Carleman estimates and controllability of linear stochastic heat equations

This work is concerned with Carleman inequalities and controllability properties for the following stochastic linear heat equation (with Dirichlet boundary conditions in the bounded domain D subset of R-d and multiplicative noise): {d(t)y(u) - Deltay(u) + ay(u)dt = f dt + 1(D0)u dt + bydbeta(t) in ]0, T] x D, y(u) = 0 on ]0,T] x partial derivativeD, y(u)(0) = y(0) in D, and for corresponding backward dual equation: {d(t)p(nu) + Deltap(nu)dt - ap(nu)dt + bk(nu)dt = 1(D0)nudt + k(nu)dbeta(t) in [0,T[ x D, p(nu) = 0 on [0,T[ x partial derivativeD, p(nu)(T) = eta in D. We prove the null controllability of the backward euation and obtain partial results for the controllability of the forward equation.