Heat Kernel Research Articles

The VMO-Teichmüller space and the variant of Beurling–Ahlfors extension by heat kernel

We give a real-analytic section for the Teichmüller projection onto the VMO-Teichmüller space by using the variant of Beurling–Ahlfors extension by heat kernel introduced by Fefferman et al. (Ann Math 134:65–124, 1991). Based on this result, we prove that the VMO-Teichmüller space can be endowed with a real Banach manifold structure that is real-analytically equivalent to its complex Banach manifold structure. We also obtain that the VMO-Teichmüller space admits a real-analytic contraction mapping.

A Note on the Heat Kernel for the Rescaled Harmonic Oscillator from Two Step Nilpotent Lie Groups

A Note on the Heat Kernel for the Rescaled Harmonic Oscillator from Two Step Nilpotent Lie Groups

Manufacturing process classification based on heat kernel signature and convolutional neural networks

Manufacturing process classification based on heat kernel signature and convolutional neural networks

Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data

Abstract In this article, we study the large-time behavior of combination of the rarefaction waves with viscous contact wave for a one-dimensional compressible Navier-Stokes system whose transport coefficients depend on the temperature. It is shown that if the adiabatic exponent γ is suitably close to 1, the unique solution global in time to ideal polytropic gas exists and asymptotically tends toward the combination of a viscous contact wave with rarefaction waves under large initial perturbation. New and subtle analysis is developed to overcome difficulties due to the smallness of γ – 1 to derive heat kernel estimates. Moreover, our results extend the studies in a previous work [F. M. Huang, J. Li, and A. Matsumura, Arch. Ration. Mech. Anal. 197 (2010), no. 1, 89–116].

The $L^{\infty }$-positivity Preserving Property and Stochastic Completeness

We say that a Riemannian manifold satisfies the $L^p$-positivity preserving property if $(-\Delta + 1)u\ge 0$ in a distributional sense implies $u \ge 0$ for all $ u \in L^p$.While geodesic completeness of the manifold at hand ensures the $L^p$-positivity preserving property for all $p \in (1, +\infty)$, when $p = + \infty$ some assumptions are needed. In this paper we show that the $L^\infty$-positivity preserving property is in fact equivalent to stochastic completeness, i.e., the fact that the minimal heat kernel of the manifold preserves probability. The result is achieved via some monotone approximation results for distributional solutions of $-\Delta + 1 \ge 0$, which are of independent interest.

Heat kernel estimates for symmetric jump processes with anisotropic jumping kernels

We show two-sided bounds of heat kernel for symmetric pure jump Markov process in R d \mathbb {R}^d with jumping kernel J ( x , y ) J(x,y) that is comparable to \frac {\bbone _{\mathcal {V}}(x-y)}{|x-y|^{d+\alpha }}, where V \mathcal {V} is a union of symmetric cones and 0 > α > 2 0>\alpha >2 .

Heat kernels for reflected diffusions with jumps on inner uniform domains

In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain D D in a length metric space. The length metric is the intrinsic metric of a strongly local Dirichlet form. When D D is an inner uniform domain in the Euclidean space, a prototype for a special case of the processes under consideration is symmetric reflected diffusions with jumps on D D , whose infinitesimal generators are non-local (pseudo-differential) operators L \mathcal {L} on D D of the form \[ L u ( x ) = 1 2 ∑ i , j = 1 d ∂ ∂ x i ( a i j ( x ) ∂ u ( x ) ∂ x j ) + lim ε ↓ 0 ∫ { y ∈ D : ρ D ( y , x ) > ε } ( u ( y ) − u ( x ) ) J ( x , y ) d y \mathcal {L} u(x)\! =\!\frac 12 \!\sum _{i, j=1}^d\! \frac {\partial }{\partial x_i}\! \left (\!\!a_{ij}(x) \frac {\partial u(x)}{\partial x_j}\!\right ) \!+ \lim _{\varepsilon \downarrow 0}\! \int _{\{y\in D: \, \rho _D(y, x)>\varepsilon \}}\!\! (u(y)-u(x)) J(x, y)\, dy \] satisfying “Neumann boundary condition”. Here, ρ D ( x , y ) \rho _D(x,y) is the length metric on D D , A ( x ) = ( a i j ( x ) ) 1 ≤ i , j ≤ d A(x)=(a_{ij}(x))_{1\leq i,j\leq d} is a measurable d × d d\times d matrix-valued function on D D that is uniformly elliptic and bounded, and \[ J ( x , y ) ≔ 1 Φ ( ρ D ( x , y ) ) ∫ [ α 1 , α 2 ] c ( α , x , y ) ρ D ( x , y ) d + α ν ( d α ) , J(x,y)≔\frac {1}{\Phi (\rho _D(x,y))} \int _{[\alpha _1, \alpha _2]} \frac {c(\alpha , x,y)} {\rho _D(x,y)^{d+\alpha }} \,\nu (d\alpha ), \] where ν \nu is a finite measure on [ α 1 , α 2 ] ⊂ ( 0 , 2 ) [\alpha _1, \alpha _2] \subset (0, 2) , Φ \Phi is an increasing function on [ 0 , ∞ ) [ 0, \infty ) with c 1 e c 2 r β ≤ Φ ( r ) ≤ c 3 e c 4 r β c_1e^{c_2r^{\beta }} \le \Phi (r) \le c_3 e^{c_4r^{\beta }} for some β ∈ [ 0 , ∞ ] \beta \in [0,\infty ] , and c ( α , x , y ) c(\alpha , x, y) is a jointly measurable function that is bounded between two positive constants and is symmetric in ( x , y ) (x, y) .

Axion anomalies

We study fermions derivatively coupled to axion-like or pseudoscalar fields, and show that the axial vector current of the fermions is not conserved in the limit where the fermion is massless. This apparent violation of the classical chiral symmetry is due to the background axion field. We compute the contributions to this anomalous Ward identity due to the pseudoscalar field alone, which arise in Minkowski space, as well as the effects due to an interaction with an external gravitational field. For the case of massless fermions, these interactions induce terms in the axion effective action that can be removed by the addition of local counterterms. We demonstrate that these counterterms are generated by the transformation of the path integral measure when transforming the theory from a form where the chiral symmetry is manifest to one where the symmetry is only apparent after using the classical equations of motion. We work perturbatively in Minkowski space and include the effects of interactions with a linearized gravitational field. Using the heat kernel method, we study the transformation properties of the path integral measure, and include the effects of non-linear gravity as well as interactions with gauge fields. Finally, we verify our relation by considering derivatively coupled fermions during pseudoscalar-driven inflation and computing the divergence of the axial current in de Sitter spacetime.

Zero-electron-mass limit of the compressible Navier–Stokes–Poisson equations with well/ill-prepared initial data

In this study, we consider the viscous compressible Navier–Stokes–Poisson equations, which consist of the balance laws for electron density and moment, and a Poisson equation for the electrostatic potential. The limit of vanishing electron mass of this system with both well/ill-prepared initial data on the whole space is rigorously justified within the framework of local smooth solution. We first make use of the symmetric hyperbolic–parabolic structure of the compressible Navier–Stokes–Poisson equation to obtain uniform estimate in the short time, by which we show uniform existence of local classical solution to the compressible Navier–Stokes–Poisson equation in $\mathbb {R}^d(d\geq 1)$. Further, with uniform estimate of time derivatives, we show the zero-electron-mass limit of the solutions for the compressible Navier–Stokes–Poisson equation with well-prepared initial data in $\mathbb {R}^d(d\geq 1)$ by using Aubin's lemma. A detailed spectral analysis on the linearized system is done so that we are able to prove the zero-electron-mass limit of the solutions with ill-prepared initial data in $\mathbb {R}^d(d\geq 3)$, where the convergence occurs away from the time $t=0$. Finally, note that the dissipation mechanism for the linearized compressible Navier–Stokes–Poisson system is different from that of the compressible Euler equations in Grenier (Commun. Partial Diff. Eqns.21 (1996), 363–394); Grenier (Commun. Pure Appl. Math.50 (1997), 821–865); Ukai (J. Math. Kyoto Univ.26 (1986), 323–331), or that of the compressible Euler–Poisson equations in Ali and Chen (Nonlinearity24 (2011), 2745–2761), since its eigenvalues are somehow similar to that of heat equation, and the fundamental solution contains a part behaving like the heat kernel, thus a big difficulty is the singularity of the heat kernel at $t=0$.

Singular solutions for fractional parabolic boundary value problems

The standard problem for the classical heat equation posed in a bounded domain Omega of {mathbb {R}}^n is the initial and boundary value problem. If the Laplace operator is replaced by a version of the fractional Laplacian, the initial and boundary value problem can still be solved on the condition that the non-zero boundary data must be singular, i.e., the solution u(t, x) blows up as x approaches partial Omega in a definite way. In this paper we construct a theory of existence and uniqueness of solutions of the parabolic problem with singular data taken in a very precise sense, and also admitting initial data and a forcing term. When the boundary data are zero we recover the standard fractional heat semigroup. A general class of integro-differential operators may replace the classical fractional Laplacian operators, thus enlarging the scope of the work. As further results on the spectral theory of the fractional heat semigroup, we show that a one-sided Weyl-type law holds in the general class, which was previously known for the restricted and spectral fractional Laplacians, but is new for the censored (or regional) fractional Laplacian. This yields bounds on the fractional heat kernel.

Bilateral Harnack Inequalities for Stochastic Differential Equation with Multiplicative Noise

By constructing a coupling with unbounded time-dependent drift, a lower bound estimate of dimension-free Harnack inequality with power is obtained for a large class of stochastic differential equation with multiplicative noise. The key is an application of the inverse Hölder inequality. Combining this with the well-known upper bound, bilateral dimension-free Harnack inequality with power is established. As a dual inequality, the bilateral shift-Harnack inequalities with power are also investigated for stochastic differential equation with additive noise. Applications to the study of heat kernel inequalities are provided to illustrate the obtained inequalities.

Homogeneous algebras via heat kernel estimates

We study homogeneous Besov and Triebel–Lizorkin spaces defined on doubling metric measure spaces in terms of a self-adjoint operator whose heat kernel satisfies Gaussian estimates together with its derivatives. When the measure space is a smooth manifold and such operator is a sum of squares of smooth vector fields, we prove that their intersection with L ∞ L^\infty is an algebra for pointwise multiplication. Our results apply to nilpotent Lie groups and Grushin settings.

Starches Properties from Soft, Medium‐Hard, and Hard Brazilian Wheat upon Annealing

AbstractWheat kernel hardness is influenced by starch and protein association into the wheat grain. Therefore, starch properties may differ and result in different properties after physical modification. Starches isolated from hard, medium‐hard, and soft Brazilian wheat are modified by annealing and evaluated by physicochemical, pasting and thermal properties, and crystallinity. Soft and hard wheat starches have the highest amylose content and are not influenced by annealing treatment. Relative crystallinity is higher in medium‐hard wheat starch due to the lowest amylose content. However, it is not influenced by annealing treatment. In contrast, soft and hard wheat starches increase the relative crystallinity after annealing. Moreover, annealing reduces the pasting properties profile of all starches by decreasing viscosity (peak and final), breakdown, and setback; consequently, gel texture is also reduced. A decrease in swelling power is also noticed after annealing treatment. Wheat kernel hardness has influenced into properties of both native and annealed starches.

Asymptotic behavior of solutions to a chemotaxis-logistic model with transitional end-states

We study Cauchy problem of a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate. Our Cauchy data connect two different end-states for the chemical signal while the cell density takes its typical carrying capacity at the far fields. We are interested in the time-asymptotic behavior of the solution. We show that in the borderline, the component representing the chemical signal converges to a permanent, diffusive background wave, which connects the two end-states monotonically. On the other hand, the cell component converges to the spatial derivative of a heat kernel. The asymptotic solution has explicit formulation and is common to all solutions sharing the same end-states. Optimal L2 and L∞ convergence rates are obtained. We first convert the model into a 2×2 hyperbolic-parabolic system via inverse Hopf-Cole transformation. Then we apply Chapman-Enskog expansion to identify the asymptotic solution. After extracting the asymptotic solution, we use a variety of analytic tools to study the remainder and obtain optimal rates. These include time-weighted energy method, spectral analysis, Green's function estimate and iterations. Our results apply to a general class of Cauchy data for the model and for its transformed system. In particular, our results apply to large data solutions.

Global classical solvability in a food metric chemotaxis system under zero boundary conditions at infinity

We consider the nutrient–chemotaxis model, derived by a food metric, on the real line. The model consists of the equation for the nutrient density of ordinary differential equation type and that for the microorganism density of parabolic type, in which the diffusion coefficient is singular at the zero nutrient density. We establish a global existence result of bounded classical solutions for a large class of initial data under zero boundary conditions at infinity. To this end, we propose a transformation related to the food metric that transforms the original system into a parabolic–hyperbolic system with linear diffusion. We obtain a proper global well-posedness result for the transformed system using standard estimates, including the heat kernel estimates. We convert this to the solution of the original equation via an inverse transformation.

On complex-time heat kernels of fractional Schrödinger operators via Phragmén–Lindelöf principle

We consider fractional Schrödinger operators with possibly singular potentials and derive certain spatially averaged estimates for its complex-time heat kernel. The main tool is a Phragmén–Lindelöf theorem for polynomially bounded functions on the right complex half-plane. The interpolation leads to possibly nonoptimal off-diagonal bounds.

On holography in general background and the boundary effective action from AdS to dS

We study quantum fields on an arbitrary, rigid background with boundary. We derive the action for a scalar in the holographic basis that separates the boundary and bulk degrees of freedom. A relation between Dirichlet and Neumann propagators valid for any background is obtained from this holographic action. As a simple application, we derive an exact formula for the flux of bulk modes emitted from the boundary in a warped background. We also derive a formula for the Casimir pressure on a (d − 1)-brane depending only on the boundary-to-bulk propagators, and apply it in AdS. Turning on couplings and using the holographic basis, we evaluate the one-loop boundary effective action in AdS by means of the heat kernel expansion. We extract anomalous dimensions of single and double trace CFT operators generated by loops of heavy scalars and nonabelian vectors, up to third order in the large squared mass expansion. From the boundary heat kernel coefficients we identify CFT operator mixing and corrections to OPE data, in addition to the radiative generation of local operators. We integrate out nonabelian vector fluctuations in AdS4,5,6 and obtain the associated holographic Yang-Mills β functions. Turning to the expanding patch of dS, following recent proposals, we provide a boundary effective action generating the perturbative cosmological correlators using analytical continuation from dS to EAdS. We obtain the “cosmological” heat kernel coefficients in the scalar case and work out the divergent part of the dS4 effective action which renormalizes the cosmological correlators. We find that bulk masses and wavefunction can logarithmically run as a result of the dS4 curvature, and that operators on the late time boundary are radiatively generated. More developments are needed to extract all one-loop information from the cosmological effective action.

Orbital functions and heat kernels of Kleinian groups

Orbital functions and heat kernels of Kleinian groups

Imaginary power operators on Hardy spaces

Let ( X , d , μ ) (X,d, \mu ) be an Ahlfors n n -regular metric measure space. Let L \mathcal {L} be a non-negative self-adjoint operator on L 2 ( X ) L^2(X) with heat kernel satisfying Gaussian estimate. Assume that the kernels of the spectral multiplier operators F ( L ) F(\mathcal {L}) satisfy an appropriate weighted L 2 L^2 estimate. By the spectral theory, we can define the imaginary power operator L i s , s ∈ R \mathcal {L}^{is}, s\in \mathbb R , which is bounded on L 2 ( X ) L^2(X) . The main aim of this paper is to prove that for any p ∈ ( 0 , ∞ ) p \in (0,\infty ) , ‖ L i s f ‖ H L p ( X ) ≤ C ( 1 + | s | ) n | 1 / p − 1 / 2 | ‖ f ‖ H L p ( X ) , s ∈ R , \begin{equation*} \big \|\mathcal {L}^{is} f\big \|_{H^p_{\mathcal {L}}(X)} \leq C (1+|s|)^{n|1/p-1/2|} \|f\|_{H^p_{\mathcal {L}}(X)}, \quad s \in \mathbb {R}, \end{equation*} where H L p ( X ) H^p_\mathcal {L}(X) is the Hardy space associated to L \mathcal {L} , and C C is a constant independent of s s . Our result applies to sub-Laplaicans on stratified Lie groups and Hermite operators on R n \mathbb {R}^n with n ≥ 2 n\ge 2 .

Hypergraph Modeling via Spectral Embedding Connection: Hypergraph Cut, Weighted Kernel k-Means, and Heat Kernel

We propose a theoretical framework of multi-way similarity to model real-valued data into hypergraphs for clustering via spectral embedding. For graph cut based spectral clustering, it is common to model real-valued data into graph by modeling pairwise similarities using kernel function. This is because the kernel function has a theoretical connection to the graph cut. For problems where using multi-way similarities are more suitable than pairwise ones, it is natural to model as a hypergraph, which is generalization of a graph. However, although the hypergraph cut is well-studied, there is not yet established a hypergraph cut based framework to model multi-way similarity. In this paper, we formulate multi-way similarities by exploiting the theoretical foundation of kernel function. We show a theoretical connection between our formulation and hypergraph cut in two ways, generalizing both weighted kernel k-means and the heat kernel, by which we justify our formulation. We also provide a fast algorithm for spectral clustering. Our algorithm empirically shows better performance than existing graph and other heuristic modeling methods.