According to Wikipedia, the analogue of a random walk for a LÃ©vy distribution is called a LÃ©vy flight. I presume it’s called a “flight” because paths are no longer guaranteed to be continuous, but can have sudden jumps.

I also spotted a survey paper, More “Normal” than Normal: Scaling Distributions and Complex Systems, which argues that in physical applications, heavy-tailed distributions such as the LÃ©vy distribution are more natural than the normal distribution. This seems to be emerging conventional wisdom in some circles, but I don’t know how true it is.