Assume a particle, at instant 0 at the origin of three dimensional euclidean space jumps at each tick of the clock exactly one unit from its current position into a random direction. By this we mean: any two patches of the same area of the unit sphere centered at the current position have equal probability to receive the particle after its next jump; i.o.w. we assume uniform distribution for the directions selected from the unit sphere.
Question: What is - as a function of $r$ - the probability to encounter the particle after exactly $n$ random jumps within the 0-centered ball $B=B(0,r)$ of radius $r.$
Together with my student I have on this problem exact results up to $n=4$: the probability distributions are in each of the cases $n=1,2,3,4$ piecewise polynomial functions of $r$ and from this we have even a precise (but slightly complicated) conjecture for general $n.$ The proof up to 4 needs laborious computations of volumes and before proceeding with more research, we would like to know whether this so natural problem is already treated in the literature (or perhaps buried in more general results). We have searched a lot in Zentralblatt and on the Internet and looked in books on random walks, but not found anything that seems decisevely of help.
( I found questions (and answers) somewhat related to this on this site but not
exactly this. See the information given in the `essay'
"Proof: Mean and Variance of the squared distance of a random walk in n-dimensional space")
This problem has traditionally been called the "problem of flights", or of "random flights", and is connected with "radial functions" and "Hankel transforms". (But if you google "problem of flights" you will be swamped by articles about aviation.)
For $u\in\mathbb R^3$ the expectation $\phi(u)=E\exp(i\langle u,X\rangle)$ for $X$ uniformly distributed on the unit sphere is given by $\phi(u)=\sin(|u|)/|u|$ (this follows from the fact that each coordinate of $X$ is uniformly distributed on $[-1,1]$); "all" you need to do is find the inverse Fourier transform of $\phi^n$.
A 1947 paper of Quenouille is one classical reference; it cites a 1943 paper of Chandrasekhar. Some recent course notes are relevant: see equation (22) and what leads up to it. This 1985 survey paper gives a wealth of historical info, including many other references. (Since this work mostly dates from the first half of the last century, before the modern notation and terminology of probability theory stabilized, these references have a somewhat quaint or antique feel to them.)