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  "body": "\n## Definition\n\nLet $d \\ge 1$ and let $D \\subseteq \\R^d$ be a [[domain|0Q4CBD3HX0]].\nLet $p^D_t(x, y)$ be the transition density of [[Brownian motion|22735FGJV6]] in $\\R^d$ killed on leaving $D$.\nLet $h \\colon D \\to (0, \\infty)$ be a [[positive harmonic function|EYMQ82Z8XJ]] on $D$.\nThe *h-conditioned Brownian motion* in $D$ is the [[Doob h-transform|ZN7HJNR2AC]] of killed Brownian motion by $h$.\nIt is the Markov process on $D$ with transition densities\n$$p^h_t(x, y) = \\frac{h(y)}{h(x)} \\, p^D_t(x, y), \\qquad t \u003e 0, \\ x, y \\in D.$$\n\n## Remarks\n\nA positive harmonic function on $D$ is excessive for killed Brownian motion, so the h-transform is defined; the h-process may have a finite lifetime.\nExample: $D = (0, \\infty)$ and $h(x) = x$.\nHere $h$ is positive and harmonic, and the h-process is the three-dimensional Bessel process: Brownian motion conditioned never to hit $0$.\nExample: for the unit disc and $h(z) = (1 - \\abs{z}^2)/\\abs{\\zeta - z}^2$, the [[Poisson kernel|TCYSWCTYYM]] with pole at $\\zeta$ on the unit circle, the h-process is Brownian motion conditioned to exit the disc at $\\zeta$.\nNon-example: $h(x) = x$ on $D = \\R$ is harmonic but takes negative values, so the formula does not give a family of positive kernels.\nMore generally, conditioning to exit at a point of the Martin boundary uses the minimal harmonic function with pole at that point.\n",
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