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Integral Varadhan formula for non-linear heat flow

Shin-ichi Ohta, Kohei Suzuki

TL;DR

This paper formulates and proves an integral Varadhan short-time formula for non-linear heat flow on measured Finsler manifolds, showing that the small-time decay of the non-linear heat-propagation probability is governed by a distance-like quantity even in the absence of metric reversibility. The authors introduce a distance functional $\bar{d}_m(A,B)$ and establish $\lim_{t\downarrow0} t\log {\sf P}_t(A,B) = -\tfrac12 \bar{d}_m(A,B)^2$ for finite-mass sets, with an open-set improvement to the standard distance when applicable. They extend the upper bound to nonsmooth spaces under infinitesimal strict convexity and Sobolev-to-Lipschitz conditions using modern metric-measure calculus and a linearised heat-flow framework, while the lower bound is derived via energy estimates and a Tauberian argument. The results bridge geometry, analysis, and probability for nonlinear diffusion beyond the linear Dirichlet-form setting, with implications for understanding heat propagation in asymmetric and non-smooth spaces.

Abstract

We prove the integral Varadhan short-time formula for non-linear heat flow on measured Finsler manifolds. To the best of the authors' knowledge, this is the first result establishing a Varadhan-type formula for non-linear semigroups. We do not assume the reversibility of the metric, and the distance function can be asymmetric. In this generality, we reveal that the probabilistic interpretation is well-suited for our formula; the probability that a particle starting from a set $A$ can be found in another set $B$ describes the distance from $A$ to $B$. One side of the estimates (the upper bound of the probability) is also established in the nonsmooth setting of infinitesimally strictly convex metric measure spaces satisfying the local Sobolev-to-Lipschitz property.

Integral Varadhan formula for non-linear heat flow

TL;DR

This paper formulates and proves an integral Varadhan short-time formula for non-linear heat flow on measured Finsler manifolds, showing that the small-time decay of the non-linear heat-propagation probability is governed by a distance-like quantity even in the absence of metric reversibility. The authors introduce a distance functional and establish for finite-mass sets, with an open-set improvement to the standard distance when applicable. They extend the upper bound to nonsmooth spaces under infinitesimal strict convexity and Sobolev-to-Lipschitz conditions using modern metric-measure calculus and a linearised heat-flow framework, while the lower bound is derived via energy estimates and a Tauberian argument. The results bridge geometry, analysis, and probability for nonlinear diffusion beyond the linear Dirichlet-form setting, with implications for understanding heat propagation in asymmetric and non-smooth spaces.

Abstract

We prove the integral Varadhan short-time formula for non-linear heat flow on measured Finsler manifolds. To the best of the authors' knowledge, this is the first result establishing a Varadhan-type formula for non-linear semigroups. We do not assume the reversibility of the metric, and the distance function can be asymmetric. In this generality, we reveal that the probabilistic interpretation is well-suited for our formula; the probability that a particle starting from a set can be found in another set describes the distance from to . One side of the estimates (the upper bound of the probability) is also established in the nonsmooth setting of infinitesimally strictly convex metric measure spaces satisfying the local Sobolev-to-Lipschitz property.
Paper Structure (27 sections, 15 theorems, 137 equations)

This paper contains 27 sections, 15 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Aim
  3. Background
  4. Main results
  5. The structure of the paper
  6. Preliminaries
  7. Finsler manifolds
  8. Uniform convexity and smoothness
  9. Gradient vectors
  10. Heat semigroup
  11. Linearised heat semigroups
  12. Truncation functions
  13. Maximal functions
  14. Upper estimate
  15. Finsler case
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $({\sf M},F)$ be a complete $C^{\infty}$-Finsler manifold equipped with a $C^{\infty}$-measure ${\sf m}$ on ${\sf M}$ with ${\sf m}({\sf M})<\infty$. Assume that the uniform convexity and smoothness constants are finite. Then, for any measurable sets $A,B \subset {\sf M}$ with $0< {\sf m}(A),{\s In particular, for any open sets $A,B \subset {\sf M}$, we have

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.1
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 18 more