Some properties of monotone gradient systems (Q2756186)

Maximal dissipativity for some gradient systems is proved in this paper. The gradient operator NEWLINE\[NEWLINEN_0\varphi= \textstyle{{1\over 2}} \text{ Tr}[D^2\varphi]- \langle x,\text{AD }\varphi\rangle- \langle DU(x), D\varphi\rangle,NEWLINE\]NEWLINE where \(A: D(A)\to H\) is a selfadjoint negative operator in a separable Hilbert space \(H\), \(U: H\to (-\infty,+\infty)\) is a convex nonnegative semicontinuous proper function is considered. It is shown that \(N_0\) is essentially selfadjoint in the weighted space \(L^2(H,\nu)\), where \(\nu\) is the measure \(\nu(dx)= \rho(x)\mu(dx)\) for some \(\rho\), and \(\mu\) is the Gaussian measure with mean zero and covariance operator \(Q= -{1\over 2} A^{-1}\). Some more assumptions, collected in hypothesis 1.1, are made. Some additional interesting results are also proved: strongly mixing property for \(\nu\), Poincaré and logarithmic Sobolev inequalities, and the strong Feller property for the transition semigroup.