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

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

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 $A$ and a cylindrical Wiener process on a convex set $\Gamma$ in a Hilbert space $H$. We prove the existence and uniqueness of a strong solution of this problem when $\Gamma$ is a regular convex set. The result is also extended to the non-symmetric case. Finally, we extend our results to the case when $\Gamma=K_\alpha$, where $K_\alpha={f\in L^2 (0,1)|f\geq -\alpha},\alpha\geq0$

Recent developments in stochastic analysis and related topics : proceedings of the First Sino-German Conference on Stochastic Analysis (A satellite conference of ICM 2002) , Beijing, China, 29 August - 3 September 2002

xv, 452 p. : ill. ; 23 cm