STATISTICAL EXPONENTIAL FORMULAS FOR HOMOGENEOUS DIFFUSION

Let Delta(1)(p) denote the 1-homogeneous p-Laplacian, for 1 <= p <= infinity. This paper proves that the unique bounded, continuous viscosity solution u of the Cauchy problem {u(t) - (p/N+p-2) Delta(1)(p)u = 0 for x is an element of R-N and t > 0, u(+, 0) = u(0) is an element of BUC(R-N) is given by the exponential formula u(t) := lim(n ->infinity) (M-p(t/n))(n) u(0), where the statistical operator M-p(h): BUC(R-N) -> BUC(R-N) is defined by (M-p(h)phi) (x) := (1 - q) median(partial derivative B (x,root 2h)) {phi} + q integral(partial derivative B(x,root 2h)) phi ds when 1 <= p <= 2, with q := N(p-1)/N+P-2, and by (M-p(h)phi) (x) := (1 - q) midrange(partial derivative B (x,root 2h)) {phi} + q integral(partial derivative B(x,root 2h)) phi ds when p >= 2, with q - N/N+P-2. Possible extensions to problems with Dirichlet boundary conditions are mentioned briefly.