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Derivative formula and applications for hyperdissipative stochastic Navier-Stokes/Burgers equations. (English) Zbl 1264.60047
The abstract stochastic equation $dX_t=Q\,dW_t-\{LX_t+B(X_t,X_t)\}$ is considered, where $$L$$ is a positively definite self-adjoint operator on a Hilbert space $$H$$, $$V=D(L^{1/2})$$, $$Q$$ is an injective Hilbert-Schmidt operator on $$H$$, $$B:V\times V\to H$$ is a bilinear operator, $$W$$ is a cylindrical Wiener process on $$H$$ and four additional hypotheses are imposed on $$Q$$, $$L$$ and $$B$$. Under these assumptions, the equation is well-posed and the solutions define a Markov semigroup $$(P_t)$$ on $$H$$. The authors study existence of directional derivatives of $$x\mapsto P_tf(x)$$, represent $$D_hP_tf$$ as an expectation of a random expression (Bismut-type derivative formula) similar but yet different from the Elworthy-Li formula, prove several estimations for $$DP_tf$$ (e.g., gradient estimates, dimension-free Harnack inequality, $$V_\theta$$-strong Feller property, heat kernel estimates) and apply the results to prove a support theorem for the invariant measure and a density theorem for $$P_t$$. It is shown that the obtained results are applicable to the stochastic hyperdissipative Navier-Stokes/Burgers equation.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60J75 Jump processes (MSC2010) 60J45 Probabilistic potential theory
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