What's the relationship between ESC pwm input and output?
Most ESCs convert the servo pulse into into a linear PWM range. 

However it's not quite as simple as that, because ESC's are designed to work with a variety of radio control transmitters which may have slightly different pulse ranges. Also, many transmitters have adjustable center and end-point settings. Several methods may be used to 'calibrate' the ESC to a particular transmitter:-


Auto-calibrate: The ESC assumes that the initial pulse width received represents 'low throttle' (motor off) and records it as the low end-point. As the throttle is advanced above half way, the highest point reached so far is taken as 'full throttle'. This procedure has to be followed each time the ESC is turned on.
Manual Calibration: The ESC is put into calibration mode by starting at full throttle (>1.5ms), then going to low throttle (&lt;1.5ms) after a few seconds. These settings are recorded in an eeprom, so the calibration only has to be done once. Default settings that should work with most transmitters (eg. 1.2 to 1.8ms) are often pre-programmed into the ESC at the factory. 
Using a programming card or PC link cable: End-points can be set to specific values and/or switched between auto-calibrate and fixed end-points. Some ESCs also allow setting different throttle curves (eg. exponential) or a 'governor mode' where the servo pulse width represents an rpm that the motor speed is to be locked onto.
Fixed end-points: The ESC may have two or more fixed end-point ranges, which be can be selected using throttle programming or with a programming card.  


The ESC's instruction manual should tell you which method is used and how to do the calibration.