The stochastic Burgers equation stands an important role in fluid dynamics and has been studied by several authors. We mainly refer to the works [3], [4], [5] and [1], where the existence and uniqueness of the global solution as well as the existence and uniqueness of the invariant measure has been established in the "classical" case of space-time white noise.

We discuss the stochastic Burgers equation driven by "rougher" space-time white noise "B dW(t)", where B is the negative Laplacian operator with power gamma in [0, 1/4). We follow the approach of [2], where the polynomial moment estimates of the solution were found in the case gamma = 0. We generalize the results for gamma in [0, 1/4) and we improve the polynomial moment estimates to the exponential moment estimates.

In the second part of our talk, we establish the existence and uniqueness of the invariant measure to our system and we also show its exponential integrability.

References:

[1] Da Prato, G.: Kolmogorov Equations for Stochastic PDEs. Advanced Courses in Mathematics - CRM Barcelona, Springer Basel AG, Basel (2004)

[2] Da Prato, G., Debussche, A.: m-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise. Potential Analysis 26, 31-55 (2007)

[3] Da Prato, G., Debussche, A., Temam, R.: Stochastic Burgers' equation. Nonlinear Differential Equations and Applications NoDEA 1, 389-402 (1994)

[4] Da Prato, G., Gatarek, D.: Stochastic Burgers equation with correlated noise. Stochastics and Stochastic Reports 52, 29-41 (1995)

[5] Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Vol. 229, Cambridge University Press, Cambridge (1996)