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We consider the dynamics of systems with arbitrary friction and diffusion.These include, as a special case, systems for which friction and diffusion areconnected by Einstein fluctuation-dissipation relation, e.g. Brownian motion.We study the limit where friction effects dominate the inertia, i.e. where themass goes to zero (Smoluchowski-Kramers limit). {Using the It\^o stochasticintegral convention,} we show that the limiting effective Langevin equationshas different drift fields depending on the relation between friction anddiffusion. {Alternatively, our results can be cast as different interpretationsof stochastic integration in the limiting equation}, which can be parametrizedby $\alpha \in \mathbb{R}$. Interestingly, in addition to the classical It\^o($\alpha=0$), Stratonovich ($\alpha=0.5$) and anti-It\^o ($\alpha=1$)integrals, we show that position-dependent $\alpha = \alpha(x)$, and evenstochastic integrals with $\alpha \notin [0,1]$ arise. Our findings aresupported by numerical simulations.

Noise-induced drift in stochastic differential equations with arbitrary friction and diffusion in the Smoluchowski-Kramers limit