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Start date: 2021 × Subject: Large deviations ×

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Now showing 1 - 3 of 3
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    Precise Laplace asymptotics for singular stochastic PDEs: The case of 2D gPAM
    (Amsterdam [u.a.] : Elsevier, 2022) Friz, Peter K.; Klose, Tom
    We implement a Laplace method for the renormalised solution to the generalised 2D Parabolic Anderson Model (gPAM) driven by a small spatial white noise. Our work rests upon Hairer's theory of regularity structures which allows to generalise classical ideas of Azencott and Ben Arous on path space as well as Aida and Inahama and Kawabi on rough path space to the space of models. The technical cornerstone of our argument is a Taylor expansion of the solution in the noise intensity parameter: We prove precise bounds for its terms and the remainder and use them to estimate asymptotically irrevelant terms to arbitrary order. While most of our arguments are not specific to gPAM, we also outline how to adapt those that are.
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    Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence
    (Berlin ; Heidelberg : Springer, 2021) Mielke, Alexander; Montefusco, Alberto; Peletier, Mark A.
    We introduce two new concepts of convergence of gradient systems (Q,Eε,Rε) to a limiting gradient system (Q,E0,R0). These new concepts are called ‘EDP convergence with tilting’ and ‘contact–EDP convergence with tilting.’ Both are based on the energy-dissipation-principle (EDP) formulation of solutions of gradient systems and can be seen as refinements of the Gamma-convergence for gradient flows first introduced by Sandier and Serfaty. The two new concepts are constructed in order to avoid the ‘unnatural’ limiting gradient structures that sometimes arise as limits in EDP convergence. EDP convergence with tilting is a strengthening of EDP convergence by requiring EDP convergence for a full family of ‘tilted’ copies of (Q,Eε,Rε). It avoids unnatural limiting gradient structures, but many interesting systems are non-convergent according to this concept. Contact–EDP convergence with tilting is a relaxation of EDP convergence with tilting and still avoids unnatural limits but applies to a broader class of sequences (Q,Eε,Rε). In this paper, we define these concepts, study their properties, and connect them with classical EDP convergence. We illustrate the different concepts on a number of test problems.
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    Large deviations for Markov jump processes with uniformly diminishing rates
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Agazzi, Andrea; Andreis, Luisa; Patterson, Robert I. A.; Renger, D. R. Michiel
    We prove a large-deviation principle (LDP) for the sample paths of jump Markov processes in the small noise limit when, possibly, all the jump rates vanish uniformly, but slowly enough, in a region of the state space. We further show that our assumptions on the decay of the jump rates are optimal. As a direct application of this work we relax the assumptions needed for the application of LDPs to, e.g., Chemical Reaction Network dynamics, where vanishing reaction rates arise naturally particularly the context of Mass action kinetics.