×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Arnaudon, M.; Thalmaier, A.; Wang, F.-Y., Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below, Bull. sci. math., 130, 3, 223-233, (2006) · Zbl 1089.58024
[2] Beznea, L.; Boboc, N.; Röckner, M., Markov processes associated with \(L^p\)-resolvents and applications to stochastic differential equations in Hilbert spaces, J. evol. equ., 6, 4, 745-772, (2006) · Zbl 1114.60060
[3] Bogachev, V.; Da Prato, G.; Röckner, M., Regularity of invariant measures for a class of perturbed ornstein – uhlenbeck operators, Nonlinear differential equations appl., 3, 261-268, (1996) · Zbl 0852.60071
[4] Cerrai, S., Second order PDE’s in finite and infinite dimensions. A probabilistic approach, Lecture notes in math., vol. 1762, (2001), Springer-Verlag
[5] Da Prato, G., Kolmogorov equations for stochastic pdes, (2004), Birkhäuser · Zbl 1066.60061
[6] Da Prato, G.; Röckner, M., Singular dissipative stochastic equations in Hilbert spaces, Probab. theory related fields, 124, 2, 261-303, (2002) · Zbl 1036.47029
[7] Da Prato, G.; Malliavin, P.; Nualart, D., Compact families of Wiener functionals, C. R. acad. sci. Paris, 315, 1287-1291, (1992) · Zbl 0782.60002
[8] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge Univ. Press · Zbl 0761.60052
[9] W. Liu, Doctor-Thesis, Bielefeld University, 2008
[10] Ma, Z.M.; Röckner, M., Introduction to the theory of (non-symmetric) Dirichlet forms, (1992), Springer-Verlag
[11] Ondreját, M., Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes math. (rozprawy mat.), 426, (2004) · Zbl 1053.60071
[12] S.-X. Ouyang, Doctor-Thesis, Bielefeld University, 2008
[13] Prevot, C.; Röckner, M., A concise course on stochastic partial differential equations, Lecture notes in math., (2007), Springer · Zbl 1123.60001
[14] Röckner, M.; Wang, F.-Y., Harnack and functional inequalities for generalized mehler semigroups, J. funct. anal., 203, 1, 237-261, (2003) · Zbl 1059.47051
[15] Stannat, W., (nonsymmetric) Dirichlet operators on \(L^1\): existence, uniqueness and associated Markov processes, Ann. sc. norm. super. Pisa cl. sci. (4), 28, 1, 99-140, (1999) · Zbl 0946.31003
[16] Wang, F.-Y., Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. theory related fields, 109, 417-424, (1997) · Zbl 0887.35012
[17] Wang, F.-Y., Harnack inequality and applications for stochastic generalized porous media equations, Ann. probab., 35, 4, 1333-1350, (2007) · Zbl 1129.60060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.