One-dimensional golf is a function $g$ on $\mathbb R$ such that $g(x)= 1+\min_\mu g(x+N(\mu,c\mu^2))$ if $|x|>1$ and 0 if $|x|\le 1.$
Here $N$ is the normal distribution, whose mean $\mu$ you can choose in each step, but the deviation will depend on it, and on your talent $c$.

What is $f$? Is there an explicit formula for it? Roughly how does it behave?

I set the dependence of the deviation as $\sigma^2=c\mu^2$, but other models might be more realistic.
Also note that I assumed the terrain to be flat.
The problem of course can be similarly defined in $\mathbb R^2$, in which case choosing the direction is obvious (if the terrain is flat), so the problem would reduce to the one-dimensional case, with a possibly different distribution function.
I wonder if this model is realistic to any extent, and how well it would fit on data from professional golf tournaments.
One would of course need to use different $c$'s, or even different distributions for different players.

My question has been inspired by this question by Nate River .