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Singular dissipative stochastic equations in Hilbert spaces. (English) Zbl 1036.47029
Probab. Theory Relat. Fields 124, No. 2, 261-303 (2002); erratum ibid. 143, No. 3-4, 659-664 (2009).
The authors construct weak solutions to SDEs of the form $dX = (AX+F_0(X))\,dt + \sqrt C\,dW_t, \qquad X(0)=x\in H$ on a Hilbert space $$H$$. In the equation, $$W_t$$ is a cylindrical Wiener process, $$C$$ is a positive definite, bounded self adjoint linear operator on $$H$$, $$A$$ is the generator of a strongly continuous semigroup on $$H$$, and $$F_0(x):=y_0$$ where $$y_0\in F(x)$$, $$| y_0| =\min_{y\in F(x)} | y|$$ and $$F$$ is a maximally dissipative map from $$H$$ to its power set.
The solution is constructed in two steps: first, the authors solve the corresponding Kolmogorov equations in a suitable $$L^2$$-space and construct thus a strong Markov diffusion semigroup. In a second step, it is then shown that the Markov semigroups have a suitable (strong) Fellerian modification which allows to get a proper conservative diffusion process for the solutions of the single starting points. The last two sections deal with uniqueness of the solution and applications, in particular gradient systems and reaction-diffusion equations.

##### MSC:
 47D07 Markov semigroups and applications to diffusion processes 35K90 Abstract parabolic equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 47B44 Linear accretive operators, dissipative operators, etc.
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