarding whether there is a canonical way to define smoothness, and whether there is a way to uniquely determine a stochastic process.

*

*By definition, a stochastic process can be considered as a random variable that is evolving along time. Now assume $K\subset{R}$ is compact. Let us consider a stochastic process $X(t, \beta)$, where $t$ is the time; and for any fixed $t$, $X(t, \beta)$ is a random variable subject to a distribution $\mathbb{F}(\beta)$ with $\beta\in K$. What is the canonical way to define the continuity/smoothness of $X(t, \beta)$ with respect to $t$ so that this stochastic process can be considered as a `dynamical system of random variable'?


*For a random variable $X$ (with the assumption that its distribution is nice enough), we can uniquely recover its distribution through its moment generating function (MGF). Is there any tool, like the MGF of random variable, to uniquely determine a stochastic process?
 A: 
By definition, a stochastic process can be considered as a random
variable that is evolving along time.

Your "definition" is not nearly precise enough to allow you to consider the questions you're asking.
Try looking up the actual standard definition and you'll find your questions have standard answers.
A stochastic process is a function
$$
X(t, \omega) : [0, \infty) \times \Omega \rightarrow \mathbb{R},
$$
where $(\Omega, \mathcal{F}, P)$ is a probability space, such that $X$ is measurable in $(t, \omega)$.
(Same definition in the discrete-time case, after replacing $[0, \infty)$ by $\mathbb{N}$.)

What is the canonical way to define the continuity...of
$X(t,\omega)$ with respect to $t$...?

$X(t,\omega)$ is said to have continuous sample paths almost surely, or continuous almost surely, if
$X(t, \omega) : [0, \infty) \rightarrow \mathbb{R}$ is continuous for almost all $\omega$, i.e. for all $\omega$ in an event with probability one.

For a random variable X (with the assumption that its distribution is
nice enough), we can uniquely recover its distribution through its
moment generating function (MGF). Is there any tool, like the MGF of
random variable, to uniquely determine a stochastic process?

No, the generalization to stochastic processes of a distribution of a r.v. is the law of the process. There's no functionals that plays the same role that MGF, or the characteristic function, plays for r.v.'s, and
characterizes the law of the process.
Indeed, you should not expect such a functional to exist.
The distribution of a r.v. is a probability measure on the real line---it is the push-forward measure given by a random variable.
In the case where $X(t, \omega)$ is continuous in $t$ for all $\omega$, the law of $X$ is a probability measure on $C[0, \infty)$, the set of continuous functions on $[0, \infty)$. This is an infinite dimensional set and very different from the real line.
("...with the assumption that its distribution is nice enough" is not quite correct.
The characteristic function always characterizes the distribution. It is true that, in order to recover the moments from the derivatives, you need to assume distribution is somewhat regular.)
