53 research outputs found
A note on stochastic semilinear equations and their associated Fokker-Planck equations
In this paper we treat semilinear stochastic partial differential equations
by two methods. First, we extend the framework of [BDR10] from a Hilbert space
to a Gelfand triple and as an application we prove the existence of solutions
for the Fokker-Planck equations associated to semilinear equations with
space-time white noise and both with polynomially growing nonlinearities and
Burgers type nonlinearities at the same time. Second we adopt the approximation
technique from [BDR10] to obtain existence of unique strong solutions to
semilinear stochastic partial differential equations driven by space-time white
noise, generalizing corresponding known results from the literature.Comment: To appear in Journal of Mathematical Analysis and Application
Sub and supercritical stochastic quasi-geostrophic equation
In this paper, we study the 2D stochastic quasi-geostrophic equation on
for general parameter and multiplicative noise.
We prove the existence of weak solutions and Markov selections for
multiplicative noise for all . In the subcritical case
, we prove existence and uniqueness of (probabilistically) strong
solutions. Moreover, we prove ergodicity for the solution of the stochastic
quasi-geostrophic equations in the subcritical case driven by possibly
degenerate noise. The law of large numbers for the solution of the stochastic
quasi-geostrophic equations in the subcritical case is also established. In the
case of nondegenerate noise and in addition exponential ergodicity
is proved.Comment: Published at http://dx.doi.org/10.1214/13-AOP887 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Stochastic quasi-geostrophic equation
In this note we study the 2d stochastic quasi-geostrophic equation in
for general parameter and multiplicative
noise. We prove the existence of martingale solutions and pathwise uniqueness
under some condition in the general case, i.e. for all . In
the subcritical case , we prove existence and uniqueness of
(probabilistically) strong solutions and construct a Markov family of
solutions. In particular, it is uniquely ergodic for provided the
noise is non-degenerate. In this case, the convergence to the (unique)
invariant measure is exponentially fast. In the general case, we prove the
existence of Markov selections
- …