Convolution power

If $x$ is a function on Euclidean space R^d and $n$ is a natural number, then one can define the convolution power as follows:

x^{*n}=\underbrace {x*x*x*\cdots *x*x} _{n},\quad x^{*0}=\delta _{0}

where * denotes the convolution operation of functions on R^d and δ₀ is the Dirac delta distribution. This definition makes sense if x is an integrable function (in L¹), a compactly supported distribution, or is a finite Borel measure.

In some cases, it is possible to define powers x^*t for arbitrary real t > 0. If μ is a probability measure, then μ is infinitely divisible provided there exists, for each positive integer n, a probability measure μ_1/n such that

\mu _{1/n}^{*n}=\mu .

That is, a measure is infinitely divisible if it is possible to define all nth roots. Not every probability measure is infinitely divisible, and a characterization of infinitely divisible measures is of central importance in the abstract theory of stochastic processes. Intuitively, a measure should be infinitely divisible provided it has a well-defined "convolution logarithm." The natural candidate for measures having such a logarithm are those of (generalized) Poisson type, given in the form

\pi _{\alpha ,\mu }=e^{-\alpha }\sum _{n=0}^{\infty }{\frac {\alpha ^{n}}{n!}}\mu ^{*n}.

In fact, the Lévy–Khinchin theorem states that a necessary and sufficient condition for a measure to be infinitely divisible is that it must lie in the closure, with respect to the vague topology, of the class of Poisson measures (Stroock 1993, §3.2).

Many applications of the convolution power rely on being able to define the analog of analytic functions as formal power series with powers replaced instead by the convolution power. Thus if $\textstyle {F(z)=\sum _{n=0}^{\infty }a_{n}z^{n}}$ is an analytic function, then one would like to be able to define

F^{*}(x)=a_{0}\delta _{0}+\sum _{n=1}^{\infty }a_{n}x^{*n}.

If x ∈ L¹(R^d) or more generally is a finite Borel measure on R^d, then the latter series converges absolutely in norm provided that the norm of x is less than the radius of convergence of the original series defining F(z). In particular, it is possible for such measures to define the complex exponential

\exp ^{*}(x)=\delta _{0}+\sum _{n=1}^{\infty }{\frac {x^{*n}}{n!}}.

It is not generally possible to extend this definition to arbitrary distributions, although a class of distributions on which this series still converges in an appropriate weak sense is identified by Ben Chrouda, El Oued & Ouerdiane (2002).

As convolution algebras are special cases of Hopf algebras, the convolution power is a special case of the (ordinary) power in a Hopf algebra. In applications to quantum field theory, the convolution exponential, convolution logarithm, and other analytic functions based on the convolution are constructed as formal power series in the elements of the algebra (Brouder, Frabetti & Patras 2008). If, in addition, the algebra is a Banach algebra, then convergence of the series can be determined as above.

Properties

If x is itself suitably differentiable, then the properties of convolution, one has

{\mathcal {D}}{\big \{}x^{*n}{\big \}}=({\mathcal {D}}x)*x^{*(n-1)}=x*{\mathcal {D}}{\big \{}x^{*(n-1)}{\big \}}

where ${\mathcal {D}}$ denotes the derivative operator. Specifically, this holds if x is a compactly supported distribution or lies in the Sobolev space W^1,1 to ensure that the derivative is sufficiently regular for the convolution to be well-defined.

References

Ben Chrouda, Mohamed; El Oued, Mohamed; Ouerdiane, Habib (2002), "Convolution calculus and applications to stochastic differential equations", Soochow Journal of Mathematics, 28 (4): 375–388, ISSN 0250-3255, MR1953702.
Brouder, Christian; Frabetti, Alessandra; Patras, Frédéric (2008), Decomposition into one-particle irreducible Green functions in many-body physics
Stroock, Daniel W. (1993), Probability theory, an analytic view, Cambridge University Press, ISBN 978-0-521-43123-1, MR1267569.

Properties

See also

References