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A renewal equation in a multidimensional space. (English. Russian original) Zbl 1099.60060
Theory Probab. Appl. 49, No. 4, 737-744 (2005); translation from Teor. Veroyatn. Primen. 49, No. 4, 779-785 (2004).
Consider the integral equation $f(x)=g(x)+\int K(x-t)f(t)\,dt\tag{*}$ in the unknown function $$f$$, where $$x\in \mathbb{R}^n$$, $$t\in \mathbb{R}^n$$, $$n>1$$, $$g\in L_1$$ and $$K$$ is a probability density on $$\mathbb{R}^n$$. The paper studies unicity and existence of the solution $$f$$. This problem is solved in the Banach spaces $$L_{\infty 1}$$ and $$ML$$. Here $$L_{\infty 1}$$ is the set of those $$h(x_1,\dots,x_n)$$ where $$k(x_1)= \int|h(x_1,\dots,x_n)|dx_2\dots dx_n$$ is essentially bounded and $$ML$$ is the set of those $$h$$ where $$k (x_1)$$ is the sum of an $$L_\infty(\mathbb{R})$$ and an $$L_1(\mathbb{R})$$ function. It is shown by Fourier methods that the homogeneous equation only has the trivial constant solution in $$L_{\infty 1}$$ and by extension also in $$ML$$. By iteration it is shown, when $$\int\|x\|K(x)dx<\infty$$ and $$\int x_1K(x)dx> 0$$, that there is a solution of (*) in $$ML$$. Rotation of the coordinate system extends this result to general nonzero moment vectors. This solution leads to the construction of the renewal density for $$g=K$$. The author applies results from his preceding paper [Stochastic Processes Appl. 85, No. 2, 237–247 (2000; Zbl 0997.60096)] on the essentially different case $$n=1$$.
##### MSC:
 60K05 Renewal theory
##### Keywords:
solvability; joint motion
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