Singular dissipative stochastic equations in Hilbert spaces
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.
stochastic differential equations on a Hilbert space
scientific article; zbMATH DE number 1837370