A weak space-time formulation for the linear stochastic heat equation
Journal article, 2017
We apply the well-known Banach–Nečas–Babuška inf–sup theory in a stochastic setting to introduce a weak space-time formulation of the linear stochastic heat equation with additive noise. We give sufficient conditions on the data and on the covariance operator associated to the driving Wiener process, in order to have existence and uniqueness of the solution. We show the relation of the obtained solution to the mild solution and to the variational solution of the same problem. The spatial regularity of the solution is also discussed. Finally, an extension to the case of linear multiplicative noise is presented.
Linear multiplicative noise
Stochastic linear heat equation