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Kolmogorov equations in infinite dimensions: wellposedness and regularity of solutions, with applications to stochastic generalized Burgers equations, to appear on The Annals of Probab
"... Abstract. We develop a new method to uniquely solve a large class of heat equations, so called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time ..."
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Abstract. We develop a new method to uniquely solve a large class of heat equations, so called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally Lipschitz drifts. Apart from general analytic interest, the main motivation is to apply this to uniquely solve martingale problems in the sense of StroockVaradhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic NavierStokes equations. In this paper this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.
L p –REGULARITY FOR PARABOLIC OPERATORS WITH UNBOUNDED TIME–DEPENDENT COEFFICIENTS
, 903
"... Abstract. We establish the maximal regularity for nonautonomous OrnsteinUhlenbeck operators in L pspaces with respect to a family of invariant measures, where p ∈ (1, +∞). This result follows from the maximal L pregularity for a class of elliptic operators with unbounded, timedependent drift coe ..."
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Abstract. We establish the maximal regularity for nonautonomous OrnsteinUhlenbeck operators in L pspaces with respect to a family of invariant measures, where p ∈ (1, +∞). This result follows from the maximal L pregularity for a class of elliptic operators with unbounded, timedependent drift coefficients and potentials acting on L p (R N) with Lebesgue measure. 1.
L ∞UNIQUENESS OF GENERALIZED SCHRÖDINGER OPERATORS
, 2007
"... The main purpose of this paper is to show that the generalized Schrödinger operator A V f = 1 2∆f + b∇f − V f, f ∈ C ∞ 0 (R d), is a pregenerator for which we can prove its L ∞ ` R d, dx ´uniqueness. Moreover, we prove the L 1 (R d, dx)uniqueness of weak solutions for the FokkerPlanck equation ..."
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The main purpose of this paper is to show that the generalized Schrödinger operator A V f = 1 2∆f + b∇f − V f, f ∈ C ∞ 0 (R d), is a pregenerator for which we can prove its L ∞ ` R d, dx ´uniqueness. Moreover, we prove the L 1 (R d, dx)uniqueness of weak solutions for the FokkerPlanck equation associated with this pregenerator. Key Words: C0semigroups; L ∞uniqueness; generalized Schrödinger operators; FokkerPlanck equation.