STOCHASTIC NONLINEAR DIFFUSION EQUATIONS WITH SINGULAR DIFFUSIVITY

摘要

In this paper we are concerned with the stochastic diffusion equation dX(t) =div[sgn(Δ(X(t)))]dt + Q～(1/2) dW(t) in (0,∞) × O, where O is a bounded open subset of Rd, d = 1, 2,W(t) is a cylindrical Wiener process on L2(O), and sgn(ΔX) = ΔX/|ΔX|_d ifΔX ≠ 0 and sgn (0) = {v ∈ Rd : |v|d ≤ 1}. The multivalued and highly singular diffusivity term sgn(ΔX) describes interaction phenomena, and the solution X = X(t) might be viewed as the stochastic flow generated by the gradient of the total variation ||DX||. Our main result says that this problem is well posed in the space of processes with bounded variation in the spatial variable ξ. The above equation is relevant for modeling crystal growth as well as for total variation based techniques in image restoration.