1st Reading February 22, 2011 16:36 WSPC/S0219-0257 102-IDAQPRT 00426 Infinite Dimensional Analysis, Quantum Probability 1 and Related Topics 2 Vol. 14, No. 1 (2011) 1–14 3 c  World Scientific Publishing Company 4 DOI: 10.1142/S0219025711004262 5 EXOTIC LAPLACIANS AND DERIVATIVES 6 OF WHITE NOISE 7 LUIGI ACCARDI 8 Centro Vito Volterra, 9 Universit` a di Roma “Tor Vergata”, 10 00133 Via Columbia 2, Roma, Italy 11 accardi@volterra.mat.uniroma2.it 12 UN CIG JI 13 Department of Mathematics, 14 Research Institute of Mathematical Finance, 15 Chungbuk National University, 16 Cheongju 361-763, Korea 17 uncigji@chungbuk.ac.kr 18 KIMIAKI SAIT ˆ O 19 Department of Mathematics, 20 Meijo University, 21 Nagoya 468, Japan 22 ksaito@meijo-u.ac.jp 23 Received 7 June 2010 24 Revised 25 January 2011 25 Communicated by N. Obata 26 In this paper, we give a relationship between the Exotic Laplacians and the L´ evy Lapla- 27 cians in terms of the higher order derivatives of white noise by introducing an injective 28 and continuous linear operator acting on white noise functionals. Moreover, we study 29 a relationship between Exotic Laplacians, acting on higher order singular functionals, 30 each other in terms of the constructed operator. 31 Keywords : White noise theory; Exotic Laplacian; L´ evy Laplacian; higher order 32 derivative. 33 AMS Subject Classification: 60H40 34 1. Introduction 35 In Ref. 19, L´ evy introduced an infinite dimensional Laplacian, called the L´ evy 36 Laplacian, which, being defined as the Ces` aro mean of the second derivatives along 37 a sequence of orthogonal axes. The L´ evy Laplacian, the associated heat equation 38 and stochastic processes have been studied by many authors from several different 39 1