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  "body": "\n## Definition\n\nLet $(E, \\mathcal{E})$ be a measurable space and let $(P_t)_{t \\ge 0}$ be a sub-Markov transition semigroup on $E$.\nSo the $P_t$ are kernels as for [[Markov kernels|2AVNDP8BJP]] but with $P_t(x, E) \\le 1$, and $P_s P_t = P_{s+t}$.\nLet $h \\colon E \\to (0, \\infty)$ be a measurable function that is [[excessive|46436KE7NT]] for $(P_t)$.\nThe *h-transform* of $(P_t)$ is the family of kernels\n$$P^h_t(x, dy) = \\frac{h(y)}{h(x)} \\, P_t(x, dy), \\qquad t \\ge 0, \\ x \\in E.$$\nA Markov process with transition semigroup $(P^h_t)$ is called an *h-process*.\n\n## Remarks\n\nThe kernels $P^h_t$ form a sub-Markov semigroup.\nIndeed, $P^h_t(x, E) = P_t h(x)/h(x) \\le 1$ since $h$ is excessive, and\n$$\\int P^h_s(x, dy) P^h_t(y, dz) = \\frac{h(z)}{h(x)} \\int P_s(x, dy) P_t(y, dz) = \\frac{h(z)}{h(x)} P_{s+t}(x, dz).$$\nIf $P_t h = h$ for all $t$, then each $P^h_t$ is a Markov kernel and the h-process has infinite lifetime.\nExample: let $(P_t)$ be [[Brownian motion|22735FGJV6]] on $(0, \\infty)$ killed at $0$, and $h(x) = x$.\nThen $P_t h(x) = E^x[B_t ; t \u003c \\tau_0] = x$ by optional stopping, so $h$ is invariant, and the h-process is the three-dimensional Bessel process, Brownian motion conditioned never to hit $0$.\nNon-example: for Brownian motion on $\\R$ and $h(x) = 1 + x^2$, one has $P_t h(x) = 1 + x^2 + t \u003e h(x)$, so $h$ is not excessive and $P^h_t(x, \\R) \u003e 1$.\nIf $L$ is the [[infinitesimal generator|VPA41PWRR1]] of $(P_t)$, the generator of the h-process acts by $L^h f = h^{-1} L(h f)$ on suitable $f$.\nFor $L = \\frac12 \\Delta$ and harmonic $h$, this is $L^h f = \\frac12 \\Delta f + \\nabla(\\log h) \\cdot \\nabla f$, a Brownian motion with drift $\\nabla \\log h$.\n",
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