A variational approach to stochastic nonlinear parabolic problems (Q638453)

The paper tackles the existence and uniqueness for solutions to the following stochastic nonlinear parabolic problems: \[ dX - \Delta X dt + \beta(t,\,X)dt \ni \sqrt{Q} dW_t\;\;\; \text{ in }\; (0,\, T) \times {\mathcal O}; \] \[ X(0) = x \;\; \text{ in }\; {\mathcal O},\;\; X= 0 \;\;\text{ on }\; (0,\, T) \times \partial{\mathcal O} \] and \[ dX - \beta(t,\,X)dt \ni \sqrt{Q} dW_t\;\;\; \text{ in }\; (0,\, T) \times {\mathcal O}; \] \[ X(0) = x \;\; \text{ in }\; {\mathcal O},\;\; X= 0 \;\;\text{ on }\; (0,\, T) \times \partial{\mathcal O}. \] The solutions \(X\) to these problems are stochastic processes with values in a separable Hilbert space \(H\). Here, \(H=L^2({\mathcal O})\) for the first problem and \(H=H^{-1}({\mathcal O})\) for the second problem. \({\mathcal O}\) is a bounded and open set of \({\mathbb R}^d\) having a smooth boundary \(\partial{\mathcal O}\), \(Q\) is a linear continuous self-adjoint positive operator on \(H\) with finite trace, \(W_t\) is a cylindrical Wiener process and \(\beta(t,\cdot): {\mathbb R} \to {\mathbb R}\) is a time-dependent maximal monotone graph. Assuming some additional conditions on \(\beta\), \(W_t\) or \(Q\), the author shows that each of these problems admits a unique strong solution. Moreover, it is proven that these problems are equivalent with stochastic convex optimal control problems with linear state systems. This fact leads to sharper existence and uniqueness results under minimal growth conditions on nonlinear diffusion coefficients.