Publications
9 results found
Weber H, Moinat A, Chandra A, 2023, A priori bounds for the φ⁴ equation in the full sub-critical regime, Archive for Rational Mechanics and Analysis, Vol: 247, Pages: 1-76, ISSN: 0003-9527
We derive a priori bounds for the Φ4 equation in the full sub-critical regime using Hairer’s theory of regularity structures. The equation is formally given by(∂t − )φ = −φ3 + ∞φ + ξ ,where the term +∞ϕ represents infinite terms that have to be removed in a renormalisation procedure. We emulate fractional dimensions d<4 by adjusting the regularity of the noise term ξ, choosing ξ∈C−3+δ. Our main result states that if ϕ satisfies this equation on a space–time cylinder D=(0,1)×{|x|⩽1}, then away from the boundary ∂D the solution ϕ can be bounded in terms of a finite number of explicit polynomial expressions in ξ. The bound holds uniformly over all possible choices of boundary data for ϕ and thus relies crucially on the super-linear damping effect of the non-linear term −ϕ3. A key part of our analysis consists of an appropriate re-formulation of the theory of regularity structures in the specific context of (*), which allows us to couple the small scale control one obtains from this theory with a suitable large scale argument. Along the way we make several new observations and simplifications: we reduce the number of objects required with respect to Hairer’s work. Instead of a model (Πx)x and the family of translation operators (Γx,y)x,y we work with just a single object (Xx,y) which acts on itself for translations, very much in the spirit of Gubinelli’s theory of branched rough paths. Furthermore, we show that in the specific context of (*) the hierarchy of continuity conditions which constitute Hairer’s definition of a modelled distribution can be reduced to the single continuity condition on the “coefficient on the constant level”.
Chandra A, Chevyrev I, Hairer M, et al., 2022, Langevin dynamic for the 2D Yang-Mills measure, Publications mathématiques de l'IHÉS, Vol: 136, Pages: 1-147, ISSN: 1618-1913
We define a natural state space and Markov process associated to the stochastic Yang–Mills heat flow in two dimensions.To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric.To construct the Markov process we show that the stochastic Yang–Mills heat flow takes values in our space of connections and use the “DeTurck trick” of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations.Our main tool for solving for the Yang–Mills heat flow is the theory of regularity structures and along the way we also develop a “basis-free” framework for applying the theory of regularity structures in the context of vector-valued noise – this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest.
Bonnefoi T, Chandra A, Moinat A, et al., 2022, A priori bounds for rough differential equations with a non-linear damping term, Journal of Differential Equations, Vol: 318, Pages: 58-93, ISSN: 0022-0396
We consider a rough differential equation with a non-linear damping drift term: dY (t) = −|Y |m−1Y (t)dt + σ (Y (t))dX(t),where m>1, X is a (branched) rough path of arbitrary regularity α>0, and where σ is smooth and satisfies an m and α-dependent growth property. We show a strong a priori bound for Y, which includes the “coming down from infinity” property, i.e. the bound on Y(t) for a fixed t>0 holds uniformly over all choices of initial datum Y(0).The method of proof builds on recent work on a priori bounds for the ϕ4 SPDE in arbitrary subcritical dimension [7]. A key new ingredient is an extension of the algebraic framework which permits to derive an estimate on higher order conditions of a coherent controlled rough path in terms of the regularity condition at lowest level.
Chandra A, Gunaratnam TS, Weber H, 2022, Phase transitions for ϕ43, Communications in Mathematical Physics, Vol: 392, Pages: 691-782, ISSN: 0010-3616
We establish a surface order large deviation estimate for the magnetisation of low temperature ϕ43. As a byproduct, we obtain a decay of spectral gap for its Glauber dynamics given by the ϕ43 singular stochastic PDE. Our main technical contributions are contour bounds for ϕ43, which extends 2D results by Glimm et al. (Commun Math Phys 45(3):203–216, 1975). We adapt an argument by Bodineau et al. (J Math Phys 41(3):1033–1098, 2000) to use these contour bounds to study phase segregation. The main challenge to obtain the contour bounds is to handle the ultraviolet divergences of ϕ43 whilst preserving the structure of the low temperature potential. To do this, we build on the variational approach to ultraviolet stability for ϕ43 developed recently by Barashkov and Gubinelli (Duke Math. J. 169(17):3339–3415, 2020).
Chandra A, Hairer M, Chevyrev I, et al., 2021, Renormalising SPDEs in regularity structures, Journal of the European Mathematical Society, Vol: 23, Pages: 869-947, ISSN: 1435-9855
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.
Chandra A, Hairer M, Shen H, 2020, The dynamical sine-Gordon model in the full subcritical regime
We prove that the dynamical sine-Gordon equation on the two dimensional torusintroduced 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 existencetheory available in the framework of regularity structures [Hai14, BHZ16, CH16,BCCH17] since it involves a non-polynomial nonlinearity and the solution isexpected 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 newequation that arises has a multiplicative dependence on highly non-Gaussiannoises which makes stochastic estimates highly non-trivial - as a result [HS16]was only able to treat part of the subcritical regime. Moreover, the cumulantsof these noises fall out of the scope of the later work [CH16]. In this work wesystematically leverage "charge" cancellations specific to this model andobtain stochastic estimates that allow us to cover the entire subcriticalregime.
Chandra A, Erhard D, Shen H, 2019, Local solution to the multi-layer KPZ equation, Journal of Statistical Physics, Vol: 175, Pages: 1080-1106, ISSN: 0022-4715
In this article we prove local well-posedness of the system of equations ∂thi=∑ij=1∂2xhj+(∂xhi)2+ξ on the circle where 1≤i≤N and ξ is a space-time white noise. We attempt to generalize the renormalization procedure which gives the Hopf-Cole solution for the single layer equation and our h1 (solution to the first layer) coincides with this solution. However, we observe that cancellation of logarithmic divergences that occurs at the first layer does not hold at higher layers and develop explicit combinatorial formulae for them.
Chandra A, Hairer M, 2018, An analytic BPHZ theorem for regularity structures, Publisher: arXiv
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.
Chandra A, Shen H, 2017, Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem, Electronic Journal of Probability, Vol: 22, ISSN: 1083-6489
Upon its inception the theory of regularity structures [7] 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 the 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, as was done in [12]. 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 [13], [16] - as an example we prove here a generalization of the Wong-Zakai Theorem found in [10].
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