We use Brownian motion ideas to study Schriidinger operators H = +A + V on Lp(R”). In particular: (a) We prove that limt+m t-l In /I e&H j111.9 is p-independent for a very large class of V’s where 11 A ll,,,n = norm of A as an operator from L’ to L”. (b) For Y > 3 and V E LY/~--E n L”Ip+c, we show that sup jl eetH Ilm,m -: (D if and only if H has no negative eigenvalues or zero energy resonance...