Branching random walks and Minkowski sum of random walks

We show that the range of a critical branching random walk conditioned to survive forever and the Minkowski sum of two independent simple random walk ranges are intersection-equivalent in any dimension \(d\ge 5\) $d\ge 5$ , in the sense that they hit any finite set with comparable probability, as their common starting point is sufficiently far away from the set to be hit. Furthermore, we extend a discrete version of Kesten, Spitzer and Whitman’s result on the law of large numbers for the volume of a Wiener sausage. Here, the sausage is made of the Minkowski sum of N independent simple random walk ranges in \(\mathbb {Z}^d\) ${\mathbb{Z}}^d$ , with \(d\ge 2N+1\) $d\ge 2N+1$ , and of a finite set \(A\subset \mathbb {Z}^d\) $A\subset {\mathbb{Z}}^d$ . When properly normalised the volume of the sausage converges to a quantity equivalent to the capacity of A with respect to the kernel \(k(x,y)=(1+\Vert x-y\Vert )^{2N-d}\) $k(x,y)={(1+\Vert x-y\Vert )}^{2N-d}$ . As a consequence, we establish a new relation between capacity and branching capacity.