The following integral equation arises while calculating the probability that, a particle which starts at the origin and moves a unit distance in a random direction on each ‘move’, will be within the unit sphere after $n$ moves: $$ f_n(x) = \begin{cases} \int_{1-x}^{1+x} \frac{df_{n-1}(t)}{dt} \left( \frac{x^2-(t-1)^2}{4t} \right) dt, & 0\le x\lt 1 \\ f_{n-1} (x-1) + \int_{x-1}^{x+1} \frac{df_{n-1}(t)}{dt} \left( \frac{x^2 -(t-1)^2}{4t} \right) dt, & 1 \le x \le n-2 \\ f_{n-1} (x-1) + \int_{x-1}^{n-1} \frac{df_{n-1}(t)}{dt} \left( \frac{x^2 -(t-1)^2}{4t} \right) dt, & n-2\lt x\lt n \\ 1, & x\ge n \end{cases} $$ Here, $n \ge 3$. At first glance, this looks quite unsolvable, as it is a mixture of a recurrence relation and an integral equation, that too with differing arguments in $x$. But just to make sure, is there a way to solve for $f_n(x)$? I’m ultimately looking for $f_n(1)$ so it’s also fine if that can be obtained without actually solving the equation.
Note: $f_n(x)$ is defined to be the probability that the particle is inside the sphere of radius $x$ centered at the origin after $n$ moves. As the ‘base case’, $$f_2(x) =\begin{cases} \frac{x^2}{4}, & 0\le x\le 2 \\ 1, & x\gt 2 \end{cases}$$
Here are the graphs of $\color{blue}{f_2(x)}, \color{green}{f_3(x)}, \color{red}{f_4(x)} $,
and some initial values:
$$f_0(1) = 1 \\ f_1(1) = 0 \\ f_2(1) = \frac 14 \\ f_3(1)= \frac 16 \\ f_4(1) = \frac{23}{192} \\ f_5(1) =\frac{11}{120} $$
NOT AN ANSWER.
I know that Borwein studied that problem in various papers. The following one seems to be relevant to you:
Jonathan M. Borwein, Armin Straub, Christophe Vignat. Densities of short uniform random walks in higher dimensions. Journal of Mathematical Analysis and Applications, Elsevier, 2016, 437 (1), pp.668-707. 10.1016/j.jmaa.2016.01.01
I hope this link works; https://hal-centralesupelec.archives-ouvertes.fr/hal-01261938/document