## 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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### References:

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