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A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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Stochastic operator on $\ell^1$ has dense range

Let $P:\ell^1(\mathbb{Z}^d) \rightarrow \ell^1(\mathbb{Z}^d)$ be given by $$(Pz)(x)=\sum_{y \tilde \ x} \frac{1}{2d} z(y)$$ where the tilde indicates that $y$ is a neighboured vertex of $x.$ I woul …