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In general, the simplest extension of (real-valued) one-parameter processes is to (real-valued) multi-parameter processes, where the index set is $\mathbb{R}^n_+$ (usually $n=2$). This includes, for example, the $n$-parameter Brownian sheet, and more generally, $n$-parameter L\'evy sheets (or processes). However, such processes are much simpler than general set-indexed processes.
In general, the simplest extension of (real-valued) one-parameter processes is to (real-valued) multi-parameter processes, where the index set is $\mathbb{R}^n_+$ (usually $n=2$). This includes, for example, the $n$-parameter Brownian sheet, and more generally, $n$-parameter L\'evy sheets (or processes). However, such processes are much simpler than general set-indexed processes.