On the subordination of spherically symmetric Lévy processes in Lie groups (Q2720352)

Let \(X:= (X(t))_{t\geq 0}\) be a Lévy process on a locally compact group \(G\) and \(T:= (T(t))_{t\geq 0}\) an independent subordinator, i.e., a Lévy process with values in \([0,\infty[\). It is well-known that \(X\circ T:= (X(T(t)))_{t\geq 0}\) then is again a Lévy process in \(G\). The generator \(L_{X\circ T}\) of the convolution semigroup of \(X\circ T\) is related to the generators \(L_X\) and \(L_T\) of \(X\) and \(T\), respectively. In particular, if \(H\) is a compact subgroup of \(G\) such that \((G,H)\) is a Gelfand pair, and if \((X(t))_{t\geq 0}\) is \(H\)-biinvariant, then \(X\circ T\) is also \(H\)-biinvariant, and the spherical Fourier transform of \(L_{X\circ T}\) may be expressed explicitly in terms of the spherical Fourier transform of \(L_X\) and the Fourier transform of \(L_T\). This is carried out in this paper in the case of a symmetric space \(G/H\), and the Lévy measures associated with \(X\) and \(X\circ T\) are compared. Moreover, a stochastic differential equation for \(X\circ T\) is given in the case of a Brownian motion \(X\).