to keep a vigil outside smb's home (house)

вести наблюдение — ( за кем-л) to conduct (maintain) surveillance (watch) ( over); keep ( smb) under observation (under surveillance); shadow; spy (on / upon); tail; ( с помощью электронных устройств) to conduct (maintain) electronic surveillance ( over); ( круглосуточное наблюдение) to watch ( smb) around the clock

устанавливать наблюдение — to set up surveillance (watch) ( over)

под наблюдением полиции — ( под надзором) under police supervision