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

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.