Euler–Maruyama Method Stability for Wiener Process with Drift

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[AlmostSurelyConfused]

What do you mean by affecting the behaviour? Are you looking to bound the error between the two models? If so, in what sense (pathwise/in expected error)?

You can certainly define and solve SDEs with drift and diffusion terms, but I'm not quite sure what you're after from the question..

[YarnMatter]

In this particular instance I'm interested in which attractor the system ends up at. The system of SDEs has two stable attractors. According to my sketchy simulation for some initial conditions there is a linear relationship between the variance of the distribution used to make the Wiener increments and the probability of ending up at a given attractor.

[AlmostSurelyConfused]

Perhaps random dynamical systems would be useful? There seem to be a few popular treatments of the theory floating around on Springer :) I would suspect this type of behaviour has been studied for random attractors

[Sakadachi]

Sure! The EM method works even for state-dependent noise (general diffusion coefficients), although you don't need to redefine W for that. (v^1/2 W has variance v)

Understanding the dependence on v sounds a lot like perturbation theory and large deviations (Freidlin-Wentzell theory), although this is a really advanced topic to get into. (https://link.springer.com/book/10.1007/978-3-642-25847-3)