Solutions of evolution equations associated to infinite-dimensional Laplacian

We study an evolution equation associated with the integer power of the Gross Laplacian Delta(p)(G) 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, as well as by Barhoumi-Kuo-Ouerdiane for the case p = 1 (See Ref. [A. Barhoumi, H. H. Kuo and H. Ouerdiane, Soochow J. Math. 32 (2006) 113.]).