Smoothing properties of nonlinear transition semigroups: case of Lipschitz nonlinearities (Q871612)

Consider, in a separable Hilbert space \(H\), a semilinear stochastic equation \[ dX(t)=[AX(t)+F(X(t))]dt+{\sqrt Q}dW(t),\;t\in]0,T];\qquad X(0)=x\in H, \] where \(A:D(A)\subset H\to H\) generates a strongly continuous semigroup \(e^{tA}\) on \(H\) such that \(\| e^{tA}\| \leq Me^{at}\) for some \(a<0\), \(W\) is a cylindrical Wiener process in \(H\), \(Q\) is linear continuous, self-adjoint and nonnegative, and \(F:H\to H\) is a nonlinear function. It is known [see \textit{G. Da Prato} and \textit{J. Zabczyk}, Stochastics Stochastics Rep. 35, No. 2, 63--77 (1991; Zbl 0726.60062)] that under appropriate additional assumptions on \(A\) and \(Q\), if \(F\) is bounded, the transition semigroup \(P_t\varphi(x):=E\varphi(X(t,x))\) (\(X(t,x)\) being the unique mild solution to the equation) transforms bounded measurable functions into Fréchet differentiable ones. The main theorem states that an analogous result holds for \(F\) unbounded and Lipschitz continuous. Namely, the semigroup \(P_t\) transforms bounded Lipschitz functions into Fréchet differentiable ones, and \(\| D_xP_t\varphi\| _0\leq\| \varphi\| _{\text{Lip}}M\exp\{(a+M\| F\| _{\text{Lip}})t\}\), where \(D_x\) is the Fréchet derivative, \(\| \cdot\| _0\) is the sup norm, and \(\| \cdot\| _{\text{Lip}}\) is the Lipschitz one. Moreover, \(u(t,x):=P_t\varphi(x)\) is a unique strong (and mild) solution of the Kolmogorov equation \[ \begin{aligned} &u_t={1\over 2}\, \text{ Tr}\, [Qu_{xx}]+\langle Ax+F(x),u_x\rangle,\quad t\in]0,T],\;x\in D(A),\\ &u(0,x)=\varphi(x),\quad x\in H.\end{aligned} \]