How can I calculate a sample path for a simple stochastic process

Question

I try to find a trajectory (sample path) for such very simple Gaussian stochastic process: $X(\omega;t)=N(\mu t,\sigma^2)$, $t\in T=(-\infty, \infty)$ ($N$, as always, is the Gaussian distribution). As for any $t$ the process is a Gaussian random variable, it is easy to find expectation, variance, finite-dimensional probability distributions of $X$. However, it is not very easy to find a trajectory. I suppose, that it has the form: $\mu t+b$, where $b$ is a real number, but it is desirable to prove this

Answer 1

Let $X$ be a random variable with $X \sim N(0, \sigma^2)$. Define $$X_t = \mu t + X$$ Then $X_t$ has the properties you desire.