×

zbMATH — the first resource for mathematics

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.
PDF BibTeX XML Cite
Full Text: DOI