Nonlinear stochastic evolution equations of second order with damping

Abstract

Convergence of a full discretization of a second order stochastic evolution equation with nonlinear damping is shown and thus existence of a solution is established. The discretization scheme combines an implicit time stepping scheme with an internal approximation. Uniqueness is proved as well.

Convergent numerical approximation of the stochastic total variation flow

“…In order to show {F t i } N i=1 -measurability of the random variables {X i ε } N i=1 , {X i ε,δ,n } N i=1 we make use of the following lemma, cf. [9,6]. Lemma 4.2.…”

“…Proof of Lemma 4.6. The proof follows along the lines of [9], [6]. We sketch the main steps of the proof for the convenience of the reader.…”

“…The following result shows that the limit Y ≡ X ε,δ n , i.e., that the numerical solution of scheme (42) converges to the unique variational solution of (7) for τ → 0. Owing to the properties (10), (11) the convergence proof follows standard arguments for the convergence of numerical approximations of monotone equations, see for instance [9], [6], and is therefore omitted. We note that the convergence of the whole sequence {X ε,δ,n τ } τ >0 follows by the uniqueness of the variational solution.…”

The BDF2-Maruyama Scheme for Stochastic Evolution Equations with Monotone Drift

“…These techniques rely on the monotonicity condition (9) and have been used to show well-posedness of the one-step BEM scheme applied to nonlinear stochastic evolution equations, see, e.g., [16,Theorem 3.3] or [19,Theorem 2.9]. Here, we adapt this approach to the multi-step BDF2-Mayurama scheme in order to prove well-posedness.…”

Numerical approximation of singular-degenerate parabolic stochastic PDEs

“…i) We deduce from Corollary 4.1 iii), iv) that u − τ u − and u τ u in L p (Ω×(0, T ); V). The limit are the same according to [25,Lemma 4.2] see also [40,proof of Prop. 3.3].…”