  We consider the dynamics of systems with arbitrary friction and diffusion.
These include, as a special case, systems for which friction and diffusion are
connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion.
We study the limit where friction effects dominate the inertia, i.e. where the
mass goes to zero (Smoluchowski-Kramers limit). {Using the It\^o stochastic
integral convention,} we show that the limiting effective Langevin equations
has different drift fields depending on the relation between friction and
diffusion. {Alternatively, our results can be cast as different interpretations
of stochastic integration in the limiting equation}, which can be parametrized
by $\alpha \in \mathbb{R}$. 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 even
stochastic integrals with $\alpha \notin [0,1]$ arise. Our findings are
supported by numerical simulations.
