Abstract.
We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models.
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Received: 23 January 2000 / Revised version: 4 October 2000 / Published online: 5 June 2001
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Bogachev, V., Röckner, M. Elliptic equations for measures on infinite dimensional spaces and applications. Probab Theory Relat Fields 120, 445–496 (2001). https://doi.org/10.1007/PL00008789
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DOI: https://doi.org/10.1007/PL00008789
- Mathematics Subject Classification (2000): Primary 46G12, 35J15, 28C20; Secondary: 60H15, 82B20, 60J60, 60K35
- Key words or phrases: Invariant measures of diffusions – Elliptic equation for measures – Lyapunov function – Gibbs distribution – Logarithmic gradient – Stochastic Burgers equation – Stochastic Navier–Stokes equation – Stochastic reaction-diffusion equation