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Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups. (English) Zbl 1193.47047
The authors continue earlier investigations [G. Da Prato and 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 F.-Y. 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 $$f\in B_b(H)$$.

##### MSC:
 47D07 Markov semigroups and applications to diffusion processes 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J25 Continuous-time Markov processes on general state spaces 60J35 Transition functions, generators and resolvents
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