arXiv subjects

Petri Laarne

Publications and source records attributed to Petri Laarne.

3 recordsLinked to original sources

arXiv2026

Invariance of $ϕ^4$ measure under nonlinear wave and Schrödinger equations on the plane

We show almost sure wellposedness of mild solution to the cubic nonlinear wave equation in a weighted Besov space over $\mathbb R^2$. To achieve this, we show that any weak limit of $ϕ^4$ measures on increasing tori is invariant under the equation. We review and slightly simplify the periodic theory and the construction of the weak limit measure, and then use finite speed of propagation to reduce the infinite-volume case to the previous setup. Our argument also gives a weaker invariance result on the nonlinear Schrödinger equation in the same setting.

math.AP↗

arXiv2025

Metastable transition times of the 1D dynamical sine-Gordon model

We study the dynamics of a stochastic heat equation with $γ\sin(βu)$ nonlinearity on one-dimensional torus. We show an Eyring--Kramers law for the jump rate between potential wells in the small-noise limit, and that the transition state undergoes a bifurcation at $γβ= 1$. The argument follows the potential-theoretic approach of Berglund and Gentz [Electron. J. Probab. 2013].

math-ph↗

arXiv2024

Eyring--Kramers law for the hyperbolic $ϕ^4$ model

We study the expected transition frequency between the two metastable states of a stochastic wave equation with double-well potential. By transition state theory, the frequency factorizes into two components: one depends only on the invariant measure, given by the $ϕ^4_d$ quantum field theory, and the other takes the dynamics into account. We compute the first component with the variational approach to stochastic quantization when $d = 2, 3$. For the two-dimensional equation with random data but no stochastic forcing, we also compute the transmission coefficient.

math.AP↗