Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups
The authors continue earlier investigations [\textit{G.\,Da\,Prato} and \textit{M.\,Röckner}, Probab.\ Theory Relat.\ Fields 124, No.\,2, 261--303 (2002; Zbl 1036.47029)] in order to prove a Harnack inequality for solutions \((X(t))\) of stochastic differential equations (resp., their transition kernels) in the sense of \textit{F.-Y.\thinspace Wang} [Probab.\ Theory Relat.\ Fields 109, No.\,3, 417--424 (1997; Zbl 0887.35012)] of the form \[ d X(t)=(A X(t) + F(X(t)))dt + \sigma d W(t), \quad X(0) = x (\in H), \] where \(H\) is a separable Hilbert space, \((W(t))\) a cylindrical Brownian motion on \(H\), \(\sigma\) a positive definite operator with bounded inverse, \((A, D(A))\) the generator of a \(C_0\)-one-parameter semigroup satisfying the growth condition \(\langle Ax,x\rangle \leq \omega \| x\|^2\) on the domain \(D(A)\), for some real \(\omega\). \(F\) is a set-valued \(m\)-dissipative map \(F:H\supseteq D(F)\to 2^H\). Let \(F_0\) denote a map \(F_0:D(F)\to H\) satisfying \(F_0(x)\in F(x)\) and \(|F_0(x)| = \min_{y\in F(x)}|y|\). The corresponding Kolmogorov operator \(L_0\), defined on a subspace \(\mathcal{E}_A(H)\subseteq B_b(H)\), the space of bounded measurable real functions, is defined by \[ L_0(\varphi)(x)= \tfrac{1}{2} \mathrm{tr}(\sigma^2 D^2 \varphi(x))+ \langle x, A^*D\varphi(x)\rangle + \langle F_0(x), D\varphi(x)\rangle \] for \(x\in D(F)\), \(\varphi\in \mathcal{E}_A(H)\). The investigations rely, as in the aforementioned paper, on several assumptions, \(H_0 - H_5\). In particular, \(H_4\) implies the existence of a infinitesimally invariant probability measure \(\mu\) concentrated on the domain \(D(F)\), and \(L_0\) generates a Markov semigroup of transition kernels, called \(p_t^\mu(\cdot, d x)\) (on \(L^2(H,\mu)\)), such that a Harnack inequality holds for \(p>1\), \(f\in B_b(H)\) (Theorem 1.6): \[ (p_t^\mu f(x))^p \leq p_t^\mu f^p(y)\cdot \exp \left[\|\sigma^{-1}\|^2 p \omega |x-y|^2/\left((p-1)(1-\mathrm{e}^{-2\omega t})\right) \right] \] for \(x, y\in \operatorname{supp} \mu =: H_0\) and \( t>0\). The authors prove four corollaries of the main result, implying, e.g., the uniqueness of \(\mu\), estimates for the \(\mu\)-densities of the kernels \(p_t(y,\cdot)\) and hyper-boundedness of the transition operators, and, furthermore, \(p_t^\mu(L^p(H,\mu))\subseteq C(H_0)\) for all \(t>0\), hence the strong Feller property. The proof runs along the following steps: first the measurable function \(F\) (resp., \(F_0\)) is approximated by resolvents \(x\mapsto F_\alpha(x) := \frac{1}{\alpha}((I-\alpha F)^{-1}-I)(x)\) (Yosida approximation), \(\alpha >0\), which are single-valued Lipschitz functions, and these are approximated by \(C^\infty\)-functions \(F_{\alpha,\beta}\) (defined by regularizations with Gaussian distributions), and analogously, at first \(f\) is assumed to be bounded Lipschitz, then the results are extended to \(f\in C_b(H)\), and finally to \(f\in B_b(H)\).
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