Markovianity and ergodicity for a surface growth PDE

Abstract

The paper analyzes a model in surface growth where the uniqueness of weak solutions seems to be out of reach. We prove existence of a weak martingale solution satisfying energy inequalities and having the Markov property. Furthermore, under nondegeneracy conditions on the noise, we establish that any such solution is strong Feller and has a unique invariant measure

Markov selections for the 3D stochastic Navier–Stokes equations

“…Other equations with lack of well-posedness may be handled by a variant of this method, see Blömker et al [2] for a model of surface growth. Finally, a preliminary version of the results presented here have been given in Flandoli and Romito [19].…”

“…Again, it holds only for regular conditions. Proposition 6.9 Under Assumption 5.10, consider two arbitrary Markov selections (P (1) x ) x∈H and (P (2) x ) x∈H . If there are x 0 ∈ W and t 0 > 0 such that P (1) x 0 = P (2) x 0 on [0, t 0 ], then P (1) x = P (2) x for all x ∈ W. Proof For every ε ∈ (0, t 0 ), the disintegrations of P (1) x 0 and P (2) x 0 at time ε are the same, up to time t 0 .…”

“…Proposition 6.9 Under Assumption 5.10, consider two arbitrary Markov selections (P (1) x ) x∈H and (P (2) x ) x∈H . If there are x 0 ∈ W and t 0 > 0 such that P (1) x 0 = P (2) x 0 on [0, t 0 ], then P (1) x = P (2) x for all x ∈ W. Proof For every ε ∈ (0, t 0 ), the disintegrations of P (1) x 0 and P (2) x 0 at time ε are the same, up to time t 0 . Since they are solutions over [0, t 0 − ε] with initial conditions chosen at random by the common law µ ε at time ε of both P (1) x 0 and P (2) x 0 , we can deduce that P (1) y and P (2) y coincide over [0, t 0 − ε] for µ ε −a.e.…”

“…Corollary 6.10 Under Assumption 5.10, let (P (1) x ) x∈H and (P (2) x ) x∈H be two Markov selections. If there is an initial distribution on W such that the two solutions corresponding to the given selections coincide, then P (1) x = P (2) x for all x ∈ W. We remark that, as in Stroock and Varadhan [25,Theorem 12.2.4], uniqueness of the martingale problem follows from uniqueness of the Markov selection (see also Remark 2.7), so the previous result implies immediately the following criterion for uniqueness in law.…”

Stochastic Partial Differential Equations: An Introduction

“…[7] and the references therein). Taking into account random noises the equation is formulated on the interval ƒ WD0; LOE as follows:…”