Non-local Dirichlet Forms

• Non-local Dirichlet forms are symmetric quadratic forms on function spaces that model the energy of jump processes with long-range interactions.
• They employ a non-negative, symmetric jump kernel to establish parabolic Harnack inequalities and derive two-sided heat kernel estimates.
• This framework extends classical diffusion analysis to irregular or fractal spaces, providing robust tools for studying anomalous diffusion and regularity.

Non-local Dirichlet forms are a class of symmetric quadratic forms defined on function spaces over a metric measure space which characterize energy associated to jump processes. In contrast to the classical local Dirichlet forms (related to diffusions), the non-local variant captures “long‐range” interactions via a non-negative, symmetric jump kernel. These forms not only provide the analytic framework for symmetric jump processes but also serve as a bridge to various aspects of geometric–functional inequalities, heat kernel estimates, and regularity theory.

1. Definitions and Framework

A typical non-local Dirichlet form is defined on a metric measure space $(M,d,\mu)$. Given a symmetric jump measure $J(dx,dy)$ on $M\times M$, one considers for suitable functions $f,g$ the bilinear form

$$\mathcal{E}(f,g)=\int_{M\times M\setminus\mathrm{diag}} (f(x)-f(y))(g(x)-g(y))\, J(dx,dy).$$

A key example is when the jumping kernel takes the form

$$J(dx,dy)=\frac{c(x,y)}{d(x,y)^{d+\alpha}}\,\mu(dx)\mu(dy),$$

with $c(x,y)$ symmetric and uniformly bounded away from zero and infinity. In this setting the parameter $\alpha>0$ governs the intensity of jumps. The underlying metric measure space typically satisfies the volume doubling property (there exists $C_\mu$ such that for all $x\in M,r>0$,

$V(x,2r) \le C_\mu\, V(x,r),$

with $V(x,r)=\mu(B(x,r))$) and the reverse volume doubling property (there exist $c_\mu>0$ and $d_1>0$ so that

$$\frac{V(x,R)}{V(x,r)}\ge c_\mu \Big(\frac{R}{r}\Big)^{d_1} \quad \text{for } 0<r\le R).$$

A scale function $\phi\colon[0,\infty)\to[0,\infty)$ is assumed strictly increasing and continuous, and satisfies a two–sided power law

$$c_1\Big(\frac{R}{r}\Big)^{\beta_1} \le \frac{\phi(R)}{\phi(r)} \le c_2\Big(\frac{R}{r}\Big)^{\beta_2} \quad \text{for } 0<r\le R,$$

which plays a central role in formulating heat kernel bounds and functional inequalities.

2. Parabolic Harnack Inequalities

The parabolic Harnack inequality (PHI) is a fundamental tool in the paper of regularity for solutions of parabolic equations. For non-local Dirichlet forms, the PHI is formulated with respect to a scale function and parabolic cylinders. In particular, one defines for a base point $x_0\in M$, time $t_0\ge0$ and radius $R>0$ the cylinder

$$Q(t_0,x_0,C_4\phi(R),R)=(t_0,t_0+C_4\phi(R))\times B(x_0,R).$$

A non-negative function $u(t,x)$ that is caloric (i.e. solves the non-local heat equation) in this cylinder is said to satisfy the parabolic Harnack inequality $PHI(\phi)$ if there exist positive constants $C_1,\dots,C_6$ such that

$\sup_{Q_-}u\le C_6\, \inf_{Q_+}u,$

where

$$Q_-=(t_0+C_1\phi(R),t_0+C_2\phi(R))\times B(x_0,C_5R) \quad\text{and}\quad Q_+=(t_0+C_3\phi(R),t_0+C_4\phi(R))\times B(x_0,C_5R).$$

This inequality quantifies the propagation of positivity in space–time for caloric functions even when the underlying process makes large jumps.

3. Equivalent Characterizations and Stability

A central result in the theory is that several analytic and probabilistic conditions are equivalent to the validity of the PHI. Under the volume doubling (VD), reverse doubling (RVD), and the scale function assumptions, one may prove that the following are equivalent:

1. $PHI(\phi)$ holds.
2. A slightly stronger version $PHI^+(\phi)$ holds.
3. Two-sided heat kernel estimates $UHK(\phi)$, near-diagonal lower bounds $NDL(\phi)$ and a uniform jump spread condition $UJS$ hold.
4. $NDL(\phi)$ and $UJS$ hold.
5. Parabolic Hölder regularity $PHR(\phi)$ (for caloric functions), together with an exit time estimate $E_\phi$ and $UJS$, hold.
6. Elliptic Hölder regularity $EHR$ for harmonic functions, exit time estimate $E_\phi$, and $UJS$ hold.
7. A combination of the Poincaré inequality $PI(\phi)$, jump kernel comparison $J_{\phi,\leq}$, cutoff Sobolev inequality $CSJ(\phi)$, and $UJS$ hold.

In this context the jump kernel comparison is given by

$$\frac{c_1}{V(x,d(x,y))\,\phi(d(x,y))}\le J(x,y)\le \frac{c_2}{V(x,d(x,y))\,\phi(d(x,y))},$$

while the uniform jump spread (UJS) condition reads

$$J(x,y)\le \frac{c}{V(x,r)}\int_{B(x,r)}J(z,y)\, d\mu(z),\quad 0<r\le \frac{d(x,y)}2.$$

It is a remarkable feature of these results that the parabolic Harnack inequality is stable under perturbations of the jump kernel provided the kernel remains comparable in the above sense. This stability result is pivotal as it allows one to deduce two-sided heat kernel bounds and space–time regularity even in irregular or fractal settings.

4. Connections with Heat Kernel Estimates and Hölder Regularity

For local Dirichlet forms the classical theory shows equivalence between Harnack inequalities and two-sided heat kernel estimates. In the non-local setting the situation is more subtle. Although PHI does imply the existence of a heat kernel with two-sided estimates, one must add the lower bound on the jump kernel; in fact one obtains

$HK(\phi)\iff PHI(\phi)+J_{\phi,\geq}.$

Moreover, the parabolic Harnack inequality automatically implies Hölder continuity for both caloric and harmonic functions. Specifically, one obtains estimates of the form

$$|u(s,x)-u(t,y)|\le c\left(\frac{\phi^{-1}(|s-t|)+d(x,y)}{r}\right)^\theta\sup_{Q}|u|,$$

for appropriately chosen parabolic cylinders $Q$. Conversely, assuming Hölder regularity, exit time estimates, and certain spread conditions on the kernel, one may recover the parabolic Harnack inequality. This interplay highlights the central role of functional inequalities and heat kernel methods in understanding non-local operators.

5. Applications and Impact

The analytic framework for non-local Dirichlet forms applies to a number of important situations. For instance, stable-like processes and jump processes on fractals (such as the Sierpiński gasket or carpet) fall under this theory. Even when the jump kernel is supported on cones or exhibits anisotropies or truncation effects, the stability of the PHI allows one to control the behavior of solutions. The equivalence results have direct implications in:

• Heat kernel analysis – yielding two-sided bounds that are crucial for probabilistic estimates.
• Regularity theory – ensuring that caloric and harmonic functions enjoy Hölder continuity despite the presence of jumps.
• Spectral theory and further geometric–functional inequalities – the characterizations imply that classical tools from diffusion theory have robust analogues in the non-local case.

This unified picture not only extends many well‐known results from local to non-local settings but also provides quantitative criteria that can be verified in a wide variety of applied contexts (e.g. in anomalous diffusion, fractal analysis, and financial mathematics).

6. Conclusion

The paper of non-local Dirichlet forms interweaves analysis, geometry, and probability by providing a framework to model symmetric jump processes and their associated PDEs. The stability of the parabolic Harnack inequality and its equivalence to other analytic conditions (including heat kernel estimates, Poincaré and cutoff Sobolev inequalities, and Hölder continuity) form the cornerstone of the theory. These equivalences not only extend classical diffusion theory into the non-local regime but also offer robust methods for handling irregular spaces and anisotropic jump dynamics. The results are applicable to a broad range of processes—from stable-like jumps on Euclidean spaces to diffusions on fractals—and thereby have significant impact on both pure and applied analysis.