that i write above and you possibly did not try.
( you might have got a feeling what fits more. )
sorry, i can not say what way is better without fully understanding what your project is about.
your basic question ( besides coding ) was
how to get arduino PWM output to processing
and that not makes sense in case
you send the PWM setpoint from processing to arduino.
in case you want read a potentiometer from arduino Ain
and set a arduino PWM output,
that can work inside arduino ( and no PC / USB / processing … required )
so pls tell us more what for you want use processing in that project?
( besides learning )