arXiv subjects

Ajay Chandra

Publications and source records attributed to Ajay Chandra.

26 records · Page 2Linked to original sources

arXiv2018

The dynamical sine-Gordon model in the full subcritical regime

We prove that the dynamical sine-Gordon equation on the two dimensional torus introduced in [HS16] is locally well-posed for the entire subcritical regime. At first glance this equation is far out of the scope of the local existence theory available in the framework of regularity structures [Hai14, BHZ16, CH16, BCCH17] since it involves a non-polynomial nonlinearity and the solution is expected to be a distribution (without any additional small parameter as in [FG17, HX18]). In [HS16] this was overcome by a change of variable, but the new equation that arises has a multiplicative dependence on highly non-Gaussian noises which makes stochastic estimates highly non-trivial - as a result [HS16] was only able to treat part of the subcritical regime. Moreover, the cumulants of these noises fall out of the scope of the later work [CH16]. In this work we systematically leverage "charge" cancellations specific to this model and obtain stochastic estimates that allow us to cover the entire subcritical regime.

math.PR↗

arXiv2018

An analytic BPHZ theorem for regularity structures

We prove a general theorem on the stochastic convergence of appropriately renormalized models arising from nonlinear stochastic PDEs. The theory of regularity structures gives a fairly automated framework for studying these problems but previous works had to expend significant effort to obtain these stochastic estimates in an ad-hoc manner. In contrast, the main result of this article operates as a black box which automatically produces these estimates for nearly all of the equations that fit within the scope of the theory of regularity structures. Our approach leverages multi-scale analysis strongly reminiscent to that used in constructive field theory, but with several significant twists. These come in particular from the presence of "positive renormalizations" caused by the recentering procedure proper to the theory of regularity structure, from the difference in the action of the group of possible renormalization operations, as well as from the fact that we allow for non-Gaussian driving fields. One rather surprising fact is that although the "canonical lift" is of course typically not continuous on any Hölder-type space containing the noise (which is why renormalization is required in the first place), we show that the "BPHZ lift" where the renormalization constants are computed using the formula given in arXiv:1610.08468, is continuous in law when restricted to a class of stationary random fields with sufficiently many moments.

math.PR↗

arXiv2017

Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem

Upon its inception the theory of regularity structures allowed for the treatment for many semilinear perturbations of the stochastic heat equation driven by space-time white noise. When the driving noise is non-Gaussian the machinery of theory can still be used but must be combined with an infinite number of stochastic estimates in order to compensate for the loss of hypercontractivity. In this paper we obtain a more streamlined and automatic set of criteria implying these estimates which facilitates the treatment of some other problems including non-Gaussian noise such as some general phase coexistence models - as an example we prove here a generalization of the Wong-Zakai Theorem found by Hairer and Pardoux.

math.PR↗

arXiv2017

Stochastic PDEs, Regularity Structures, and Interacting Particle Systems

These lecture notes grew out of a series of lectures given by the second named author in short courses in Toulouse, Matsumoto, and Darmstadt. The main aim is to explain some aspects of the theory of "Regularity structures" developed recently by Hairer in arXiv:1303.5113 . This theory gives a way to study well-posedness for a class of stochastic PDEs that could not be treated previously. Prominent examples include the KPZ equation as well as the dynamic $Φ^4_3$ model. Such equations can be expanded into formal perturbative expansions. Roughly speaking the theory of regularity structures provides a way to truncate this expansion after finitely many terms and to solve a fixed point problem for the "remainder". The key ingredient is a new notion of "regularity" which is based on the terms of this expansion.

math.AP↗

arXiv2013

Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions

In this article we provide the complete proof of the result announced in arXiv:1210.7717 about the construction of scale invariant non-Gaussian generalized stochastic processes over three dimensional p-adic space. The construction includes that of the associated squared field and our result shows this squared field has a dynamically generated anomalous dimension which rigorously confirms a prediction made more than forty years ago, in an essentially identical situation, by K. G. Wilson. We also prove a mild form of universality for the model under consideration. Our main innovation is that our rigourous renormalization group formalism allows for space dependent couplings. We derive the relationship between mixed correlations and the dynamical systems features of our extended renormalization group transformation at a nontrivial fixed point. The key to our control of the composite field is a partial linearization theorem which is an infinite-dimensional version of the Koenigs Theorem in holomorphic dynamics. This is akin to a nonperturbative construction of a nonlinear scaling field in the sense of F. J. Wegner infinitesimally near the critical surface. Our presentation is essentially self-contained and geared towards a wider audience. While primarily concerning the areas of probability and mathematical physics we believe this article will be of interest to researchers in dynamical systems theory, harmonic analysis and number theory. It can also be profitably read by graduate students in theoretical physics with a craving for mathematical precision while struggling to learn the renormalization group.

math.PR↗

arXiv2012

Rigorous quantum field theory functional integrals over the p-adics: research announcement

In this short note we announce the construction of scale invariant non-Gaussian generalized stochastic processes over three dimensional p-adic space. The construction includes that of the associated squared field and our result shows this squared field has a dynamically generated anomalous dimension which rigorously confirms a prediction made more than forty years ago, in an essentially identical situation, by Kenneth G. Wilson. We also prove a mild form of universality for the model under consideration.

math.PR↗

arXiv2008

Efficiency and influence function of estimators for ARCH models

This paper proposes a closed-form optimal estimator based on the theory of estimating functions for a class of linear ARCH models. The estimating function (EF) estimator has the advantage over the widely used maximum likelihood (ML) and quasi-maximum likelihood (QML) estimators that (i) it can be easily implemented, (ii) it does not depend on a distributional assumption for the innovation, and (iii) it does not require the use of any numerical optimization procedures or the choice of initial values of the conditional variance equation. In the case of normality, the asymptotic distribution of the ML and QML estimators naturally turn out to be identical and, hence, coincides with ours. Moreover, a robustness property of the EF estimator is derived by means of influence function. Simulation results show that the efficiency benefits of our estimator relative to the ML and QML estimators are substantial for some ARCH innovation distributions.

math.ST↗

arXiv2008

Testing the equality of error distributions from k independent GARCH models

In this paper we study the problem of testing the null hypothesis that errors from k independent parametrically specified generalized autoregressive conditional heteroskedasticity (GARCH) models have the same distribution versus a general alternative. First we establish the asymptotic validity of a class of linear test statistics derived from the k residual-based empirical distribution functions. A distinctive feature is that the asymptotic distribution of the test statistics involves terms depending on the distributions of errors and the parameters of the models, and weight functions providing the flexibility to choose scores for investigating power performance. A Monte Carlo study assesses the asymptotic performance in terms of empirical size and power of the three-sample test based on the Wilcoxon and Van der Waerden score generating functions in finite samples. The results demonstrate that the two proposed tests have overall reasonable size and their power is particularly high when the assumption of Gaussian errors is violated. As an illustrative example, the tests are applied to daily individual stock returns of the New York Stock Exchange data.

math.ST↗