Space-Time Duality in Relativistic Diffusion via Subordination

Technical Summary: Space-Time Duality in Relativistic Diffusion via Subordination

Problem Statement
The paper addresses the relationship between two fundamental frameworks for relativistic modifications of normal diffusion: the Cattaneo-Vernotte model, which leads to the telegraph equation (a damped wave equation with finite propagation speed), and the relativistic Schrödinger operator, which leads to spatially nonlocal diffusion equations. While it is established that spatially nonlocal diffusion arises from subordinating normal diffusion, the intrinsic connection between the telegraph equation and the spatially nonlocal diffusion equation remains less explored. The paper seeks to establish a dual relationship between these two relativistic models, utilizing normal diffusion as a bridge, and to generalize this duality to broader classes of operator families.

Methodology
The authors employ a combination of probability theory (specifically the theory of subordinators and inverse subordinators) and functional analysis (abstract evolutionary integral equations and Bernstein function calculus).

1. Subordination and Inverse Subordination: The core methodology relies on the duality between a subordinator $D(t)$ and its inverse $E(t) = \inf\{z > 0 : D(z) > t\}$. The paper utilizes the fact that if a process $Y(t)$ is obtained by subordinating a process $B(t)$ via $D(t)$, then $B(t)$ can be recovered by subordinating $Y(t)$ via the inverse process $E(t)$.
2. Laplace Transforms and Bernstein Functions: The authors use Laplace transforms to analyze the generators of the semigroups associated with the wave, telegraph, and diffusion equations. They identify specific Bernstein functions (e.g., $\mu(\lambda) = -a + \sqrt{a^2 + \lambda}$) that serve as Laplace exponents for the relevant subordinators.
3. Abstract Operator Theory: The results are first established in an abstract setting involving a $C_0$-semigroup $T(t)$ generated by an operator $A$. The paper defines "generalized dual relativistic diffusion families" where one family is obtained by space-subordination (using the subordinator) and the other by time-subordination (using the inverse subordinator).
4. Analytic Continuation: The paper reviews how the telegraph equation and the spatially nonlocal equation can be derived from the free relativistic Schrödinger equation via different analytic continuations (e.g., $t \to it$ vs. $m \to im$ or $\hbar \to -i\hbar$).

Key Contributions and Results

1. Reciprocal Subordination Mechanism: The paper reveals a novel reciprocal mechanism: while spatially nonlocal diffusion is obtained by subordinating normal diffusion, the normal diffusion process can be recovered by subordinating the telegraph process via the corresponding inverse subordinator.

   • Let $u(x,t)$ be the solution to the telegraph equation (TE).
   • Let $v(x,t)$ be the solution to the normal diffusion equation (DE).
   • Let $f(x,t)$ be the solution to the spatially nonlocal equation (NLE).
   • The paper establishes the chain: $(TE) \xrightarrow{E_\mu} (DE) \xrightarrow{D_\mu} (NLE)$, where $E_\mu$ is the inverse subordinator and $D_\mu$ is the subordinator associated with the Bernstein function $\mu(\lambda) = -a + \sqrt{a^2 + \lambda}$.

2. Duality Between TE and NLE: By leveraging the duality between the subordinator $D_\mu(t)$ and the inverse subordinator $E_\mu(t)$, the authors establish a distinct duality between the telegraph equation and the spatially nonlocal diffusion equation. Normal diffusion acts as the "anchor" or bridge between these two relativistic models.

   • Specifically, $v(x,t) = \int_0^\infty p_\mu(s,t) u(x,s) ds$ (recovering diffusion from telegraph via inverse subordinator density $p_\mu$).
   • And $f(x,t) = \int_0^\infty q_\mu(s,t) v(x,s) ds$ (obtaining nonlocal diffusion from diffusion via subordinator density $q_\mu$).

3. Harmonic Extension and PDE Connections: The paper connects these stochastic subordination relations to partial differential equations. Using harmonic extension techniques, it shows that the spatially nonlocal operator $a - \sqrt{a^2 - c^2\Delta}$ can be realized as the Dirichlet-to-Neumann map of a specific hyperbolic equation in an extended domain, further linking the telegraph and nonlocal equations.

4. Asymptotic Analysis: The paper analyzes the asymptotic behaviors of both models under various limits ($c \to \infty$, $t \to \infty$, $m \to 0$, $t \to 0$). It demonstrates that both the telegraph equation and the spatially nonlocal equation converge to the normal diffusion equation in the limits of infinite speed of light ($c \to \infty$) or long time ($t \to \infty$), validating their role as relativistic corrections to normal diffusion.

5. Generalization to Abstract Dual Families: The specific duality is generalized to a broader class of space-time dual operator families anchored on $C_0$-semigroups. The authors define generalized dual relativistic diffusion families $R_\phi(t)$ and $S_\phi(t)$ for a general Laplace exponent $\phi$.

   • $S_\phi(t)$ (space-dual) corresponds to the subordinated semigroup (e.g., fractional or relativistic diffusion).
   • $R_\phi(t)$ (time-dual) corresponds to the process recovered via the inverse subordinator, satisfying a nonlocal-in-time evolution equation $\Phi_t u(t) = Au(t)$.
   • The paper provides a step-by-step procedure to construct the integro-differential operator $\Phi_t$ for general $\phi$.

6. Specific Applications:

   • Relativistic Stable Processes: The theory is applied to relativistic stable operators of the form $a - (a^2/\beta - \Delta)^{\beta/2}$. For specific values of $\beta$ (e.g., $\beta=1, 2/3$), the time-dual equation recovers the telegraph equation or higher-order hyperbolic equations.
   • Higher-Order Cauchy Problems: The paper investigates whether general higher-order Cauchy problems possess the structure of generalized dual relativistic diffusion. It provides conditions on the coefficients of the characteristic polynomial to ensure the inverse function is a Bernstein function, which is necessary for the probabilistic interpretation.

Significance and Claims
The paper claims to establish a "distinct duality" between the telegraph equation and spatially nonlocal diffusion equations, a relationship that was previously not explicitly formulated in terms of subordination and inverse subordination. The significance lies in:

• Unification: It unifies two seemingly different relativistic diffusion models (one with finite propagation speed, one with infinite speed but nonlocal spatial structure) under a single probabilistic framework centered on normal diffusion.
• Reciprocity: It highlights a reciprocal mechanism where normal diffusion is not just a limit but an intermediate state recoverable from the telegraph process via inverse subordination.
• Generalization: It extends these specific physical models to a general abstract theory of dual operator families, providing a systematic method to generate time-dual equations for various semigroups.
• Stochastic Interpretation: It offers a clear stochastic interpretation for the connection between wave-like, diffusive, and nonlocal diffusive processes, linking them through the properties of subordinators and their inverses.

The authors note that while the duality is established, the well-posedness of the resulting higher-order time-dual equations depends on specific conditions (e.g., the Bernstein property of the inverse Laplace exponent), which they explore in the final examples.