3 research outputs found

    Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations

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    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 Stroock--Varadhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic Navier--Stokes 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.Comment: Published at http://dx.doi.org/10.1214/009117905000000666 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    The stochastic reflection problem on an infinite dimensional convex set and BV functions in a Gelfand triple

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    In this paper, we introduce a definition of BV functions in a Gelfand triple which is an extension of the definition of BV functions in [2] by using Dirichlet form theory. By this definition, we can consider the stochastic reflection problem associated with a self-adjoint operator AA and a cylindrical Wiener process on a convex set Ī“\Gamma in a Hilbert space HH. We prove the existence and uniqueness of a strong solution of this problem when Ī“\Gamma is a regular convex set. The result is also extended to the non-symmetric case. Finally, we extend our results to the case when Ī“=KĪ±\Gamma=K_\alpha, where KĪ±=fāˆˆL2(0,1)āˆ£fā‰„āˆ’Ī±,Ī±ā‰„0K_\alpha={f\in L^2 (0,1)|f\geq -\alpha},\alpha\geq0
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