arXiv subjects

Kaj Nyström

Publications and source records attributed to Kaj Nyström.

At least 19 recordsLinked to original sources

arXiv2026

Higher regularity of regular free boundaries for the Kolmogorov obstacle problem

We establish higher regularity of the regular free boundary for non-negative solutions of the Kolmogorov obstacle problem \[ \mathcal K u=χ_{\{u>0\}}, \qquad \mathcal K=Δ_x+x\cdot\nabla_y-\partial_t. \] Assume that the contact set satisfies a quantitative thickness condition at a free-boundary point. We prove that, in a neighborhood of that point, the free boundary is a non-characteristic $C_{\mathcal K}^{1,β}$ hypersurface for some $β\in(0,1)$. Equivalently, its inward non-degenerate normal is Hölder continuous with respect to the intrinsic Kolmogorov distance, and the deviation of the graph from its tangent hyperplane is of order $O(r^{1+β})$ at intrinsic scale $r$. The proof combines the half-space blow-up and differentiability theory at regular points with boundary Harnack estimates in asymptotically cylindrical intrinsic Lipschitz domains. A principal difficulty is that the diffusion derivatives determining the normal do not satisfy the homogeneous Kolmogorov equation. We introduce a $\mathcal K$-harmonic-replacement argument that converts the commutator sources into an $O(r)$ error under intrinsic blow-up and yields a contraction estimate for the oscillations of derivative quotients. This gives Hölder continuity of the non-degenerate normal. In addition, a boundary decay estimate for the transport derivative $Yu=(x\cdot\nabla_y-\partial_t)u$ provides the required improvement in the coupled time-transport direction. Combining these estimates with the available control in the degree-three variables yields a full intrinsic improvement of flatness.

math.AP↗

arXiv2026

Weighted Kolmogorov equations: transfer estimates, local boundedness, and Harnack inequalities

Let $m\geq1$, $k\geq0$, $n=m+k$, and write $x=(v,z)\in\mathbb R^m\times \mathbb R^k$, where $v$ is the active velocity variable and $z$ is passive with respect to $Y=v\cdot\nabla_y+\partial_t$. We consider \[ \operatorname{div}_x(A\nabla_xu)-Y(wu)=0, \] where $w=w(x)\in A_2(\mathbb R^n)$ and the measurable matrix $A$ has ellipticity and size controlled by $w$. For arbitrary such weights, including weights depending on the active variables, we prove a weighted hypoelliptic transfer estimate, a kinetic Sobolev inequality, local boundedness, the weak Harnack and Harnack inequalities, local Hölder continuity, and the strong minimum principle. The Harnack argument uses a full-variable weighted measure-data compactness theorem. An $\mathrm L^p$ velocity-averaging estimate gives compactness of smooth moments of $wu$, and weighted Poincaré inequalities in all diffusive variables reconstruct $u$. At a two-phase limit, testing with non-negative profiles produces transport directions given by weighted mean velocities. A $\mathrm{BV}$ argument applied to the limiting subsolution inequality yields the no-jump principle needed for expansion of positivity and a weighted kinetic ink-spots lemma. The theory includes the weight $|λ|^{1-2s}$ in the extension of Garofalo and Tralli for fractional powers of the Kolmogorov operator.

math.AP↗

arXiv2026

Solvability of the $\mathrm L^2$ Dirichlet problem for parabolic operators with spatially periodic coefficients

We prove global reverse Hölder estimates in $\mathrm L^2$ for the Poisson kernel of scalar divergence-form parabolic operators in the upper half-space. The coefficient matrix is real, symmetric, and periodic in every spatial variable $X=(λ,x)$, while no periodicity in time is assumed. If its boundary trace $A^0=A^0(x,t)$ has a quantitative uniform modulus of continuity in time and its convergence to $A^0$ in the transverse variable is controlled by a square-Dini modulus, then the Poisson kernel satisfies a uniform $\mathrm{RH}_2$ estimate at scales below the period. Full spatial periodicity is then used to propagate this estimate to every scale. The main new ingredient in the large-scale argument is a positive adjoint height coordinate $Φ^*=λ+O(1)$. Boundary comparison with $Φ^*$ yields the required large-scale decay of the Green function and hence the global $\mathrm{RH}_2$ estimate. Consequently, the $\mathrm L^2$ Dirichlet problem is uniquely solvable among weak solutions $u$ satisfying $N_\ast u\in\mathrm L^2$. We also establish, for fully spatially periodic coefficients, a nonsymmetric $A_\infty$ theorem under square-Dini convergence to a transverse-independent boundary trace, with no temporal regularity beyond measurability. We explain how the results transfer to time-independent Lipschitz graph domains under the stated coefficient and periodicity-compatibility hypotheses, and we derive scale-uniform boundary estimates for the periodic homogenization families considered below. The condition $D_t^{1/2}A^0\in\mathrm L^\infty_x(\mathrm{BMO}_t)$ provides a natural sufficient criterion for the temporal continuity required in the symmetric theorem.

math.AP↗

arXiv2026

Boundary Harnack inequalities for Kolmogorov equations in asymptotically cylindrical Lipschitz domains

We establish boundary Harnack inequalities for non-negative solutions of the Kolmogorov equation \[ \mathcal Ku=Δ_xu+x\cdot\nabla_yu-\partial_tu=0 \] in non-characteristic intrinsic Lipschitz graph domains whose defining functions may depend on the higher-order variable $y_m$. Earlier theory required independence of $y_m$. We replace this symmetry by quantitative asymptotic cylindricality, measured by a scale-invariant $C^{0,1/3}_{y_m}$ defect. The perturbative comparison theorem applies whenever this defect is sufficiently small, and hence at all sufficiently small scales for $C^{0,α}_{y_m}$ graphs with $α>1/3$, as well as under a vanishing critical cylindrical modulus. We obtain local comparability and intrinsic Hölder continuity of quotients of positive solutions, together with a Hölder estimate for their logarithmic quotient. The proof combines local boundary estimates, valid without $y_m$-independence, with a global rigidity argument for cylindrical blow-up limits. The central issue is uniform balance of forward and backward reference values. If this balance fails at collapsing scales, maximal-scale selection and a second rescaling produce a non-negative polynomial-growth Dirichlet solution in an unbounded cylindrical domain that vanishes at the backward reference point but not at the forward one. We exclude this configuration using boundary-energy and mean-value estimates, finite-dimensionality of spaces of polynomial-growth solutions, and translation invariance in $y_m$. The resulting finite-dimensional translation representation yields real-analytic dependence on $y_m$. Propagation of zeros, analytic continuation along invariant fibres, and a Tikhonov uniqueness argument then force the limiting solution to vanish identically. The exponent $1/3$ is critical because $y_m$ has homogeneous degree three.

math.AP↗

arXiv2026

A Commutator Framework for Selective Spectral Alignment in Deep Neural Networks

We develop a finite-width geometric framework describing how learned feature geometries are organized, transported, and selectively aligned in deep neural networks. Incompatibility among weight-generated covariance, gates, and backward sensitivities is quantified through three families of commutators: between gates and covariance, between sensitivities and covariance, and between average gradient outer products (AGOPs) and neural feature matrices (NFMs). An exact layerwise identity decomposes the sensitivity-covariance commutator into four sources: downstream transport, adjacent-layer imbalance, pointwise sensitivity fluctuations, and nonlinear gate-covariance interactions. The AGOP-NFM commutator is a singular-value-weighted transport of the internal commutator, explaining why observed feature-side alignment alone does not determine the internal geometry from which it emerges. Buffered localized energies resolve mixing between separated covariance subspaces. We establish spectral-gap, projector-evolution, and stabilization estimates, and formulate conditional Lyapunov principles that yield decay under explicit geometric error-bound or intrinsic-damping assumptions. These criteria do not follow from gradient flow alone and clarify why risk reduction need not imply commutator collapse. Analytic examples and numerical experiments exhibit factorization of spectral and activation geometry, transient growth, and cancellation among nonzero sources. In tested finite-time regimes, cancellation dominated by a negative transport-imbalance interaction persists across depths, widths, and two regression benchmarks. Spectral alignment therefore appears as a layer- and scale-dependent compatibility phenomenon governed by transport, interaction, cancellation, and possible damping, rather than a universal consequence of training.

stat.ML↗

arXiv2026

Fokker-Planck Analysis and Invariant Laws for a Continuous-Time Stochastic Model of Adam-Type Dynamics

We develop a continuous-time model for the long-term dynamics of adaptive stochastic optimization, focusing on bias-corrected Adam-type methods. Starting from a finite-sum setting, we identify a canonical scaling of learning rates, decay parameters, and gradient noise that yields a coupled, time-inhomogeneous stochastic differential equation for the parameters $x_t$, first-moment tracker $z_t$, and second-moment tracker $y_t$. Bias correction persists via explicit time-dependent coefficients, and the dynamics becomes asymptotically time-homogeneous. We analyze the associated Fokker-Planck equation and, under mild regularity and dissipativity assumptions on $f$, prove existence and uniqueness of invariant measures. Noise propagation is governed by $A(x)=\mathrm{Diag}(\nabla f(x))H_f(x)$. Hypoellipticity may fail on $\mathcal D_A\times\mathbb R^m\times(\mathbb R_+)^m$, where \[ \mathcal D_A=\{x\in\mathbb R^m:\exists j,\ e_j^\top A(x)=0\}\subset\{x:\det A(x)=0\}=\mathcal D_A^\dagger, \] and critical points of $f$ lie in $\mathcal D_A$. We show $\mathcal D_A^\dagger\neq\mathbb R^m$ and use this to prove exponential convergence of the Markov semigroup $μ_0P_t$ to a unique invariant measure, uniformly in $μ_0$. The proof uses a Harris-type argument, minorization on Lyapunov sublevel sets, control constructions, and hypoellipticity on $(\mathbb R^m\setminus\mathcal D_A)\times\mathbb R^m\times(\mathbb R_+)^m$. This provides a transparent continuous-time view of Adam-type dynamics.

math.AP↗

arXiv2025

Solvability of the $L^p$ Dirichlet problem for the heat equation implies parabolic uniform rectifiability

Let $Ω\subset \mathbb{R}^{n+1}$ be an open set in space-time with boundary $Σ= \partial Ω$. Under minimal and natural background assumptions - namely, that $Σ$ is time-symmetrically parabolic Ahlfors--David regular and that $Ω$ satisfies an interior corkscrew condition - we treat a one-phase parabolic free boundary problem which establishes the necessity of parabolic uniform rectifiability for $L^p(dσ)$ solvability of the Dirichlet problem for the heat equation. More precisely, we prove that if the caloric measure associated with $Ω$ satisfies a weak-$A_\infty$ condition with respect to the surface measure $σ= \mathcal{H}_{\mathrm{par}}^{n+1}\!\lfloor_Σ$, then $Σ$ is parabolically uniformly rectifiable, hence equivalently, that solvability of the Dirichlet problem for the heat (or adjoint heat) equation in $Ω$ with boundary data in $L^p(dσ)$, for some $p \in (1,\infty)$, implies parabolic uniform rectifiability. Our main theorem thus identifies parabolic uniform rectifiability as the correct geometric framework for boundary regularity, and $L^p$ solvability, in the parabolic setting.

math.AP↗

arXiv2025

The Kato square root problem for parabolic operators with an anti-symmetric part in BMO

We solve the Kato square root problem for parabolic operators whose coefficients can be written as the sum of a complex part, which is coercive, and a real anti-symmetric part, which is in BMO. In particular, we allow for certain unbounded coefficients.

math.AP↗

arXiv2024

Solvability of the $L^p$ Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph

We prove that if a parabolic Lipschitz (i.e., Lip(1,1/2)) graph domain has the property that its caloric measure is a parabolic $A_\infty$ weight with respect to surface measure (which in turn is equivalent to $L^p$ solvability of the Dirichlet problem for some finite $p$), then the function defining the graph has a half-order time derivative in the space of (parabolic) bounded mean oscillation. Equivalently, we prove that the $A_\infty$ property of caloric measure implies, in this case, that the boundary is parabolic uniformly rectifiable. Consequently, by combining our result with the work of Lewis and Murray we resolve a long standing open problem in the field by characterizing those parabolic Lipschitz graph domains for which one has $L^p$ solvability (for some $p <\infty$) of the Dirichlet problem for the heat equation. The key idea of our proof is to view the level sets of the Green function as extensions of the original boundary graph for which we can prove (local) square function estimates of Littlewood-Paley type.

math.AP↗

arXiv2024

On fundamental solutions and Gaussian bounds for degenerate parabolic equations with time-dependent coefficients

We consider second order degenerate parabolic equations with real, measurable, and time-dependent coefficients. We allow for degenerate ellipticity dictated by a spatial $A_2$-weight. We prove the existence of a fundamental solution and derive Gaussian bounds. Our construction is based on the original work of Kato \cite{Kato}.

math.AP↗

arXiv2024

Higher order interpolative geometries and gradient regularity in evolutionary obstacle problems

We prove new optimal $C^{1,α}$ regularity results for obstacle problems involving evolutionary $p$-Laplace type operators in the degenerate regime $p > 2$. Our main results include the optimal regularity improvement at free boundary points in intrinsic backward $p$-paraboloids, up to the critical exponent, $α\leq 2/(p-2)$, and the optimal regularity across the free boundaries in the full cylinders up to a universal threshold. Moreover, we provide an intrinsic criterion by which the optimal regularity improvement at free boundaries can be extended to the entire cylinders. An important feature of our analysis is that we do not impose any assumption on the time derivative of the obstacle. Our results are formulated in function spaces associated to what we refer to as higher order or $C^{1,α}$ intrinsic interpolative geometries.

math.AP↗

arXiv2023

A Galerkin type method for kinetic Fokker Planck equations based on Hermite expansions

In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the domain $(0, T) \times D \times \mathbb{R}^d$, where $D$ is either $\mathbb{T}^d$ or $\mathbb{R}^d$. Our approach is based on a Hermite expansion in the velocity variable only, with a hyperbolic system that appears as the truncation of the Brinkman hierarchy, as well as ideas from $\href{arXiv:1902.04037v2}{AAMN21}$ and additional energy-type estimates that we have developed. We also establish the regularity of the solution based on the regularity of the initial data and the source term.

math.AP↗

arXiv2023

Solving the Dirichlet problem for the Monge-Ampère equation using neural networks

The Monge-Ampère equation is a fully nonlinear partial differential equation (PDE) of fundamental importance in analysis, geometry and in the applied sciences. In this paper we solve the Dirichlet problem associated with the Monge-Ampère equation using neural networks and we show that an ansatz using deep input convex neural networks can be used to find the unique convex solution. As part of our analysis we study the effect of singularities, discontinuities and noise in the source function, we consider nontrivial domains, and we investigate how the method performs in higher dimensions. We investigate the convergence numerically and present error estimates based on a stability result. We also compare this method to an alternative approach in which standard feed-forward networks are used together with a loss function which penalizes lack of convexity.

stat.ML↗

arXiv2023

Corona Decompositions for Parabolic Uniformly Rectifiable Sets

We prove that parabolic uniformly rectifiable sets admit (bilateral) corona decompositions with respect to regular Lip(1,1/2) graphs. Together with our previous work, this allows us to conclude that if $Σ\subset\mathbb{R}^{n+1}$ is parabolic Ahlfors-David regular, then the following statements are equivalent. (1) $Σ$ is parabolic uniformly rectifiable. (2) $Σ$ admits a corona decomposition with respect to regular Lip(1,1/2) graphs. (3) $Σ$ admits a bilateral corona decomposition with respect to regular Lip(1,1/2) graphs. (4) $Σ$ is big pieces squared of regular Lip(1,1/2) graphs.

math.MG↗

arXiv2022

The Kato square root problem for weighted parabolic operators

We give a simplified and direct proof of the Kato square root estimate for parabolic operators with elliptic part in divergence form and coefficients possibly depending on space and time in a merely measurable way. The argument relies on the nowadays classical reduction to a quadratic estimate and a Carleson-type inequality. The precise organization of the estimates is different from earlier works. In particular, we succeed in separating space and time variables almost completely despite the non-autonomous character of the operator. Hence, we can allow for degenerate ellipticity dictated by a spatial $A_2$-weight, which has not been treated before in this context.

math.AP↗

arXiv2022

Tug-of-war with Kolmogorov

We introduce a new class of strongly degenerate nonlinear parabolic PDEs $$((p-2)Δ_{\infty,X}^N+Δ_X)u(X,Y,t)+(m+p)(X\cdot\nabla_Yu(X,Y,t)-\partial_tu(X,Y,t))=0,$$ $(X,Y,t)\in\mathbb R^m\times \mathbb R^m\times \mathbb R$, $p\in (1,\infty)$, combining the classical PDE of Kolmogorov and the normalized $p$-Laplace operator. We characterize solutions in terms of an asymptotic mean value property and the results are connected to the analysis of certain tug-of-war games with noise. The value functions for the games introduced approximate solutions to the stated PDE when the parameter that controls the size of the possible steps goes to zero. Existence and uniqueness of viscosity solutions to the Dirichlet problem is established. The asymptotic mean value property, the associated games and the geometry underlying the Dirichlet problem, all reflect the family of dilation and the Lie group underlying operators of Kolmogorov type and this makes our setting different from the context of standard parabolic dilations and Euclidean translations applicable in the context of the heat operator and the normalized parabolic infinity Laplace operator.

math.AP↗

arXiv2022

Square function estimates for the evolutionary p-Laplace equation

We prove novel (local) square function/Carleson measure estimates for non-negative solutions to the evolutionary $p$-Laplace equation in the complement of parabolic Ahlfors-David regular sets. In the case of the heat equation, the Laplace equation as well as the $p$-Laplace equation, the corresponding square function estimates have proven fundamental in symmetry and inverse/free boundary type problems, and in particular in the study of (parabolic) uniform rectifiability. Though the implications of the square function estimates are less clear for the evolutionary $p$-Laplace equation, mainly due its lack of homogeneity, we give some initial applications to parabolic uniform rectifiability, boundary behaviour and Fatou type theorems for $\nabla_Xu$.

math.AP↗

arXiv2022

Deep learning, stochastic gradient descent and diffusion maps

Stochastic gradient descent (SGD) is widely used in deep learning due to its computational efficiency, but a complete understanding of why SGD performs so well remains a major challenge. It has been observed empirically that most eigenvalues of the Hessian of the loss functions on the loss landscape of over-parametrized deep neural networks are close to zero, while only a small number of eigenvalues are large. Zero eigenvalues indicate zero diffusion along the corresponding directions. This indicates that the process of minima selection mainly happens in the relatively low-dimensional subspace corresponding to the top eigenvalues of the Hessian. Although the parameter space is very high-dimensional, these findings seems to indicate that the SGD dynamics may mainly live on a low-dimensional manifold. In this paper, we pursue a truly data driven approach to the problem of getting a potentially deeper understanding of the high-dimensional parameter surface, and in particular, of the landscape traced out by SGD by analyzing the data generated through SGD, or any other optimizer for that matter, in order to possibly discover (local) low-dimensional representations of the optimization landscape. As our vehicle for the exploration, we use diffusion maps introduced by R. Coifman and coauthors.

stat.ML↗