A renewal equation in a multidimensional space

The following renewal equation in a multidimensional space (REMS) is considered
$$ f(x)=g(x)+\int_{R^n}K(x-t)\,f(t)\,dt, $$
where $K$ is the density of a distribution in $R^n$. Assuming that $g\in L_1(R^n)$ and that the nonzero vector of the first moment of $K$ is finite we prove the existence and uniqueness of a solution of an REMS within a certain class of functions. The renewal density for the solution of this equation is constructed and its properties are investigated. We give a probabilistic interpretation for our results by means of an example from the theory of random walks in $R^n$.