Measuring the irreversibility of numerical schemes for reversible stochastic differential equations

Abstract. For a Markov process the detailed balance condition is equivalent to the time-reversibility of the process. For stochastic differential equations (SDE’s) time discretization numerical schemes usually destroy the property of time-reversibility. Despite an extensive literature on the numerical analysis for SDE’s, their stability properties, strong and/or weak error estimates, large deviations and infinite-time estimates, no quantitative results are known on the lack of reversibility of the discrete-time approximation process. In this paper we provide such quantitative estimates by using the concept of entropy production rate, inspired by ideas from non-equilibrium statistical mechanics. The entropy production rate for a stochastic process is defined as the relative entropy (per unit time) of the path measure of the process with respect to the path measure of the time-reversed process. By construction the entropy production rate is nonnegative and it vanishes if and only if the process is reversible. Crucially, from a numerical point of view, the entropy production rate is an a posteriori quantity, hence it can be computed in the course of a simulation as the ergodic average of a certain functional of the process (the so-called Gallavotti-Cohen (GC) action functional). We compute the entropy production for various numerical schemes such as explicit Euler-Maruyama and explicit Milstein’s for reversible SDEs with additive or multiplicative noise. Additionally, we analyze the entropy production for th

Pseudo-Majoron as Light Mediator of Singlet Scalar Dark Matter

In the singlet-triplet majoron model of neutrino mass, lepton number is spontaneously broken. If it is also softly broken, then a naturally light pseudoscalar particle $\eta_I$ exists. It may then act as a light mediator for a real singlet scalar $\chi$ with odd dark parity. It is itself unstable, but decays dominantly to two neutrinos through its triplet scalar component, thereby not disturbing the cosmic microwave background (CMB). It also mixes with the standard-model Higgs boson only in one loop, thereby not contributing significantly to the elastic scattering of $\chi$ off nuclei in dark-matter direct-search experiments.Comment: 10 page