Lévy processes and homogeneous killing transformation (Q1375343)

Let \(X^M\) be the subprocess of a Lévy process \(X\) killed by a homogeneous multiplicative functional \(M\), of which the bivariate Revuz measure is \(\mu (F)=\lim_{t \to 0} \frac 1t P^m \int^t_0 F(X_{s-},X_s)d(-M_s)\) with Lebegue measure \(m\). A measure on \(R^d \times R^d\) is identified to be a bivariate Revuz measure of a homogeneous multiplicative functional \(M\) when it is translation invariant with marginal for the second entry \(\sigma\)-finite and is dominated by the jumping measure \(J(dy-x)m(dx)\). The bivariate Revuz measure is then written into \(\xi (dy-x)m(dx)\) with \(I_{R^d \setminus \{0 \}} \xi \leq J\). The Lévy exponent \(\psi \) of \(X^M\) is proven to be the Lévy exponent \(\varphi\) of \(X\) plus the Fourier transform of \(\xi\). Finally, it is shown that \(\psi\) inherits the sector condition of \(\varphi\), i.e. the imaginary part is dominated by a constant times its real part.