Maximum principles for infinite dimensional diffusion equations (Q874890)

The author is concerned with the following differential operator: \[ \mathcal{L}\varphi(x)=\tfrac{1}{2} \operatorname{Tr}[Q(x)D^2\varphi(x)]+\langle Ax+g(x), D\varphi(x)\rangle,\quad x\in D(A). \] Here \(Q(x)\) is a symmetric positive linear operator on a Hilbert space \(H\), \(A:D(A)\subset H\to H\) a linear operator and \(g:H\to H\) a nonlinear mapping. Elliptic equations \[ \lambda\varphi(x)-\mathcal{L}\varphi(x)=f(x),\quad x\in D(A) \] and parabolic equations \[ \begin{aligned} \frac{\partial u}{\partial t}(t, x)&=\mathcal{L}u(t, x), \quad t\geq 0,\;x\in D(A),\\ u(0, x)&=u_0(x), \quad x\in H \end{aligned} \] are studied. Validity of the maximum principle is proved.