The exotic (higher order Lévy) Laplacians generate the Markov processes given by distribution derivatives of white noise (Q2857627)

The paper represents a conclusion to two previous papers by the same authors [ibid. 12, No. 1, 1--19 (2009); errata ibid. 13, No. 2, 345 (2010; Zbl 1172.60019)] and [ibid. 14, No. 1, 1--14 (2011; Zbl 1213.60114)], where exotic Laplacians were proven to be related to Volterra-Gross Laplacians, and exotic Laplacians of order \(2p+1\) (\(p\in\mathbb N\)) were proven to be generators of infinite-dimensional Brownian motions corresponding to the \(p\)th-distributional derivative of the standard Brownian motion. The current paper extends this result to exotic Laplacians of even order and for fractional order. The authors prove that any exotic Laplacian of order \(2a\) (\(a\in\mathbb R_+\)) is the generator of infinite-dimensional white noise corresponding to the \(a\)th-distributional derivative of standard white noise. In particular, for \(a=\frac12\), one retrieves the classical Lévy Laplacian.