Let's say I have a function $f: \mathbb{R} \to \mathbb{R}$ for simplicity. Assume $f$ to be analytic (or can be represented by $n$th order Taylor expansion). I also have a random variable $\epsilon \sim \mathcal{N}(0, \sigma^2)$. Given $f(x + \epsilon)$, I can write:
$ \begin{align} f(x+ \epsilon) = f(x) + f'(x) \epsilon + \frac{1}{2} f''(x) \epsilon^2 + \frac{1}{6} f'''(x)\epsilon^3 + \frac{1}{24} f''''(x) \epsilon^4 + o(\epsilon^5) \end{align} $
By taking the expectation, I have $\mathbb{E}[f(x + \epsilon)] = f(x) + \frac{1}{2} f''(x) \sigma^2 + ...$. Is there some name for these sort of analysis, where they study the Taylor series with Gaussian pertrubation? Any insights would be greatful!
Such Taylor expansions are indeed studied. Not only for "Gaussian perturbations" but for arbitrary random variables as well. There is even a Wikipedia article about this subject.