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On subordination of convolution semigroups. (English) Zbl 1062.60004

Let \(\mu =(\mu_t)\) be a transient convolution semigroup of measures on \(\mathbb R^d\), and \(EX(\mu )\) be the set of all exit laws of \(\mu\), that is Borel functions \(\varphi :\;(0,\infty )\times \mathbb R^d\to [0,\infty ]\), such that \(\varphi_t=\varphi (t,\cdot )\) satisfies the relation \(\mu_s*\varphi_t=\varphi_{s+t}\) almost everywhere with respect to the Lebesgue measure \(\lambda\). The authors construct a linear isomorphism of \(EX(\mu )\) onto the set of all those positive Borel measures \(\omega\) on \(\mathbb R^d\), for which \(\mu_t*\omega\) is absolutely continuous with respect to \(\lambda\). If \(\beta_t\) is a convolution semigroup on \([0,\infty )\), the subordinated semigroup \(\mu_t^\beta =\int_0^\infty \mu_sd\beta_t(s)\) is considered. The authors find conditions, under which this subordination relations hold for exit laws of \((\mu_t^\beta )\) and \((\mu_s)\).

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
31C25 Dirichlet forms
35K05 Heat equation
47D07 Markov semigroups and applications to diffusion processes
Full Text: DOI

References:

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