The literature on stochastic processes (Ornstein–Uhlenbeck, Langevin) is not very clear as to the motivation behind using the Brownian motion or other types of noise in the dynamics. Are there any real-world applications of Brownian noise or other types of noise in the dynamics in macro (not particles with small dimensions) dynamical systems?
Are there any real-world applications of Brownian noise or other types of noise in the dynamics in macro (not particles with small dimensions) dynamical systems?
There are three ways to answer your question, stemming from 'where' the noise terms should come from.
First, you can think of any kind of population dynamics, be it crowds, cells, birds, economic agents... where the noise arises in the modeling of the sub-parts of the system. One way to model all these systems is to assume some stochastic model for the agents 1, and then try to extract meaningful macro quantities of the whole thing. Some would call this 'emergent behavior'. If you'd like to dive deeper, the relevant field is probably Active Matter Theory.
Second, when dealing with already macroscopic dynamical systems in real life, one should remember that any macro object dynamic quantity (position, speed...) must be measured before being dealt with. This, in turn, implies that there is some inherent measurement noise in your system that you have to take into account. I would recommend taking a look at Kalman filters, which are a way to deal with what I just described, especially for the guidance of vehicles.
Third, when modeling macroscopic systems, some factor impacting the dynamics is either inherently "noisy" or too tricky to model exactly. E.g., taking into account the wind changes (gusts) when modeling the mechanical behavior of high metallic structures. Another example could be the number of cars and their speed on a bridge when studying the mechanical response of such infrastructure.
1 Usually, exactly as you described, i.e. some deterministic interaction or inertial terms, plus a term to account for the randomness of each agent. This can be a bird, changing direction randomly every so often (of course there might a real reason for the changes, but you don't want to end up modeling the bird's decision process).