What is the process for getting sponsored by big companies for PhD research?
In order to do research with a big company's research arm you need to make contact with them. A blind request is probably not going to go anywhere. You need to use some intermediary who is trusted by them. Much of their research is proprietary and so they are required to be secretive about it. 

Your advisor or another professor at your university who already has contact with the company might be a good way to get a proposal heard. Another way, oddly enough, is another graduate student who is employed by that company in an important position. 

Such companies also have occasional public events, even at your own university, where their researchers look for talent and ideas. Sometimes their researchers will be presenters at conferences (as happens frequently in CS) and you can meet people there. 

However, these days, many big companies that did basic research in the past have focused quite closely on product oriented applied research. There is less of an opportunity now for them to spend time and effort on your ideas unless they already align with their own.