Non relativistic and ultra relativistic limits in 2D stochastic nonlinear damped Klein–Gordon equation

Abstract

We study the non relativistic and ultra relativistic limits in the two-dimensional nonlinear damped Klein–Gordon equation driven by a space-time white noise on the torus. In order to take the limits, it is crucial to clarify the parameter dependence in the estimates of solution. In this paper we present two methods to confirm this parameter dependence. One is the classical, simple energy method. Another is the method via Strichartz estimates.

Corrigendum: Non relativistic and ultra relativistic limits in 2D stochastic nonlinear damped Klein–Gordon equation (2022 Nonlinearity 35 2878)

“…Proof. The above estimate follows from the similar argument to the former part in the proof of theorem 4 of [1]-the convergence (U N , ε∂ t U N ) → (U, ε∂ t U) in C([0, T * ); H σ ) in probability and Fatou's lemma. This estimate implies that there exists a ρ-measurable set M T such that ρ(M T ) = 1 and that for (ψ, φ) ∈ M T , the solution U exists up to time…”

“…almost surely. Since (U, ε∂ t U) is locally well-posed for all initial values (u, v) ∈ H σ (corollary 2.1 of[1]), we have…”

“…Lemma 0.1. Let any T > 0 be fixed, and let σ < 1 be sufficiently close to 1 and δ = 1 − σ as in corollary 2.1 of [ 1] . There exists a constant C T independent of N such that…”

“…By using proposition 7 of [1] (the invariance of the law of Ψ N (t) under ρ N ) and lemma 5.1 of [1] (N-uniform L 2 (dμ)-bounds of dρ N /dμ), we have sup…”