Cutoff Thermalization for Ornstein–Uhlenbeck Systems with Small Lévy Noise in the Wasserstein Distance

This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein–Uhlenbeck systems $$(X^\varepsilon _t(x))_{t\geqslant 0}$$ ( X t ε ( x ) ) t ⩾ 0 with $$\varepsilon $$ ε -small additive Lévy noise and initial value x. The driving noise processes include Brownian motion, $$\alpha $$ α -stable Lévy flights, finite intensity compound Poisson processes, and red noises, and may be highly degenerate. Window cutoff thermalization is shown under mild generic assumptions; that is, we see an asymptotically sharp $$\infty /0$$ ∞ / 0 -collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure $$\mu ^\varepsilon $$ μ ε along a time window centered on a precise $$\varepsilon $$ ε -dependent time scale $$\mathfrak {t}_\varepsilon $$ t ε . In many interesting situations such as reversible (Lévy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. That is, for generic initial data x we obtain the stronger result $$\mathcal {W}_p(X^\varepsilon _{t_\varepsilon + r}(x), \mu ^\varepsilon ) \cdot \varepsilon ^{-1} \rightarrow K\cdot e^{-q r}$$ W p ( X t ε + r ε ( x ) , μ ε ) · ε - 1 → K · e - q r for any $$r\in \mathbb {R}$$ r ∈ R as $$\varepsilon \rightarrow 0$$ ε → 0 for some spectral constants $$K, q>0$$ K , q > 0 and any $$p\geqslant 1$$ p ⩾ 1 whenever the distance is finite. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of $$\mathcal {Q}$$ Q . Precise error bounds are given. Using these results, this article provides a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to $$\varepsilon $$ ε -small Brownian motion or $$\alpha $$ α -stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.

Are there jumps in evidence accumulation, and what, if anything, do they reflect psychologically? An analysis of Lévy-Flights models of decision-making

According to existing theories of simple decision-making, decisions are initiated by continuously sampling and accumulating perceptual evidence until a threshold value has been reached. Many models, such as the diffusion decision model, assume a noisy accumulation process, described mathematically as a stochastic Wiener process with Gaussian distributed noise. Recently, an alternative account of decision-making has been proposed in the Lévy Flights (LF) model, in which accumulation noise is characterized by a heavy-tailed power-law distribution, controlled by a parameter, α. The LF model produces sudden large “jumps” in evidence ac- cumulation that are not produced by the standard Wiener diffusion model, which some have argued provide better fits to data. It remains unclear, however, whether jumps in evidence accumulation have any real psychological meaning. Here, we investigate the conjecture by Voss et al. (2019) that jumps might reflect sudden shifts in the source of evidence people rely on to make decisions. We reason that if jumps are psychologically real, we should observe systematic reductions in jumps as people become more practiced with a task (i.e., as people converge on a stable decision strategy with experience). We fitted four versions of the LF model to behavioral data from a study by Evans and Brown (2017), using a five-layer deep inference neural network for parameter estimation. The analysis revealed systematic reductions in jumps as a function of practice, such that the LF model more closely approximated the standard Wiener model over time. This trend could not be attributed to other sources of parameter variability, speaking against the possibility of trade-offs with other model parameters. Our analysis suggests that jumps in the LF model might be capturing strategy instability exhibited by relatively inexpe- rienced observers early on in task performance. We conclude that further investigation of a potential psychological interpretation of jumps in evidence accumulation is warranted.

Mutations as Levy flights

Data from a long time evolution experiment with Escherichia Coli and from a large study on copy number variations in subjects with European ancestry are analyzed in order to argue that mutations can be described as Levy flights in the mutation space. These Levy flights have at least two components: random single-base substitutions and large DNA rearrangements. From the data, we get estimations for the time rates of both events and the size distribution function of large rearrangements.