Stochastic system vs. stochastic process I work on a project on stochastic diffusive systems described by stochastic differential equations (SDEs).
My background is from dynamical systems, so I tend to call the system under consideration a stochastic system. However my collaborator has a statistics background, and they naturally name the systems under consideration a stochastic process.
Since we are currently finishing our article, there are parts of the text where we refer to systems and parts where we refer to processes.
So I was wondering whether I can use these two terms interchangeably, or whether there are some subtle formalities that I have to consider?
Or is stochastic system equivalent to stochastic process in every respect?
 A: TL;DR: "Stochastic system" is probably best, but either one is fine.
First, one can investigate a deterministic (non-stochastic) system using statistical tools that treat the variables as random (even though they aren't) so, since your system truly contains a random element, this fact is made more clear by using the term "stochastic system", which makes it arguably preferable to "stochastic process".
Besides, a given system can be described by different models, and a given model can yield different measurements (variables) — hence, one can ascribe different stochastic processes to the same system: So, calling it "system" highlights it's the focus of the work and the "process" is just being used to study it.
Now, that's of course a very fine distinction, and these terms often remain loosely defined, so it's also OK to just add a remark in the beginning of the paper along the lines of "the terms 'stochastic system' and 'stochastic process' are used interchangeably", or even to choose either term and stick to it throughout the text.
Here are some excerpts that, besides my own experience, might support my statements above:

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*Book chapter:


Variables that fluctuate randomly in time are called stochastic processes or random functions.


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*Lecture notes:


Although any (stationary) stochastic process can be considered as a dynamical system (with phase space the set of all possible trajectories and transformation given by the time shift), there are however some important differences to mention. First of all, it is often the case that a dynamical system has many invariant measures.


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*Scholarpedia:


However, in nonlinear systems where noise acts as a driving force, noise can drastically modify the deterministic dynamics. We discuss these issues using a basic level of description which couples a stochastic process to a deterministic equation of motion: the stochastic differential equation (SDE).


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*Wikipedia entry:


a stochastic (/stoʊˈkæstɪk/) or random process is a mathematical object usually defined as a family of random variables. [...]
Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. [...]
historically the word process denoted an evolution of some system in time

A: Such terminology may vary from a (sub)field to a (sub)field. However, I would not equate a system with a process.

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*Process is a mathematical abstraction, particularly well-defined in case of stochastic processes - it is a space of time/space variables with particular statistical properties.

*System is a composition of physical objects, whose behavior might be described by a stochastic process - in which case it is natural to call it stochastic behavior and to call the system a stochastic system.

Ultimately, mixing the two terms is unlikely to result in any confusion - it is more about good writing style and clear language than about physics.
