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1. Large deviations for small noise diffusions over long time

   https://doi.org/10.1090/btran/172  

   Budhiraja, Amarjit; Zoubouloglou, Pavlos (January 2024, Transactions of the American Mathematical Society, Series B)

   We study two problems. First, we consider the large deviation behavior of empirical measures of certain diffusion processes as, simultaneously, the time horizon becomes large and noise becomes vanishingly small. The law of large numbers (LLN) of the empirical measure in this asymptotic regime is given by the unique equilibrium of the noiseless dynamics. Due to degeneracy of the noise in the limit, the methods of Donsker and Varadhan [Comm. Pure Appl. Math. 29 (1976), pp. 389–461] are not directly applicable and new ideas are needed. Second, we study a system of slow-fast diffusions where both the slow and the fast components have vanishing noise on their natural time scales. This time the LLN is governed by a degenerate averaging principle in which local equilibria of the noiseless system obtained from the fast dynamics describe the asymptotic evolution of the slow component. We establish a large deviation principle that describes probabilities of divergence from this behavior. On the one hand our methods require stronger assumptions than the nondegenerate settings, while on the other hand the rate functions take simple and explicit forms that have striking differences from their nondegenerate counterparts. 

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2. Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching

   https://doi.org/10.1017/apr.2020.47  

   Arapostathis, Ari; Pang, Guodong; Zheng, Yi (March 2021, Advances in Applied Probability)

   null (Ed.)

   Abstract We study ergodic properties of a class of Markov-modulated general birth–death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the ‘averaged’ fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth–death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied. 

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3. An Oceanic Ultra-Violet Catastrophe, Wave-Particle Duality and a Strongly Nonlinear Concept for Geophysical Turbulence

   https://doi.org/doi:10.3390/fluids2030036  

   Polzin, Kurt L.; Lvov, Yuri (June 2017, Fluids)

   Abstract: There is no theoretical underpinning that successfully explains how turbulent mixing is fed by wave breaking associated with nonlinear wave-wave interactions in the background oceanic internal wavefield. We address this conundrum using one-dimensional ray tracing simulations to investigate interactions between high frequency internal waves and inertial oscillations in the extreme scale separated limit known as “Induced Diffusion”. Here, estimates of phase locking are used to define a resonant process (a resonant well) and a non-resonant process that results in stochastic jumps. The small amplitude limit consists of jumps that are small compared to the scale of the resonant well. The ray tracing simulations are used to estimate the first and second moments of a wave packet’s vertical wavenumber as it evolves from an initial condition. These moments are compared with predictions obtained from the diffusive approximation to a self-consistent kinetic equation derived in the ‘Direct Interaction Approximation’. Results indicate that the first and second moments of the two systems evolve in a nearly identical manner when the inertial field has amplitudes an order of magnitude smaller than oceanic values. At realistic (oceanic) amplitudes, though, the second moment estimated from the ray tracing simulations is inhibited. The transition is explained by the stochastic jumps obtaining the characteristic size of the resonant well. We interpret this transition as an adiabatic ‘saturation’ process which changes the nominal background wavefield from supporting no mixing to the point where that background wavefield defines the normalization for oceanic mixing models. 

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4. On the small noise limit in the Smoluchowski-Kramers approximation of nonlinear wave equations with variable friction

   https://doi.org/10.1090/tran/8946  

   Cerrai, Sandra; Xie, Mengzi (July 2023, Transactions of the American Mathematical Society)

   We study the validity of a large deviation principle for a class of stochastic nonlinear damped wave equations, including equations of Klein-Gordon type, in the joint small mass and small noise limit. The friction term is assumed to be state dependent. We also provide the proof of the Smolchowski-Kramers approximation for the case of variable friction, non-Lipschitz nonlinear term and unbounded diffusion. 

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5. Geometric sharp large deviations for random projections of ℓpn spheres and balls

   https://doi.org/10.1214/23-EJP1020  

   Liao, Yin-Ting; Ramanan, Kavita (January 2024, Electronic Journal of Probability)

   Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics and asymptotic geometric analysis. Whereas large deviation principles identify the asymptotic exponential decay rate of probabilities, sharp large deviation estimates also provide the “prefactor” in front of the exponentially decaying term. For fixed p ∈ (1, ∞), consider independent sequences (X(n,p))_{n∈N} and (Θ_n)_{n∈N} of random vectors with Θn distributed according to the normalized cone measure on the unit l^n_2 sphere, and X(n,p) distributed according to the normalized cone measure on the unit lnp sphere. For almost every realization (θn)_{n∈N} of (Θ_n)_{n∈N}, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of X(n,p) onto θ_n, that are asymptotically exact (as the dimension n tends to infinity). Furthermore, the case when (X(n,p))_{n∈N} is replaced with (X(n,p))_{n∈N}, where X(n,p) is distributed according to the uniform (or normalized volume) measure on the unit lnp ball, is also considered. In both cases, in contrast to the (quenched) large deviation rate function, the prefactor exhibits a dependence on the projection directions (θ_n)_{n∈N} that encodes additional geometric information that enables one to distinguish between projections of balls and spheres. Moreover, comparison with numerical estimates obtained by direct computation and importance sampling shows that the obtained analytical expressions for tail probabilities provide good approximations even for moderate values of n. The results on the one hand provide more accurate quantitative estimates of tail probabilities of random projections of \ell^n_p spheres than logarithmic asymptotics, and on the other hand, generalize classical sharp large deviation estimates in the spirit of Bahadur and Ranga Rao to a geometric setting. The proofs combine Fourier analytic and probabilistic techniques. Along the way, several results of independent interest are obtained including a simpler representation for the quenched large deviation rate function that shows that it is strictly convex, a central limit theorem for random projections under a certain family of tilted measures, and multidimensional generalized Laplace asymptotics. 

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