Subordination

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If $T_t$ is a strongly continuous semigroup of contraction operators on a Banach space and $\mu_t$ is a semigroup of sub-probabilistic measures on $[0, \infty)$, then \[ \tilde{T}_t u = \int_{[0, \infty)} T_r u \mu_t(\mathrm d r) \] defines another semigroup of operators $\tilde{T}_t$, which is said to be subordinate (in the sense of Bochner) to $\tilde{T}_t$.[1]

Relation to Bernstein functions

For some Bernstein function $f$, \[ \int_{[0, \infty)} e^{-r z} \mu_t(\mathrm d r) = e^{-t f(z)} . \] Conversely, for any Bernstein function $f$ there exists a semigroup of sub-probabilistic measures $\mu_t$ satisfying the above equality.

One often writes $\tilde{L} = f(L)$, where $-L$ and $-\tilde{L}$ are the generators of $T_t$ and $\tilde{T}_t$, respectively. This notation agrees with spectral-theoretic definition of $f(L)$ if the underlying Banach space is a Hilbert space.

References

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