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  "body": "\n## Summary\n\nThe potential operator of a Markov process integrates the transition semigroup over all times.\nApplied to the indicator of a set, it gives the expected total time that the process spends in that set.\nIt is the probabilistic analogue of the Newtonian potential, and formally it inverts minus the generator of the process.\nIts finiteness distinguishes transient processes from recurrent ones.\n\n## Explanations\n\n### L1 human\n\nStart a random particle at a point and ask how long, in total over its whole life, it will spend inside a given region.\nThe potential operator answers this question, on average, for every starting point and every region.\nFor Brownian motion in three-dimensional space the answer is finite for bounded regions, because the particle drifts away for good.\nIn the plane the particle keeps coming back, and the expected time in any disc is infinite.\n",
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