  The habitability of exoplanets hosted by M-dwarf stars dramatically depends
on their space weather. We present 3D magneto-hydrodynamic simulations to
characterise the magneto-plasma environment and thus the habitability of the
Earth-like planet Proxima b when it is subject to both calm and extreme
(CME-like) space weather conditions. We study the role of the stellar wind and
planetary magnetic field, and determine the radio emission arising from the
interaction between the stellar wind of Proxima and the magnetosphere of its
planet Proxima b. We find that if Prox b has a magnetic field similar to that
of the Earth ($B_{\rm p} = B_\oplus \approx 0.32$ G) or larger, the
magnetopause standoff distance is large enough to shield the surface from the
stellar wind for essentially any planetary tilt but the most extreme values
(close to $90^{\circ} $), under a calm space weather. Even if Proxima b is
subject to more extreme space weather conditions, the planet is well shielded
by an Earth-like magnetosphere ($B_{\rm p} \approx B_\oplus$; $ \approx
23.5^{\circ}$), or if it has tilt smaller than that of the Earth. For calm
space weather conditions, the radio emission caused by the day-side
reconnection regions can be as high as 7$\times10^{19}$ erg s$^{-1}$ in the
super-Alfv\'enic regime, and is on average almost an order of magnitude larger
than the radio emission in the sub-Alfv\'enic cases, due to the much larger
contribution of the bow shock. We also find that the energy dissipation at the
bow shock is independent of the angle between the planet's magnetic dipole and
the incident stellar wind flow. If Prox b is subject to extreme space weather
conditions, the radio emission is more than two orders of magnitude larger than
under calm space weather conditions. This result yields expectations for a
direct detection--from Earth--in radio of giant planets in close-in orbits.
