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→Markov kernel defined by a kernel function and a measure: order of the kernel function |
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:<math> \int_Y k(y, x)\nu(\mathrm{d} y) = 1, \qquad \forall x \in X </math>
then <math> \kappa(dy |x) = k(y, x)\nu(dy) </math> i.e. the mapping
:<math>\begin{cases} \kappa:\mathcal B \times X \to [0,1] \\ \kappa(B|x)=\int_{B}k(
defines a Markov kernel.<ref>{{cite book|last1=Erhan|first1=Cinlar|title=Probability and Stochastics|date=2011|publisher=Springer|___location=New York|isbn=978-0-387-87858-4|pages=37–38}}</ref>. This example generalises the countable Markov process example where <math>\nu</math> was the [[counting measure]], but other important examples are convolution kernels like the Markov kernel defined by the heat equation e.g. the Gaussian on <math>X = Y = \mathbb R</math> with <math>\nu(dx) = dx </math> standard Lebesgue measure and
:<math>k_t(y, x) = \frac{1}{\sqrt{2\pi}t}e^{-(y - x)^2/(2t^2)}</math>.
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