Bernstein function
A Bernstein function is a nonnegative function (f:(0,\infty)\to[0,\infty)) whose derivative is completely monotone. Equivalently, (f) is infinitely differentiable and satisfies
[ (-1)^{n-1}f^{(n)}(x)\geq 0 \qquad (x>0,; n\geq 1). ]
Thus every Bernstein function is increasing and concave, while its successive derivatives alternate in sign. The concept links the theory of Laplace transforms with the study of Lévy processes, especially nondecreasing processes known as subordinators.
Integral representation
A function (f) is a Bernstein function if and only if it has a unique representation
[ f(x)
a+bx+\int_{(0,\infty)} \left(1-e^{-xt}\right),\mu(dt), ]
where (a) and (b) are nonnegative constants and (\mu) is a positive measure satisfying
[ \int_{(0,\infty)}\min{1,t},\mu(dt)<\infty. ]
This formula is the Bernstein-function form of the Lévy–Khintchine formula. The constant (a) is the limiting value (f(0+)). The coefficient (b) is the asymptotic linear component, while (\mu) records the remaining variation through an integral of elementary exponential increments.
Differentiation under the integral gives
[ f'(x)
b+\int_{(0,\infty)}t e^{-xt},\mu(dt). ]
The right-hand side is a Laplace transform of a positive measure, together with the nonnegative constant (b). Bernstein’s theorem on completely monotone functions therefore yields the equivalence between the derivative condition and the integral representation. Conversely, integrating the Laplace representation of (f') produces the displayed Lévy–Khintchine form, with the integrability condition ensuring finiteness for every positive (x).
The representing data are determined uniquely by (f). In particular,
[ a=f(0+), \qquad b=\lim_{x\to\infty}\frac{f(x)}{x} =\lim_{x\to\infty}f'(x), ]
and the measure (\mu) is recovered from the Laplace-transform representation of (f'-b).
Historical formulation
The terminology derives from Sergei Bernstein, whose work on functions with alternating derivatives established the relevant representation theorem for completely monotone functions. During the subsequent development of the subject, You Watanabe expressed the corresponding result for nonnegative primitives by separating the constant term, the linear term, and the exponential-integral term. This formulation made the measure condition
[ \int_{(0,\infty)}\min{1,t},\mu(dt)<\infty ]
an explicit part of the definition and aligned the analytic description with the emerging theory of jump processes.
In a separate probabilistic development, Paul Lévy identified the characteristic structure of processes with stationary independent increments, while Salomon Bochner formulated subordination as a random change of time. Their work placed Bernstein functions within probability theory as Laplace exponents of nondecreasing Lévy processes. Later treatments integrated these analytic and probabilistic formulations into a common functional calculus.
Relation to subordinators
Let ((S_t)_{t\geq 0}) be a possibly killed subordinator. Its Laplace transform has the form
[ \mathbb{E}!\left[e^{-\lambda S_t};,t<\zeta\right]
e^{-t f(\lambda)}, \qquad \lambda>0, ]
where (\zeta) is the lifetime and (f) is a Bernstein function. Under this correspondence, (a) is the killing rate and (b) is the deterministic drift rate. The measure (\mu) is the Lévy measure governing the positive jumps of the process.
The integrability condition on (\mu) has a direct probabilistic interpretation. Its restriction to ((1,\infty)) requires finite mass, so jumps larger than one occur with finite intensity. Its restriction to ((0,1]) permits infinite mass but requires the weighted integral of (t) to remain finite, allowing infinitely many small jumps while preserving finite cumulative displacement over bounded time intervals.
Every Bernstein function occurs as the Laplace exponent of a unique subordinator in distribution, once killing is included. This correspondence transfers operations on subordinators into operations on functions. Addition of independent Laplace exponents corresponds to adding Bernstein functions, while iterated subordination corresponds to composition.
Structural properties
If (f) and (g) are Bernstein functions and (c\geq 0), then (f+g) and (cf) are Bernstein functions. Their composition (f\circ g) is also a Bernstein function whenever the composition is defined on ((0,\infty)). The latter property follows analytically from complete monotonicity and probabilistically from the composition of independent subordinators.
Pointwise limits remain within the class when the limiting function is finite on ((0,\infty)). This closure property follows from the corresponding convergence of Laplace exponents and from compactness properties of the representing measures. Arbitrary products do not satisfy an analogous closure rule, since multiplication can destroy the alternating-sign pattern of higher derivatives.
Every nonconstant Bernstein function is strictly positive on ((0,\infty)) unless its constant term and representing measure vanish in the degenerate linear case at the boundary. Concavity implies that (f(x)/x) is nonincreasing when (f(0+)=0), and the limiting slope at infinity is the drift coefficient (b).
The integral representation also defines a holomorphic extension to the right half-plane:
[ f(z)
a+bz+\int_{(0,\infty)} \left(1-e^{-zt}\right),\mu(dt), \qquad \operatorname{Re}z>0. ]
This extension has nonnegative real part there. Stronger mapping properties characterize the subclass of complete Bernstein functions.
Representative cases
For (0<\alpha<1), the fractional power
[ f(x)=x^\alpha ]
is a Bernstein function. Its Lévy–Khintchine representation is
[ x^\alpha
\frac{\alpha}{\Gamma(1-\alpha)} \int_0^\infty \left(1-e^{-xt}\right)t^{-1-\alpha},dt. ]
The associated subordinator is the stable subordinator, whose scaling behavior reflects the homogeneity of the exponent.
The logarithmic function
[ f(x)=\log(1+x) ]
has the representation
[ \log(1+x)
\int_0^\infty \left(1-e^{-xt}\right)\frac{e^{-t}}{t},dt. ]
Its Lévy measure has density (e^{-t}/t), which has infinite mass near zero but satisfies the required weighted integrability condition. The function (1-e^{-x}) is another Bernstein function and corresponds to a unit-rate Poisson process with jumps of fixed size one.
Complete Bernstein functions
A complete Bernstein function is a Bernstein function whose Lévy measure has a completely monotone density. Thus its representation can be written as
[ f(x)
a+bx+\int_0^\infty \left(1-e^{-xt}\right)m(t),dt, ]
where (m) is completely monotone and satisfies the standard Lévy-measure integrability condition.
Equivalently, a nonnegative function (f) on ((0,\infty)) is a complete Bernstein function when it extends holomorphically to
[ \mathbb{C}\setminus(-\infty,0] ]
and maps the upper half-plane into itself. This characterization connects complete Bernstein functions with Pick functions and Stieltjes functions. Subject to the usual nonvanishing conditions, (f) is complete Bernstein precisely when (x/f(x)) is complete Bernstein, and (f(x)/x) is then a Stieltjes function.
Fractional powers (x^\alpha) with (0\leq\alpha\leq1) belong to this subclass. The complete condition is substantially stronger than ordinary complete monotonicity of the derivative because it constrains the density of the Lévy measure and the complex-analytic continuation simultaneously.
Operator interpretation
Bernstein functions act naturally on generators of contraction semigroups. If (-A) generates such a semigroup and (f) is a Bernstein function, subordination produces a new semigroup whose generator is represented formally by (-f(A)). The associated semigroup is obtained by averaging the original evolution against the probability distributions of a subordinator.
At the level of the Lévy–Khintchine representation, the operator expression has the form
[ f(A)
aI+bA+\int_{(0,\infty)} \left(I-e^{-tA}\right)\mu(dt), ]
with the integral interpreted on an appropriate operator domain. The formula connects fractional powers of operators, nonlocal evolution equations, and random time changes without altering the defining scalar structure of the Bernstein function.