Smoothing properties of nonlinear stochastic equations in Hilbert spaces
In a Hilbert space \(H\), let \(X(t,x)\) be the solution of a stochastic semilinear equation \[ dX(t)= AX(t)dt+ F\bigl(X(t)\bigr) dt+Q^{1/2} dW(t), \quad X(0)= x\in H, \] where \(A\) generates a strongly continuous semigroup in \(H\), \(F\) is a Lipschitz function on \(H\), \(Q\) is a bounded linear self-adjoint nonnegative (but possibly degenerate) operator in \(H\), and \(W\) is a cylindrical Wiener process in \(H\). It is proved that under some additional regularity assumptions on the coefficients the transition semigroup \((P_t\varphi)(x) =E[\varphi (X(t,x))]\) is Fréchet differentiable for any real bounded and uniformly continuous function \(\varphi\) on \(H\); moreover, the Fréchet derivative \((P_t\varphi)'\) satisfies the estimate \(|(P_t\varphi)'(x) |\leq C_t \sup_{x\in H} |\varphi (x)|\). Under somewhat weaker assumption, this semigroup is proved to satisfy the Lipschitz condition \[ \bigl|(P_t \varphi) (x_1)-(P_t \varphi) (x_2) \bigr|\leq K_t \sup_{x\in H} \bigl|\varphi (x)\bigr||x_1- x_2| \] for any bounded measurable \(\varphi\). The main tools in the proofs are Malliavin calculus and Girsanov theorem.
scientific article; zbMATH DE number 952686