Bernstein function

From nonlocal pde
Revision as of 05:30, 19 July 2012 by imported>Mateusz (Created page with "A function $f : [0, \infty) \to [0, \infty)$ is said to be a Bernstein function if $(-1)^k f^{(k)}(x) \le 0$ for $x > 0$ and $k = 1, 2, ...$.<ref name="SSV"/> ==Relation to comp...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

A function $f : [0, \infty) \to [0, \infty)$ is said to be a Bernstein function if $(-1)^k f^{(k)}(x) \le 0$ for $x > 0$ and $k = 1, 2, ...$.[1]

Relation to complete monotonicity

Clearly, $f$ is a Bernstein function if and only if it is nonnegative, and $f'$ is a completely monotone function.

Representation

By Bernstein's theorem, $f$ is a Bernstein function if and only if: \[ f(z) = a z + b + \int_{(0, \infty)} (1 - e^{-t z}) \mu(\mathrm d t) \] for some $a, b \ge 0$ and a Radon measure $\mu$ such that $\int_{(0, \infty)} \min(1, t) \mu(\mathrm d t) < \infty$.

Subordination

Bernstein functions are closely related to Bochner's subordination of semigroups. Namely, for a nonnegative definite self-adjoint operator $L$ and a Bernstein function $f$, the operator $-f(L)$ (defined by means of spectral theory) is the generator of some semigroup of operators which is subordinate to the semigroup $e^{-t L}$ generated by $-L$. Conversely, every generator of a semigroup subordinate to $e^{-t L}$ is equal to $-f(L)$ for some Bernstein function $f$.

Bernstein functions of the Laplacian

Bernstein functions of the Laplacian are translation invariant non-local operators in $\R^n$. More precisely, $A = f(-\Delta)$ for a Bernstein function $f$ if and only if \[ -A u(x) = a \Delta u(x) + b u(x) + \int_{\R^n} (u(x + z) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{|z| < 1}) k(z) \mathrm d z \] for some $a, b \ge 0$ and $k(z)$ of the form \begin{align*} k(z) &= \int_0^\infty (4 \pi t)^{-n/2} e^{-|z|^2 / (4 t)} \mu(\mathrm d t) . \end{align*}

References

  1. Schilling, R.; Song, R.; Vondraček, Z. (2010), Bernstein functions. Theory and Applications, Studies in Mathematics, 37, de Gruyter, Berlin, doi:10.1515/9783110215311, http://dx.doi.org/10.1515/9783110215311 

This article is a stub. You can help this nonlocal wiki by expanding it.