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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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