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  "body": "\n## Summary\n\nDoob's h-transform changes the transition probabilities of a Markov process by weighting them with a positive excessive function $h$.\nThe new process, the h-process, behaves like the old one conditioned on an event described by $h$.\nTypical events are leaving a domain at a given boundary point or never being killed.\nIt links positive harmonic functions, the Martin boundary, and conditioned processes.\n\n## Explanations\n\n### L1 human\n\nTake a random motion and a positive function $h$ that never increases on average along the motion.\nTilt the motion so that moves towards places where $h$ is large become more likely, in proportion to $h$.\nThe tilted motion is the h-transform.\nFor example, Brownian motion on the half-line $x \u003e 0$, tilted by $h(x) = x$, becomes Brownian motion conditioned never to hit $0$.\n",
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