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Ajay Chandra

Bio: Ajay Chandra is an academic researcher from Imperial College London. The author has contributed to research in topics: Physics & Mathematics. The author has an hindex of 6, co-authored 11 publications receiving 275 citations.

Papers
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TL;DR: In this article, a general theorem on the convergence of appropriately renormalized models arising from nonlinear stochastic PDEs was proved, and the theory of regularity structures gave a fairly automated framework for studying these problems.
Abstract: 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 Holder-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.

152 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT.
Abstract: The formalism recently introduced in arXiv:1610.08468 allows one to assign a regularity structure, as well as a corresponding "renormalisation group", to any subcritical system of semilinear stochastic PDEs. Under very mild additional assumptions, it was then shown in arXiv:1612.08138 that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to such a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT. The present work completes this programme by constructing an action of the renormalisation group onto a suitable class of stochastic PDEs which is intertwined with its action on the corresponding space of models. This shows in particular that solutions constructed from the BPHZ lift of a smooth driving noise coincide with the classical solutions of a modified PDE. This yields a very general black box type local existence and stability theorem for a wide class of singular nonlinear SPDEs.

76 citations

Posted Content
TL;DR: In this paper, the authors provided 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.
Abstract: 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.

39 citations

Journal ArticleDOI
TL;DR: In this article, the renormalization group on a suitable class of stochastic PDEs which is intertwined with its action on the corresponding space of models is constructed, which yields a general black box type local existence and stability theorem for a wide class of singular non-linear SPDEs.
Abstract: The formalism recently introduced in [BHZ19] allows one to assign a regularity structure, as well as a corresponding “renormalisation group”, to any subcritical system of semilinear stochastic PDEs. Under very mild additional assumptions, it was shown in [CH16] that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to such a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT. The present work completes this programme by constructing an action of the renormalisation group on a suitable class of stochastic PDEs which is intertwined with its action on the corresponding space of models. This shows in particular that solutions constructed from the BPHZ lift of a smooth driving noise coincide with the classical solutions of a modified PDE. This yields a very general black box type local existence and stability theorem for a wide class of singular non-linear SPDEs.

31 citations

Posted Content
TL;DR: In this article, it was shown that the dynamical sine-Gordon equation on the two-dimensional torus is locally well-posed for the entire subcritical regime of the torus.
Abstract: 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.

29 citations


Cited by
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Book ChapterDOI
31 Oct 2006

1,424 citations

Journal ArticleDOI
TL;DR: In this article, a canonical renormalization procedure for stochastic PDEs containing nonlinearities involving generalised functions is given, which is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of automorphisms.
Abstract: We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions This theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms This subgroup is sufficiently large to be able to implement a version of the BPHZ renormalisation prescription in this context This is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms, which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations Our construction is based on bialgebras of decorated coloured forests in cointeraction More precisely, we have two Hopf algebras in cointeraction, coacting jointly on a vector space which represents the generalised functions of the theory Two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic Birkhoff factorisation that arises naturally in perturbative quantum field theory

197 citations

Posted Content
TL;DR: In this article, a general theorem on the convergence of appropriately renormalized models arising from nonlinear stochastic PDEs was proved, and the theory of regularity structures gave a fairly automated framework for studying these problems.
Abstract: 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 Holder-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.

152 citations

01 Jan 2000
TL;DR: In this paper, the convergence of the Fcynman integrand is proved by an application of the power counting theorem, and the convergence is proved for the non-algebraic version of renormalization in momentum space.
Abstract: Bogoliubov’s method of renormalization is formulated in momentum space. The convergence of the rcnonnalizcd Fcynman integrand is proved by an application of the power counting theorem.

146 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT.
Abstract: The formalism recently introduced in arXiv:1610.08468 allows one to assign a regularity structure, as well as a corresponding "renormalisation group", to any subcritical system of semilinear stochastic PDEs. Under very mild additional assumptions, it was then shown in arXiv:1612.08138 that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to such a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT. The present work completes this programme by constructing an action of the renormalisation group onto a suitable class of stochastic PDEs which is intertwined with its action on the corresponding space of models. This shows in particular that solutions constructed from the BPHZ lift of a smooth driving noise coincide with the classical solutions of a modified PDE. This yields a very general black box type local existence and stability theorem for a wide class of singular nonlinear SPDEs.

76 citations