The paper constructs Markov processes for stochastic heat equations on infinite strings with manifold values.
problem Constructing Markov processes for stochastic heat equations on infinite strings with manifold values.
method Constructing conservative Markov processes corresponding to martingale solutions to stochastic heat equations on R+ or R with values in a Riemannian manifold. result The process exhibits exponential ergodicity if the Ricci curvature is strictly positive and non-ergodicity if the sectional curvature is negative.
New Brownian motion defined in Minkowski normed spaces.
problem Constructing Brownian motion in non-Euclidean spaces.
method Singular McKean--Vlasov stochastic differential equation.
result Pathwise uniqueness of solutions to the stochastic differential equation.
In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits Wiener (Brownian bridge) measure on the Riemannian path (loop) space as an invariant measure using a suitable Dirichlet form. Using the Andersson-Driver approximation, we he…
The main result of this note is the existence of martingale solutions to the stochastic heat equation (SHE) in a Riemannian manifold by using suitable Dirichlet forms on the corresponding path/loop space. Moreover, we present some characterizations of the lower bound of the Ricci curvature by functional inequalities of…
Quantum stochastic flow computes heat kernel traces for Ricci flat manifolds.
problem Computing heat kernel traces for Ricci flat manifolds.
method Quantum stochastic differential equation (qsde) on Fock space over L2 differential 1-forms, adapted flow construction. result Trace of the connection Laplacian heat kernel can be computed over any compact Ricci-flat Riemannian manifold.
New model generates images by reversing heat equation, revealing disentanglement.
problem Image generation without considering image structure.
method Stochastically reverses the heat equation to generate images, using variational approximation.
result Emergent disentanglement of overall colour and shape in images.
Neural networks solve SPDEs using Wiener chaos expansion.
problem Solving stochastic partial differential equations (SPDEs) numerically.
method Using neural networks in the truncated Wiener chaos expansion.
result Approximation rates for learning SPDE solutions with noise.
The paper characterizes stochastic incompleteness in Riemannian manifolds.
problem Stochastic incompleteness of Riemannian manifolds and its characterization.
method Characterization through solutions to nonlinear parabolic equations.
result Stochastic incompleteness is equivalent to the nonuniqueness of bounded solutions to certain nonlinear parabolic equations.
ANNs overcome curse of dimensionality for heat equation uniform errors.
problem Approximating high-dimensional heat equations with neural networks.
method Developed techniques to estimate uniform L∞-error. result ANNs' parameters grow polynomially with dimension and precision.
We give a proof of Gaussian upper bound for the heat kernel coupled with the Ricci ow. Previous proofs by Lei Ni [5] use Harnack inequality and doubling volume property, also the recent proof by Zhang and Cao [6] uses Sobolev type inequality that is conserved along Ricci ow. We will use a horizontal coupling of curve […
New heat equation method solves intertwining problems in CR geometry.
problem Intertwining problems in conformal CR geometry.
method Heat equation and extension problems approach.
result New intertwining formulas derived.
Derives gradient estimates for CR heat equation on pseudo-Hermitian manifolds.
problem Estimating solutions to CR heat equation on complex manifolds.
method Local and global Li-Yau type gradient estimates.
result Gradient estimates and Harnack inequality for positive solutions.
In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on an Euclidean space and a compact manifold. We study the "cut locus" case, namely, the case where energy-minimizing paths which join the two points under consideration form not a finite set, but a compact manifold. Under mild assum…
Novel framework discovers SPDEs from limited data.
problem Discovering SPDEs from limited data.
method Combines stochastic calculus, variational Bayes, and sparse learning.
result Accurately identifies SPDEs from limited data.
Bounds on Hessian of heat equation coupled with Ricci flow.
problem Estimating the Hessian of a solution to the conjugate heat equation coupled with Ricci flow.
method Obtained upper bounds for the Hessian.
result Local and global upper bounds for the Hessian of a positive solution.
Deep learning approximates SPDE solutions from noise trajectories.
problem Approximating solutions to stochastic partial differential equations (SPDEs).
method Uses neural networks to approximate SPDE solutions based on noise realizations.
result Accurately estimates SPDE solutions and functionals like mean and variance.
Sharp time analyticity proved for heat equation on Ricci solitons.
problem Analyticity of heat equation solutions on gradient shrinking Ricci solitons.
method Proved analyticity for solutions with quadratic exponential growth.
result Sharp growth condition for analyticity is established.
This paper classifies symmetries of biharmonic heat equations on surfaces of revolution.
problem Investigating symmetries of biharmonic heat equations on surfaces of revolution.
method Lie symmetry analysis to classify symmetries and derive invariant solutions.
result The biharmonic heat equation on a surface of revolution has the same Lie symmetries as the harmonic heat equation.
In this paper, we study the Poisson equation and heat equation in a model matrix geometry Mn. Our main results are about the Poisson equation and global behavior of the heat equation on Mn. We can show that if c0 is the initial positive definite matrix in Mn, then c(t) exists for all time and is positive …
In this paper, we study the gradient estimates of Li-Yau-Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem ut−Δu=aulogu, u>0 on the compact Riemannian manifold (M,g) of dimension n and with non-negative (Bakry-Emery)-Ricci curvature. Here…
Estimates heat equation on shrinking Ricci solitons with uniform bounds.
problem Analyzing heat equation on shrinking Ricci solitons.
method Proved L2 estimate with time-dependent Gaussian weight. result Uniform bounds for heat equation along Ricci flow.
Derives matrix Harnack inequalities for semilinear heat equations on manifolds.
problem Bounding solutions of semilinear heat equations on manifolds with geometric constraints.
method Applies Li-Yau estimates to derive Harnack inequalities for positive solutions.
result Derives matrix Harnack inequalities for positive solutions of semilinear heat equations.
Article proves Liouville theorem for heat equation in super Ricci flow.
problem Proving Liouville theorem for heat equation in super Ricci flow.
method Formulated under a growth condition concerning Perelman's reduced distance.
result Established Liouville theorem for heat equation in ancient super Ricci flow.
Ancient solutions to heat equations on graphs are shown to be analytic in time.
problem Analyzing the analyticity of ancient solutions to heat equations on graphs.
method Proving time analyticity under a sharp growth condition.
result Ancient solutions to heat equations on graphs are time analytic under certain conditions.
This paper proposes an unsupervised learning method to solve heat equations on chips.
problem Critical need for solving heat transfer equations on chips for 5G and AI.
method Hybrid framework of Auto Encoder and Image Gradient for unsupervised learning.
result Framework can solve heat transfer problems with a single training process and predict unseen cases.
Extends gradient estimates for heat equation under Finsler geometric flows.
problem Global gradient estimates for positive solutions to heat equation.
method General compact Finsler CD(−K,N) geometric flow. result Derives Harnack inequality for positive solutions.
Study on biharmonic heat equation on manifolds with curvature constraints.
problem Analyzing entire solutions of biharmonic heat equation on manifolds.
method Exponential decay estimates for biharmonic heat kernel under Ricci curvature and noncollapsing conditions. Proving uniqueness criteria for Cauchy problem.
result Conservation law for biharmonic heat kernel and uniform L-infinity estimate for entire solutions.
Proves uniqueness of heat equation solutions on Riemannian manifolds.
problem Proving uniqueness of solutions to the heat equation on Riemannian manifolds.
method Analyzes Lp solutions for 0<p<1 and improves L1 uniqueness result. result Improves curvature assumption for L1 uniqueness result. Ancient solutions of heat equation with exponential growth are analytic in time.
problem Analyticity of solutions to the heat equation in time.
method Proving analyticity for ancient solutions with exponential growth.
result Ancient solutions with exponential growth are analytic in time.
In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's …
Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.
problem Bounding ancient solutions to biharmonic heat equation.
method Using polynomial volume growth and dimensions of biharmonic functions.
result Ancient solutions are bounded by polynomial dimensions.
We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the ε-Ricci flow on a closed surface. These new Harnac…
In this short note, we study the gradient estimate of positive solutions to Poisson equation and the non-homogeneous heat equation in a compact Riemannian manifold (M^n,g). Our results extend the gradient estimate for positive harmonic functions and positive solutions to heat equations.
In this paper, we prove a differential Harnack inequality for positive solutions of time-dependent heat equations with potentials. We also prove a gradient estimate for the positive solution of the time-dependent heat equation.
We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we …
Survey on heat equation estimates on manifolds.
problem Estimating heat equations on manifolds.
method Recalling and discussing Li-Yau, Hamilton, Perelman's estimates and their applications.
result Sharp constants and improved curvature conditions for heat equations on manifolds.
The paper extends Li-Yau-Hamilton estimates to evolving Kähler metrics and nonlinear heat equations.
problem Deriving estimates for nonlinear heat equations on evolving Kähler metrics.
method Generalized matrix Li-Yau-Hamilton estimates to Kähler manifolds with evolving metrics and nonlinear heat equations.
result Extended Li-Yau-Hamilton estimates to evolving Kähler metrics and nonlinear heat equations.
In this note, we prove some new entropy formula for linear heat equation on static Riemannian manifold with nonnegative Ricci curvature. The results are analogies of Cao and Hamilton's entropies for Ricci flow coupled with heat-type equations.
Paper proves estimates for heat and conjugate heat equations under Ricci flow, leading to monotonicity of parabolic frequencies.
problem Establishing estimates for heat and conjugate heat equations under Ricci flow.
method Proving matrix Li-Yau-Hamilton estimates for positive solutions to the heat and conjugate heat equations coupled with Ricci flow.
result Monotonicity of parabolic frequencies established up to correction factors.
The paper establishes sub-gradient estimates and entropy formulas for quaternionic contact geometry heat equations.
problem Developing sub-gradient estimates and entropy formulas for quaternionic contact geometry.
method Establishing sub-gradient estimates and entropy formulas for the quaternionic contact heat equation.
result Two Perelman-type entropy formulas and sub-gradient estimates for the quaternionic contact heat equation.
A quaternionic contact (qc) heat equation and the corresponding qc energy functional are introduced. It is shown that the qc energy functional is monotone non-increasing along the qc heat equation on a compact qc manifold provided certain positivity conditions are satisfied.
In this short note we present local derivative estimates for heat equations on Riemannian manifolds following the line of W.-X. Shi. As an application we generalize a second derivative estimate of R. Hamilton for heat equations on compact manifolds to noncompact case.
The paper improves heat equation estimates under weaker Ricci curvature conditions.
problem Improving heat equation estimates under weaker Ricci curvature conditions.
method Establishing Li-Yau-type and Hamilton-type estimates for positive solutions of the heat equation under generalized Ricci flow.
result Deriving Harnack-type inequalities and monotonicity of parabolic frequency.
New Hessian estimates for heat equations on manifolds.
problem Estimating Hessian matrices for heat-type equations on Riemannian manifolds.
method Using Bismut-Stroock Hessian formula, with explicit coefficients and delay/growth rate functions.
result Novel backward weak Harnack inequality and precise pointwise Hessian estimates for eigenfunctions.
Sharp conditions found for solving heat equation on Riemannian manifolds.
problem Solving semilinear heat equation on Riemannian manifolds.
method Sharp conditions derived for local-in-time solvability.
result Sharp conditions on solvability given for complete and connected manifolds.
Article provides Bernstein gradient estimates for heat equations with potential terms.
problem Gradient estimates for heat equations with potential terms on weighted Riemannian manifolds.
method Derived Bernstein type gradient estimates for two systems of heat equations with linear, exponential, and combined potentials.
result Resolves part of the problem raised by Bhattacharyya et al. in \cite{SB-1}.
Derives heat equation estimates linked to Ricci flow on compact and noncompact manifolds.
problem Estimating heat equation coupled to Ricci flow on noncompact manifolds.
method Local derivative estimates for the heat equation coupled to the Ricci flow.
result Extends results on distance distortion and backward pseudolocality to noncompact manifolds.
The paper derives Harnack inequalities for positive heat equation solutions on Finsler manifolds.
problem Deriving inequalities for heat equation solutions on Finsler manifolds.
method Generalizing Li-Yau type gradient estimates to Finsler geometry and applying to heat equation solutions.
result General gradient estimate for positive solutions of the heat equation on Finsler manifolds under curvature assumptions.