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On a class of gradient systems with irregular potentials. (English) Zbl 1055.60064

The authors consider a stochastic gradient system in a real Hilbert space \(H\) defined by the following stochastic differential equation \[ dX(t)= [AX(t)- DU(X(t))]\,dt+ dW_t,\quad X(0)= x\in H,\tag{\(*\)} \] where \(A:{\mathcal D}(A)\subset H\to H\) is a selfadjoint operator, \(U\) a real convex function on \(H\) and \(W_t\) a cylindrical Wiener process on \(H\). \(U\) is usually called a potential and \(DU\) denotes the gradient of \(U\). If equation \((*)\) has a solution \(X(\cdot,x)\), the authors consider the transition semigroup \({\mathcal P}_t\) defined by \({\mathcal P}_t(\varphi)(x)= \mathbb{E}(\varphi(X(t,x)))\) for \(\varphi\) in a suitable class of functions on \(H\). Its infinitesimal generator \(N_0\) is formally given by \[ N_0\varphi=\textstyle{{1\over 2}} \text{\,Tr}[D^2\varphi]+\langle Ax,D\rangle-\langle DU(x),D\varphi\rangle. \] The authors prove that \(N_0\) is dissipative in \(L^1(H,\nu)\), and that its closure is \(m\)-dissipative.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
47D07 Markov semigroups and applications to diffusion processes
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