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Ilya Chevyrev

Publications and source records attributed to Ilya Chevyrev.

At least 19 recordsLinked to original sources
arXiv2026

Scaling limit of the 3D abelian Yang--Mills Langevin dynamics

We study the continuum scaling limit of the Langevin dynamics for three-dimensional U(1) lattice Yang--Mills theory. The model is defined on the discrete 3D torus with a general class of plaquette actions that are suitably normalized, including Wilson, Manton, and Villain actions. Under the weak-coupling scaling and in the DeTurck gauge, we prove that, locally in time and in probability, the rescaled logarithmic field converges to the solution of the one-form stochastic heat equation. In particular, the limiting dynamics are universal and do not depend on the higher-order details of the plaquette action.

math.PR↗
arXiv2026

Subcritical non-linear heat equations via spectral gap

We establish local well-posedness for a class of non-linear heat equations on the torus $\mathbb{T}^n$ whose highest order non-linearities have scaling-critical dimension $-1/k, \, k\in \mathbb{N}$, when the initial condition is a random field satisfying a spectral gap inequality involving a suitable Lebesgue norm. The corresponding subcriticality condition matches that of the Gaussian free field in dimension $d<2+2/k$. This extends previous subcritical well-posedness results beyond the Gaussian setting. Moreover, our proof is simpler and avoids diagrammatic arguments.

math.PR↗
arXiv2026

A PDE approach to the 2D Yang-Mills measure

We introduce the concept of rough additive functions, which extends rough paths theory to line integrals of distributional 1-forms. In the context of gauge theory, we use this notion to define controlled gauge transformations and holonomies via RDEs. One of our main results is a rough version of Uhlenbeck compactness for distributional connections based on rough additive functions. The main ingredient is a singular elliptic PDE to obtain a Coulomb gauge. We solve this PDE using regularity structures and a method inspired by the implicit function theorem. Surprisingly, the model needed to solve the equation is determined entirely from rough additive functions, which is a simpler and geometrically more natural object. We apply our results to the Yang-Mills measure on the unit square, showing that it has a gauge-fixed representation with optimal regularity. Although we focus on the unit square in this article, we expect our results to apply to more general surfaces.

math.AP↗
arXiv2026

Locality of rough path lifts

Every H\"older continuous path $X$ admits a geometric rough path lift $\boldsymbol{X}$ by the Lyons--Victoir extension theorem. A natural question that emerges when lifting more than one path segment at once is that of \emph{locality}, namely whether the lift $\boldsymbol{X}_{s,t}$ only depends on the increments $X_{s,u}$, $u \in [s,t]$. We investigate the locality of rough path lifts in deterministic and stochastic settings. On the deterministic side, we show that no local, homogeneous rough path lift can be defined on $\gamma$-H\"older paths for all $\gamma \leq 1/2$. More strongly, we show that no L\'evy area can be defined which is at the same time bounded, with no further regularity assumptions, and either local and homogeneous or time translation--invariant. We moreover show that the boundedness requirement is sharp: an unbounded, local, time translation--invariant, and bilinear L\'evy area can be defined on all continuous paths. On the stochastic side, we classify all local, square-integrable rough path lifts of $d$-dimensional fractional Brownian motion with Hurst parameter $H \in (0,1/2]$. For $H \leq 1/4$, we show that no such lifts exist when $d \geq 2$, while for $H>1/4$, we show that all such lifts are stochastic translations of the canonical rough path. We further refine the classification by requiring invariance in law under time translation, scaling, and coordinate permutation, and show that only the canonical lift satisfies these constraints except at \(H=1/3\), for which there is a one-parameter family of lifts. Finally, we present an argument showing that scale-invariance forces a local $\eta$-H\"older rough path lift of fractional Brownian motion to be square-integrable, allowing scale-invariance to replace square-integrability in our classification and non-existence results.

math.PR↗
arXiv2026

From path integral quantization to stochastic quantization: a pedestrian's journey

We give two novel proofs that the path integral and stochastic quantizations of generic scalar Euclidean quantum field theories are equivalent. Our proofs rely on Taylor interpolations indexed by forests, in the fashion of constructive field theory. The first proof works at the level of individual terms in the Feynman expansion, with the forests appearing as spanning forests in Feynman graphs. The second one works at the level of the path integral and avoids the full expansion of the Feynman perturbation series.

math-ph↗
arXiv2026

Orthogonal polynomials on path-space

We consider the orthogonalisation of the signature of a stochastic process as the analogue of orthogonal polynomials on path-space. Under an infinite radius of convergence assumption, we prove density of linear functions on the signature in $L^p$ functions on grouplike elements, making it possible to represent a square-integrable function on (rough) paths as an $L^2$-convergent series. By viewing the shuffle algebra as commutative polynomials on the free Lie algebra, we revisit much of the theory of classical orthogonal polynomials in several variables, such as the recurrence relation and Favard's theorem. Finally, we restrict our attention to the case of Brownian motion with and without drift, and prove that dimension-independent orthogonal signature exists with drift but not without. We end with numerical examples of how orthogonal signature polynomials of Brownian motion can be applied for the approximation of functions on paths sampled from the Wiener measure.

math.PR↗
arXiv2025

Large field problem in coercive singular PDEs

We derive a priori estimates for singular differential equations of the form \[ \mathcal{L} \phi = P(\phi,\nabla\phi) + f(\phi,\nabla\phi)\xi \] where $P$ is a polynomial, $f$ is a sufficiently well-behaved function, and $\xi$ is an irregular distribution such that the equation is subcritical. The differential operator $\mathcal L$ is either a derivative in time, in which case we interpret the equation using rough path theory, or a heat operator, in which case we interpret the equation using regularity structures. Our only assumption on $P$ is that solutions with $\xi=0$ exhibit coercivity. Our estimates are local in space and time, and independent of boundary conditions. One of our main results is an abstract estimate that allows one to pass from a local coercivity property to a global one using scaling, for a large class of equations. This allows us to reduce the problem of deriving a priori estimates to the case when $\xi$ is small.

math.AP↗
arXiv2025

Uniqueness of gauge covariant renormalisation of stochastic 3D Yang-Mills-Higgs

Local solutions to the 3D stochastic quantisation equations of Yang-Mills-Higgs were constructed in (arXiv:2201.03487), and it was shown that, in the limit of smooth mollifications, there exists a mass renormalisation of the Yang-Mills field such that the solution is gauge covariant. In this paper we prove uniqueness of the mass renormalisation that leads to gauge covariant solutions. This strengthens the main result of (arXiv:2201.03487), and is potentially important for the identification of the limit of other approximations, such as lattice dynamics. Our proof relies on systematic short-time expansions of singular stochastic PDEs and of regularised Wilson loops. We also strengthen the recently introduced state spaces to allow finer control on line integrals appearing in expansions of Wilson loops.

math.PR↗
arXiv2025

Rates of memory loss for null recurrent Markov chains

Orey (1962) proved that for an irreducible, aperiodic, and recurrent Markov chain with transition operator $P$, the sequence $P^n (\mu - \nu)$ converges to zero in total variation for any two probability measures $\mu$ and $\nu$. In other words, all such Markov chains exhibit memory loss. While the rates of memory loss have been extensively studied for positive recurrent chains, there is a surprising lack of results for null recurrent chains. In this work, we prove the first estimates of memory loss rates in the null recurrent case.

math.PR↗
arXiv2025

Non-Gaussianity of invariant measures to SPDEs in Da Prato-Debussche regime

We propose an elementary method to show non-Gaussianity of invariant measures of parabolic stochastic partial differential equations with polynomial non-linearities in the Da Prato--Debussche regime. The approach is essentially algebraic and involves using the generator equation of the SPDE at stationarity. Our results in particular cover the $\Phi^4_\delta$ measures in dimensions $\delta<\frac{14}{5}$, which includes cases where the invariant measure is singular with respect to the invariant measure of the linear solution.

math.PR↗
arXiv2024

A multiplicative surface signature through its Magnus expansion

In the last decade, the concept of path signature has achieved significant success in data science applications. It offers a powerful set of features that effectively capture and describe the characteristics of paths or sequential data. This is partly explained by the fact that the signature of a path can be computed in linear time, using a dynamic programming principle based on Chen's identity. The path signature can be viewed as a specific example of a product or time-/path-ordered integral. In other words, it represents a one-parameter object built on iterated integrals over a path. Defining a signature over surfaces requires considering iterated integrals over these surfaces, effectively introducing an additional parameter, resulting in a two-parameter signature. This extended signature is intrinsically connected to a non-commutative generalization of Stokes' theorem, which is fundamentally connected to the concept of crossed modules of groups. The latter provides a well-established framework in higher gauge theory, where crossed modules with feedback maps exhibiting non-trivial kernels, combined with multiparameter iterated integrals, play a pivotal role. Building on Kapranov's work, we explore the surface analog of the log-signature for paths by introducing a Magnus-type formula for the logarithm of the surface signature. This expression takes values in a free crossed module of Lie algebras, defined over a free Lie algebra. We furthermore prove a non-commutative sewing lemma applicable to the crossed module setting and give a definition of rough surface in the so-called Young-H\"older regularity regime along with a corresponding continuous extension theorem. This approach enables the analysis and computation of surface features that go beyond what can be expressed by computing line integrals along the boundary of a surface.

math.RA↗
arXiv2024

Villain action in lattice gauge theory

We prove that Villain interaction applied to lattice gauge theory can be obtained as the limit of both Wilson and Manton interactions on a larger graph which we call the {\em carpet graph.} This is the lattice gauge theory analog of a well-known property for spin $O(N)$ models where Villain type interactions are the limit of $\mathbb{S}^{N-1}$ spin systems defined on a {\em cable graph}. Perhaps surprisingly in the setting of lattice gauge theory, our proof also applies to non-Abelian lattice theory such as $SU(3)$-lattice gauge theory and its limiting Villain interaction. In the particular case of an Abelian lattice gauge theory, this allows us to extend the validity of Ginibre inequality to the case of the Villain interaction.

math.PR↗
arXiv2024

Rough path theory

The theory of rough paths arose from a desire to establish continuity properties of ordinary differential equations involving terms of low regularity. While essentially an analytic theory, its main motivation and applications are in stochastic analysis, where it has given a new perspective on It\^o calculus and a meaning to stochastic differential equations driven by irregular paths outside the setting of semi-martingales. In this survey, we present some of the main ideas that enter rough path theory. We discuss complementary notions of solutions for rough differential equations and the related notion of path signature, and give several applications and generalisations of the theory.

math.CA↗
arXiv2023

Superdiffusive limits beyond the Marcus regime for deterministic fast-slow systems

We consider deterministic fast-slow dynamical systems of the form \[ x_{k+1}^{(n)} = x_k^{(n)} + n^{-1} A(x_k^{(n)}) + n^{-1/\alpha} B(x_k^{(n)}) v(y_k), \quad y_{k+1} = Ty_k, \] where $\alpha\in(1,2)$ and $x_k^{(n)}\in{\mathbb R}^m$. Here, $T$ is a slowly mixing nonuniformly hyperbolic dynamical system and the process $W_n(t)=n^{-1/\alpha}\sum_{k=1}^{[nt]}v(y_k)$ converges weakly to a $d$-dimensional $\alpha$-stable L\'evy process $L_\alpha$. We are interested in convergence of the $m$-dimensional process $X_n(t)=x_{[nt]}^{(n)}$ to the solution of a stochastic differential equation (SDE) \[ dX = A(X)\,dt + B(X)\, dL_\alpha. \] In the simplest cases considered in previous work, the limiting SDE has the Marcus interpretation. In particular, the SDE is Marcus if the noise coefficient $B$ is exact or if the excursions for $W_n$ converge to straight lines as $n\to\infty$. Outside these simplest situations, it turns out that typically the Marcus interpretation fails. We develop a general theory that does not rely on exactness or linearity of excursions. To achieve this, it is necessary to consider suitable spaces of ``decorated'' c\`adl\`ag paths and to interpret the limiting decorated SDE. In this way, we are able to cover more complicated examples such as billiards with flat cusps where the limiting SDE is typically non-Marcus for $m\ge2$.

math.DS↗
arXiv2023

Norm inflation for a non-linear heat equation with Gaussian initial conditions

We consider a non-linear heat equation $\partial_t u = Δu + B(u,Du)+P(u)$ posed on the $d$-dimensional torus, where $P$ is a polynomial of degree at most $3$ and $B$ is a bilinear map that is not a total derivative. We show that, if the initial condition $u_0$ is taken from a sequence of smooth Gaussian fields with a specified covariance, then $u$ exhibits norm inflation with high probability. A consequence of this result is that there exists no Banach space of distributions which carries the Gaussian free field on the 3D torus and to which the DeTurck-Yang-Mills heat flow extends continuously, which complements recent well-posedness results in arXiv:2111.10652 and arXiv:2201.03487. Another consequence is that the (deterministic) non-linear heat equation exhibits norm inflation, and is thus locally ill-posed, at every point in the Besov space $B^{-1/2}_{\infty,\infty}$; the space $B^{-1/2}_{\infty,\infty}$ is an endpoint since the equation is locally well-posed for $B^η_{\infty,\infty}$ for every $η>-\frac12$.

math.AP↗
arXiv2023

Wilson-Itô diffusions

We introduce Wilson-Itô diffusions, a class of random fields on $\mathbb{R}^d$ that change continuously along a scale parameter via a Markovian dynamics with local coefficients. Described via forward-backward stochastic differential equations, their observables naturally form a pre-factorization algebra à la Costello-Gwilliam. We argue that this is a new non-perturbative quantization method applicable also to gauge theories and independent of a path-integral formulation. Whenever a path-integral is available, this approach reproduces the setting of Wilson-Polchinski flow equations.

math.PR↗
arXiv2023

Deterministic homogenization under optimal moment assumptions for fast-slow systems. Part 2

We consider deterministic homogenization for discrete-time fast-slow systems of the form $$ X_{k+1} = X_k + n^{-1}a_n(X_k,Y_k) + n^{-1/2}b_n(X_k,Y_k)\;, \quad Y_{k+1} = T_nY_k\;$$ and give conditions under which the dynamics of the slow equations converge weakly to an Itô diffusion $X$ as $n\to\infty$. The drift and diffusion coefficients of the limiting stochastic differential equation satisfied by $X$ are given explicitly. This extends the results of [Kelly-Melbourne, J. Funct. Anal. 272 (2017) 4063--4102] from the continuous-time case to the discrete-time case. Moreover, our methods (càdlàg $p$-variation rough paths) work under optimal moment assumptions. Combined with parallel developments on martingale approximations for families of nonuniformly expanding maps in Part 1 by Korepanov, Kosloff & Melbourne, we obtain optimal homogenization results when $T_n$ is such a family of maps.

math.PR↗