Red, black, and white are common wire colors for electric scooter, bike, and go kart throttles. Red, black, and brown are not common throttle connector wire colors for speed controllers though.
If you have a multimeter then I would test the Voltage of the controller's throttle connector red and black wires to make sure that there is around +5 Volts DC at the red wire using the black wire as the ground. If this is the case then you would connect red to red, black to black, and white to brown. If the +5 Volt signal is not at the red wire and is at another wire then you can use the following specifications to wire the throttle to the controller:
Throttle Wire Specifications
Red - Goes to Controller's +5 Volts Throttle Wire
Black - Goes to Controller's Ground Throttle Wire
White - Goes to Controller's Throttle Wire with Zero Voltage
If you do not have a multimeter then you could try wiring red to red, black to black, and white to brown. There is a very good chance that this will work because the red and black throttle wires have been the same on every controller that I have seen. There is however a small chance that wiring it this way could burn out the throttle from miswiring it. The most important thing that I have ever learned about electronics it is to never assume anything and to always test everything if you are not 100% certain what it is.
Please let us know how it goes.