  Using probabilistic tools, we prove that any weak* continuous semigroup
$(T_t)_{t \geq 0}$ of selfadjoint unital completely positive measurable Schur
multipliers acting on the space $\mathrm{B}(\mathrm{L}^2(X))$ of bounded
operators on the Hilbert space $\mathrm{L}^2(X)$, where $X$ is a suitable
measure space, can be dilated by a weak* continuous group of Markov
$*$-automorphisms on a bigger von Neumann algebra. We also construct a Markov
dilation of these semigroups. Our results imply the boundedness of the
McIntosh's $\mathrm{H}^\infty$ functional calculus of the generators of these
semigroups on the associated Schatten spaces and some interpolation results
connected to $\mathrm{BMO}$-spaces. We also give an answer to a question of
Steen, Todorov and Turowska on completely positive continuous Schur
multipliers.
