Lévy Laplacian and Brownian Particles in Hilbert Spaces

Let H be a separable Hilbert space and {Gn(t)} a sequence of independent Brownian motions on some probability space. Suppose {Kϵ, ϵ > 0} is a family of Hilbert-Schmidt operators on H such that, for all x ∈ H, ||Kϵx − x|| → 0 as ϵ → 0. Define W(ϵ, t) = Z(ϵ)−1 ∑ Kϵen Gn(t). Here Z(ϵ) = (Trace K*ϵ Kϵ)1/2 and {en} is a complete orthonormal system for H. Using this H-valued diffusion process W(ϵ, t), we show that the Lévy Laplacian ΔF(x) of F is given by ΔF(x) = 2 limϵ → 0 limt →0 t−1[E(F(x + W(ϵ, t))) − F(x)]. We study also the harmonic functionals on H.