Stationary solutions of damped stochastic 2-dimensional Euler’s equation

Infinitesimal Invariance of Completely Random Measures for 2D Euler Equations

“…Weak Solutions and Stochastic Integrals. The following definitions are analogous to the ones given in the aforementioned [14,17] in the purely Gaussian or Poissonian case: in fact they are a generalization. We write (X , m) denoting the law of the restriction of M ∼ [a, q, ν] to S (D) on its support X ⊂ S ′ (D).…”

“…In fact, regularity of W discussed above might be improved with a finer Sobolev embedding (see [14,17]). For our purposes, it suffices to identify (X , m) with a standard Borel probability space contained (topologically) in S ′ (D), so to have both a well-behaved underlying probability space and a natural way to couple samples of M and smooth functions.…”

“…(3.1) as stochastic integrals with respect to the random vorticity ω t . We refer once again to [14,17] for a discussion on why this gives a sensible definition of weak solution, in particular obtaining this kind of weak solutions as limits of classical ones.…”

“…(1.1) were originally introduced and studied in a series of works by Albeverio and others [8,9]: in fact, the idea of Gaussian and Poissonian invariant measures as special cases of the general, independently scattered one, dates back to [7]. The works [10,19,20,17] concern relations, in the form of scaling limits, between Gaussian and Poissonian random fields preserved by Euler's dynamics.…”

Limit Theorems and Fluctuations for Point Vortices of Generalized Euler Equations

“…This is because random measures can be interpreted as random distributions (see for instance [22] for more details on the argument). To this end fix ψ ∈…”