I would like to know what is the following process on the real line called.
Let us fix some $X_0$ and let $X_{i+1} = (1-\gamma)X_i + Y_i$ where $\gamma$ is a fixed real number and $Y_i$'s are i.i.d. random variables.
I found a reference to this in the book "Stochastic Population Dynamics in Ecology and Conservation". Specifically, it is supposed to model that the population of species is limited by the amount of resources of the environment. I would like to know if someone has studied this mathematically.
This probably doesn't answer your question but I know about ergodic Markov chains that have this form where $Y_i$ take values on a certain state space and $f(x)=(1-\gamma)x$ is a bijection on that state space.
Example: Let $Y_{i}\in\{-1,0,1\}$, $1-\gamma=2$ (so the state space of the Markov chain is $\mathbb{Z}_n$). Notice that $f(x):=2x\mod n$ is a bijection on $\mathbb{Z}_n$ when $n$ is odd. The most recent result on mixing time of this chain is here.
In general, if $\Pi$ is a permutation matrix, $P$ a probability transition matrix, then it is shown here that the Markov chain $\Pi P$ mixes in $O(\log n)$ steps where $n$ is the size of the state space if $\Pi$ satisfies a certain set expansion condition. This paper also has several references about random walks of this form. They are commonly referred to as Markov chains with deterministic jumps.
More recently, this Markov chain of the form $\Pi P$ was applied to speedup mixing of a random walk on a hypercube here. In fact, this paper shows that there is cutoff/phase transition at time $n$ which is quite interesting.