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Log-Harnack inequality for stochastic differential equations in Hilbert spaces and its consequences. (English) Zbl 1207.60053
The paper is devoted to establishing the log-Harnack inequality
\[ P_t \log f (x) \leq \log P_t f (y) + \frac{K \rho(x,y)^2}{2(1-\exp(-Kt))}, \] for any bounded strictly positive \(f\) and for a semigroup that corresponds to the Itô stochastic differential equation in \(\mathbb R^n\),
\[ dX_t = b(X_t) dt + \sigma(X_t) dB_t, \] under Lipschitz condition on (non-degenerate) \(\sigma\) and one-sided Lipschitz condition on \(b\) and with a certain metric \(\rho\) related to \(\sigma\) and a constant \(K\) from the unified Lipschitz condition. Then, by using approximation techniques, the inequality is extended to equations in Hilbert space
\[ dX_t = (AX_t + F(X_t)) dt + \sigma(X_t) dW_t, \] with a cylindrical Brownian motion \(W\) and under a series of assumptions. Earlier similar results – including those by both authors – in Hilbert space are available only for a constant diffusion coefficient. As a corollary, a strong Feller property is derived with supplementary useful bounds.

60J60 Diffusion processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI
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