Existence of limits of analytic one-parameter semigroups of copulas

Abstract

A 2-copula $F$ is idempotent if $F*F=F$.  Here $*$ denotes the product      defined in [1].  An idempotent copula $F$ is said to be a unit      for a 2-copula $A$ if $F*A=A*F=A$.  An idempotent copula is said to      annihilate a 2-copula $A$ if $F*A=A*F=F$.
If $F$ is a unit for $A$ and $s$ is a non-negative real number, define$$\exp _F(sA)=F+sA+ {\frac{s^2}{2!}}A*A + {\frac{s^3}{3!}}A*A*A +\dots .$$ For any copula $A$ and any idempotent copula $F$ which is a unit for $A$, the set$$C_s=e^{-s}\exp _F(sA),\quad s\in [0,\infty )$$ is a semigroup of copulas under the $*$ operation, which is homomorphic to the semigroup $[0,\infty )$ under addition.  We call this set an analyticone-parameter semigroup of copulas. $C_s$ can be defined also for $s<0$, and$C_{-s}*C_s=C_s*C_{-s}=F$, but in general $C_s$ is not a copula for $s<0$.
We show that for any such analytic one-parameter semigroup, the limit $\lim_{s\to \infty}C_s=E$ exists.  We show also that the limit $E$ has the followingproperties:
(i) $E$ is idempotent.
(ii) $E$ annihilates $A$, $F$ and $C_s$.
(iii) $E$ is the greatest annihilator of $A$ and of $C_s$, $s\in (0,\infty )$.
\noindent It is also true that $F$ is the least unit for $C_s$, $s\in [0,\infty)$.  We give a geometrical interpretation of this result, and we comment on theuse of analytic semigroups to construct Markov processes with continuousparameter.