Special Issue on Quantum Theory: From Foundations to Technologies; Guest Editors: B. Coecke and A. KhrennikovNo Access

Solutions of evolution equations associated to infinite-dimensional Laplacian

Habib Ouerdiane

University of Tunis El Manar, Tunisia

E-mail Address: habib.ouerdiane@gmail.com

https://doi.org/10.1142/S0219749916400189Cited by:3 (Source: Crossref)

Abstract

We study an evolution equation associated with the integer power of the Gross Laplacian $Δ_{G}^{p}$ $Δ_{G}^{p}$ and a potential function V on an infinite-dimensional space. The initial condition is a generalized function. The main technique we use is the representation of the Gross Laplacian as a convolution operator. This representation enables us to apply the convolution calculus on a suitable distribution space to obtain the explicit solution of the perturbed evolution equation. Our results generalize those previously obtained by Hochberg [K. J. Hochberg, Ann. Probab.6 (1978) 433.] in the one-dimensional case with $V = 0$ $V = 0$ , as well as by Barhoumi–Kuo–Ouerdiane for the case $p = 1$ $p = 1$ (See Ref. [A. Barhoumi, H. H. Kuo and H. Ouerdiane, Soochow J. Math.32 (2006) 113.]).

Keywords:

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