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  "body": "\n## Summary\n\nA conditioned Brownian motion in a domain is the Doob h-transform of Brownian motion killed on leaving the domain, for a positive harmonic function $h$.\nTake for $h$ the Poisson kernel or the Martin kernel with pole at a boundary point.\nThen the process is Brownian motion conditioned to leave the domain at that point.\nConditioned Brownian motion gives probabilistic proofs of boundary limit theorems, such as Fatou-type theorems and the relative Fatou theorem.\n\n## Explanations\n\n### L1 human\n\nRun a Brownian particle inside a region, but only keep the runs that leave the region at one chosen boundary point.\nThe resulting random motion is drawn towards that point and eventually exits there.\nA standard example is Brownian motion on the half-line conditioned never to reach $0$: it is pushed away from $0$ and drifts off to infinity.\n",
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