Some examples on unimodality of Lévy processes (Q1330235)

A stationary process \((X_ t)_{t\geq 0}\) with independent increments is called unimodal if for each \(t>0\) the distribution of \(X_ t\) is unimodal. The present paper contains certain counterexamples to open problems of \textit{K. Sato} (1992), on unimodality and mode behavior of Lévy processes. For instance, the author shows that there exists an increasing Lévy process \((X_ t)_{t\geq 0}\) which is not unimodal so that \((X_ t+ \sigma B_ t )_{t\geq 0}\) is unimodal for some \(\sigma>0\), where \((B_ t )_{t\geq 0}\) is a Brownian motion independent of \((X_ t )_{t\geq 0}\).