A continuous kernel for the transition semigroup associated with a diffusion process in a Hilbert space (Q818939)

Let \(\{P_t;\,t\geq 0\}\) be the transition semigroup of the diffusion \(\{X_t;\, t\geq 0\}\) on a separable Hilbert space \(H\), solving \(dX_t[AX_t+ F(X_t)]dt+ dW_t\), \(X_0= x\), where \(A\) is a selfadjoint operator with trace class inverse, \(F\) the Gâteaux derivative \(DU\) of a function \(U\in W^{2,2}(H,\gamma)\), \(\gamma\) the centered Gaussian measure on \(H\) with covariance operator \(-(1/2)A^{-1}\), and \(\{W_t;\,t\geq 0\}\) a cylindrical Wiener process. Let \(\lambda_i\), \(i= 1,2,\dots\) be the eigenvalues of \(A\), and \(e_i\), \(i= 1,2,\dots\), the corresponding normed eigenelements in \(H\), set \(x_i:=\langle x,e_i\rangle\), \(x\in H\), and suppose that \[ \exists C_2: \Biggl|\sum_{1\leq i\leq n}(\partial^2_1 U(x)+ \lambda_i x_i\partial_i U(x))\Biggr|\leq C_2 (1+\| x\|)^2\;\forall x\in H. \] Let the image of \(DU\) be contained in the domain of \(A\) and suppose that \[ \exists C_1:\| ADU(x)\|\leq C_1(1+\| x\|)\;\forall x\in H. \] Define \(L_0\) as the Ornstein-Uhlenbeck operator, given on smooth cylindrical functions by \[ L_0f(x):= (1/2)\text{Tr\,}D^2 f(x)+\langle x,ADf(x)\rangle \] and closed in \(L^2(H,\gamma)\), and let \(L_0U+ (1/2)\| DU\|^2\in L^2(H,\gamma)\) have a representative \(V\) bounded from below. Then, for all \(t\geq 0\), \(x\in H\), and every bounded Borel function \(\phi\), \(P_t\phi(x)= \int_H\phi(x) p_t(x, y)\,d\gamma(y)\) with \[ p_t(x, y)= q_t(x,y) e^{U(y)} e^{-U(x)}{\mathbf E}\exp\Biggl\{-\int^t_0 V(X^{0,X}_{t,y}(s))\,ds\Biggr\}, \] \[ q_t(x,y):= \prod_{n\geq 1}(1-\exp[- 2\lambda_n])^{-1/2}\exp\{- (y_n- x_n\exp[-\lambda_n t])/\lambda^{-1}_n(1- \exp[-2\lambda_n])- y^2_n/\lambda^{-1}_n\}. \] If, in addition, \(e^U\in L^p(H,\gamma)\) for some \(p> 1\), then \(P_t\phi\) is continuous, \(t> 0\), \(\phi\) bounded Borel.