Laplace Duality in Analysis & Probability

• Laplace duality is a framework where exponential kernels bridge two representations of mathematical objects across various fields such as analysis, probability, and geometry.
• It leverages integral transforms and semigroup identities to establish correspondences in Markov processes, log-concave functions, and operator theory, enabling effective classification and inversion.
• This unifying approach connects applications from stochastic duality and PDE methods to geometric transforms and differential privacy, highlighting the power of exponential-intertwining principles.

Searching arXiv for recent and foundational papers relevant to Laplace duality across analysis, probability, geometry, and operator theory. Laplace duality denotes a family of duality principles in which the Laplace kernel, Laplace transform, or Laplace-transform-like functionals provide the bridge between two representations of the same object. In the literature, the term covers several mathematically distinct but structurally related phenomena: semigroup duality of positive Markov processes with kernel $H(x,y)=e^{-xy}$; axiomatic characterizations of the Laplace transform on log-concave functions; dual representations of Laplace transforms of finite-dimensional distributions of Markov processes; duality methods for elliptic and nonlocal equations with rough data; and transform-based correspondences in integration, exponential families, differential privacy, and inverse problems. Across these settings, the recurring pattern is that exponential test functions convert dynamics, geometry, or constraints into an alternative representation in which generators, boundary conditions, or optimization variables are exchanged or reweighted (Foucart et al., 13 Jul 2025, Li, 2023, Kuznetsov et al., 2023).

1. Kernel-based and transform-based meanings of Laplace duality

A central probabilistic formulation uses the Laplace kernel

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$

and says that two positive Markov processes $X$ and $Y$ are in Laplace duality if

$$E_x\!\left[e^{-X_t y}\right]=E^y\!\left[e^{-xY_t}\right],\qquad t\ge 0,\ (x,y)\in[0,\infty]^2.$$

In this framework, the duality function is the one-dimensional Laplace kernel, and the semigroups of the two processes are intertwined by exponential test functions. The 2025 general framework for positive Markov processes shows that such duality is governed by complete monotonicity, boundary conventions at $0$ and $\infty$, and generator identities on exponentials (Foucart et al., 13 Jul 2025).

A second meaning concerns the Laplace transform as an operator on function spaces. For super-coercive log-concave functions,

$Lf(x)=\int_{\mathbb{R}^n} e^{x\cdot y}f(y)\,dy,$

the transform is characterized as a continuous $SL(n)$-contravariant valuation satisfying a pair of translation relations. In that setting, Laplace duality does not mean semigroup duality between stochastic processes, but rather the fact that the Laplace transform is singled out by valuation behavior, $SL(n)$-contravariance, continuity, and compatibility with ordinary translations and exponential tilts (Li, 2023).

A third meaning appears in operator-theoretic representations of finite-dimensional distributions. There, the joint Laplace transform of one Markov process can be rewritten as a Laplace-type transform of another process, with the roles of time indices and Laplace coefficients interchanged. The general theorem is formulated through spectral representations of semigroups linked by a unitary transform $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$0, and it recovers identities arising in Brownian excursion, Brownian meander, open ASEP, the open KPZ equation, Lévy processes, and birth-and-death chains (Kuznetsov et al., 2023).

These formulations are distinct, but they share a common mechanism: the exponential map $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$1 converts either nonlinear structure, operator action, or geometric constraints into a transform domain in which the relevant objects become comparable. This suggests that “Laplace duality” is best understood as a class of exponential-intertwining principles rather than a single theorem.

2. Axiomatic Laplace duality on log-concave functions

In convex and log-concave analysis, the central setting is

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$2

where $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$3 denotes proper super-coercive lower semicontinuous convex functions. A transform $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$4 on this space is a valuation if

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$5

whenever the four terms are defined, and is $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$6-contravariant if

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$7

Continuity is hypo-convergence continuity: if $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$8 hypo-converges to $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$9, then $X$0 pointwise (Li, 2023).

The characteristic translation relations are

$X$1

These relations express that ordinary translation of the input becomes multiplication by an exponential factor on the output, while multiplication of the input by an exponential factor becomes translation of the output. Under these hypotheses, Theorem 1.3 states that for $X$2, a transform

$X$3

is a continuous and $X$4-contravariant valuation satisfying those two identities if and only if there exist constants $X$5 such that

$X$6

for every $X$7 (Li, 2023).

The paper places this characterization beside the Legendre-transform characterization on convex functions. For convex $X$8,

$X$9

and if $Y$0, then the associated dual log-concave function is

$Y$1

The Legendre transform is characterized by the stronger two-sided translation conjugation

$Y$2

leading to $Y$3 in the super-coercive convex setting. By contrast, on the log-concave side the analogous structure leads to the Laplace transform, up to constants, rather than to a pure order duality (Li, 2023).

This comparison is conceptually important. The paper explicitly summarizes the difference as “Legendre = pointwise duality of convex analysis; Laplace = integral duality of log-concave analysis.” A common misconception is therefore to treat Laplace duality here as merely a disguised Legendre transform. The classification shows otherwise: the Laplace operator arises as an integral transform compatible with the same symmetry pattern, not as an order-reversing conjugacy.

3. Semigroup Laplace duality for positive Markov processes

For positive Markov processes, Laplace duality is a semigroup relation with exponential kernel. The general 2025 framework characterizes when a positive Markov process admits such a dual. The key criterion is complete monotonicity: a positive Markov process $Y$4 is called completely monotone if its semigroup preserves the cone

$Y$5

Equivalently, for every $Y$6, $Y$7. Depending on the conventions adopted for $Y$8 and $Y$9, existence of a Laplace dual is equivalent to complete monotonicity together with specific weak continuity and absorptivity conditions at the boundaries (Foucart et al., 13 Jul 2025).

The four boundary regimes are explicit. Under $$E_x\!\left[e^{-X_t y}\right]=E^y\!\left[e^{-xY_t}\right],\qquad t\ge 0,\ (x,y)\in[0,\infty]^2.$$0, $$E_x\!\left[e^{-X_t y}\right]=E^y\!\left[e^{-xY_t}\right],\qquad t\ge 0,\ (x,y)\in[0,\infty]^2.$$1 has a Laplace dual iff $$E_x\!\left[e^{-X_t y}\right]=E^y\!\left[e^{-xY_t}\right],\qquad t\ge 0,\ (x,y)\in[0,\infty]^2.$$2 is completely monotone, weakly $$E_x\!\left[e^{-X_t y}\right]=E^y\!\left[e^{-xY_t}\right],\qquad t\ge 0,\ (x,y)\in[0,\infty]^2.$$3-continuous at $$E_x\!\left[e^{-X_t y}\right]=E^y\!\left[e^{-xY_t}\right],\qquad t\ge 0,\ (x,y)\in[0,\infty]^2.$$4, and $$E_x\!\left[e^{-X_t y}\right]=E^y\!\left[e^{-xY_t}\right],\qquad t\ge 0,\ (x,y)\in[0,\infty]^2.$$5 is absorbing. Under $$E_x\!\left[e^{-X_t y}\right]=E^y\!\left[e^{-xY_t}\right],\qquad t\ge 0,\ (x,y)\in[0,\infty]^2.$$6, $$E_x\!\left[e^{-X_t y}\right]=E^y\!\left[e^{-xY_t}\right],\qquad t\ge 0,\ (x,y)\in[0,\infty]^2.$$7 has a Laplace dual iff $$E_x\!\left[e^{-X_t y}\right]=E^y\!\left[e^{-xY_t}\right],\qquad t\ge 0,\ (x,y)\in[0,\infty]^2.$$8 is completely monotone, weakly $$E_x\!\left[e^{-X_t y}\right]=E^y\!\left[e^{-xY_t}\right],\qquad t\ge 0,\ (x,y)\in[0,\infty]^2.$$9-continuous at $0$0, and $0$1 is absorbing. Under $0$2, both boundaries must be absorbing. Under $0$3, $0$4 must be weakly continuous at both boundaries. In each case, the dual boundary behavior is complementary: absorption and non-stickiness are exchanged (Foucart et al., 13 Jul 2025).

At the generator level, if exponential functions lie in the domains of both generators, Laplace duality becomes

$0$5

This infinitesimal identity is encoded by the Laplace symbol

$0$6

The Courrège-type formula derived in this setting shows that whenever the domain contains the exponentials, the generator on $0$7 has Lévy–Khintchine form

$0$8

with analogous boundary forms at $0$9 and $\infty$0. Duality of processes then corresponds to symmetry of Laplace symbols: $\infty$1 (Foucart et al., 13 Jul 2025).

Several major classes fit this template. For a continuous-state branching process with branching mechanism $\infty$2,

$\infty$3

and the classical Laplace representation

$\infty$4

produces a deterministic dual $\infty$5, where $\infty$6 solves

$\infty$7

The framework also includes continuous-state branching with immigration, continuous-state branching processes with collisions, logistic continuous-state branching processes, continuous-state branching in random environments, and more general decomposable symbols (Foucart et al., 13 Jul 2025).

A particularly sharp characterization appears for continuous-state branching processes with collisions. The generator of the CBC process $\infty$8 is

$\infty$9

while the dual diffusion $Lf(x)=\int_{\mathbb{R}^n} e^{x\cdot y}f(y)\,dy,$0 on $Lf(x)=\int_{\mathbb{R}^n} e^{x\cdot y}f(y)\,dy,$1 has generator

$Lf(x)=\int_{\mathbb{R}^n} e^{x\cdot y}f(y)\,dy,$2

The basic identity

$Lf(x)=\int_{\mathbb{R}^n} e^{x\cdot y}f(y)\,dy,$3

yields the semigroup duality

$Lf(x)=\int_{\mathbb{R}^n} e^{x\cdot y}f(y)\,dy,$4

The converse theorem shows that, under domain assumptions, CBCs are exactly the positive Feller processes without negative jumps and absorbing $Lf(x)=\int_{\mathbb{R}^n} e^{x\cdot y}f(y)\,dy,$5 whose generator is Laplace-dual to a one-dimensional diffusion generator (Foucart et al., 2022).

The 2026 CBDI extension replaces collisions by a Lévy–Khintchine drift-interaction mechanism. In the interior,

$Lf(x)=\int_{\mathbb{R}^n} e^{x\cdot y}f(y)\,dy,$6

and the duality is obtained by exchanging $Lf(x)=\int_{\mathbb{R}^n} e^{x\cdot y}f(y)\,dy,$7 and $Lf(x)=\int_{\mathbb{R}^n} e^{x\cdot y}f(y)\,dy,$8. On exponentials,

$Lf(x)=\int_{\mathbb{R}^n} e^{x\cdot y}f(y)\,dy,$9

For the minimal processes,

$SL(n)$0

This identity is then used for uniqueness in law, Fellerian extensions at $SL(n)$1 and $SL(n)$2, and boundary classification through scale functions and potential measures (Foucart et al., 7 May 2026).

4. Dual representations of Laplace transforms beyond one-time semigroup identities

Laplace duality also appears in a broader transform-theoretic form in which finite-dimensional Laplace transforms of one Markov process are represented through those of another process. The general framework assumes semigroups

$SL(n)$3

and a unitary operator

$SL(n)$4

such that both semigroups admit diagonalized forms with spectral multipliers $SL(n)$5 and $SL(n)$6. Under this assumption, one can interchange the roles of time increments and Laplace coefficients in finite-dimensional Laplace transforms (Kuznetsov et al., 2023).

The abstract theorem considers times

$SL(n)$7

and defines

$SL(n)$8

$SL(n)$9

If $SL(n)$0 is the transform of $SL(n)$1, then $SL(n)$2 is the transform of $SL(n)$3. The structural content is that the $SL(n)$4-functional is weighted by increments in the $SL(n)$5-grid, whereas the $SL(n)$6-functional is weighted by increments in the $SL(n)$7-grid. The “swap” of coefficients and time indices is the defining feature of this version of Laplace duality (Kuznetsov et al., 2023).

The same paper gives a continuous-time integral analogue: in the homogeneous case with excessive measures,

$SL(n)$8

$SL(n)$9

If $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$00, then $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$01 (Kuznetsov et al., 2023).

This framework unifies examples from several areas. For Brownian excursion, the paper recovers the identity

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$02

where $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$03 is the radial part of a 3D Cauchy process. A similar representation is obtained for Brownian meander. New examples include general Lévy processes via the Fourier transform, CIR diffusions via Laguerre transforms, and birth-and-death chains through Karlin–McGregor spectral theory. The Banach-space extension then accommodates the open KPZ example using the Kontorovich–Lebedev transform and the continuous dual Hahn process (Kuznetsov et al., 2023).

A related but more specialized transform story appears in integrable probability. Duality for q-TASEP and ASEP implies closed ODE systems for expectations of duality functionals, and these ODE systems are then resummed into “Laplace transform-like generating functions,” such as

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$04

Residue calculus converts these generating functions into Fredholm determinants. The paper explicitly describes this as a rigorous version of the replica trick in physics (Borodin et al., 2012).

5. PDE duality methods, inverse problems, and convex-dual Laplace-type operators

In PDE theory, “Laplace duality” frequently refers not to transform inversion but to a duality method for defining or characterizing solutions. For the fractional Laplacian with compactly supported bounded Radon measure data,

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$05

the paper develops a Stampacchia-style duality method on the whole space. For each test function $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$06, the dual auxiliary function is the Riesz potential

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$07

which satisfies

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$08

A function $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$09 is then a duality solution if

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$10

This identity yields existence, uniqueness, local $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$11 regularity for every $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$12, and local fractional Sobolev regularity

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$13

(Karlsen et al., 8 Aug 2025).

In inverse problems for the Laplace equation, the relevant duality is operator-theoretic. The range test (RT) and no-response test (NRT) for recovering an obstacle $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$14 from one set of Cauchy data are shown to be dual because their pre-indicators coincide: $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$15 where

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$16

The RT tests range solvability for a single-layer operator $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$17, whereas the NRT uses the dual operator $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$18. The equality of pre-indicators implies that reconstruction by RT implies reconstruction by NRT and vice versa (Lin et al., 2020).

Convex-duality formulations of Laplace-type equations produce yet another meaning. For the $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$19-Laplace equation, the operator

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$20

is realized as the derivative $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$21 of the energy

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$22

on

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$23

The theorem that $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$24 is continuous, bounded, strictly monotone, of type $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$25, and a homeomorphism from $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$26 onto $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$27 is the paper’s operator-theoretic duality statement (Liu et al., 2012).

A closely related convex-dual framework appears in the $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$28-Laplace problem. There the primal and dual functionals are

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$29

with strong duality

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$30

The duality gap equals a sum of Bregman divergences: $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$31 which the paper presents as a nonlinear analogue of the Prager–Synge identity (Gazca-Orozco, 3 Jun 2026).

These PDE formulations show that “Laplace duality” can mean a duality method for the Laplace or fractional Laplace operator, or more generally a primal-dual variational structure for nonlinear Laplace-type equations. A common misconception is to collapse these uses into transform theory alone. The papers do not support that reduction.

6. Integration, statistics, information-theoretic functionals, and geometric transforms

A 2025 paper formulates a Laplace duality for integration over sublevel sets. For

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$32

with $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$33 continuous, nonnegative, and $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$34 compact for all $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$35, the Laplace transform of the value function is

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$36

Under additional assumptions, for every $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$37 there exists a distinguished scalar $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$38 such that

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$39

The paper interprets $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$40 as the analogue of a Lagrange multiplier, with the Laplace transform playing the role that Legendre–Fenchel duality plays in optimization. In the positively homogeneous case,

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$41

and if $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$42 is a quadratic form, the dual representation reduces the original integral to Gaussian integration (Lasserre, 28 Feb 2025).

In the theory of real and multivariate exponential families, Laplace duality is defined through the log-Laplace transform

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$43

A dual measure $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$44 satisfies

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$45

When such a dual exists,

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$46

so the Hessian of the dual log-Laplace transform is the inverse variance function of the original family. The paper emphasizes that duals do not always exist, that the correct invariant object is the $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$47-exponential family obtained by considering all translations of a given family, and that examples include self-dual Gaussian, Gamma, dilogarithm, and Wishart families, while several hyperbolic and multivariate negative-binomial-type families have no dual (Letac, 2021).

In differential privacy, the privacy profile and the Rényi DP curve are connected by Laplace and inverse-Laplace transforms. For neighboring output distributions $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$48 and $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$49, the privacy loss distribution $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$50 governs both

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$51

The paper proves that

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$52

and conversely

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$53

Thus the $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$54-Rényi DP curve and the $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$55-DP curve function as Laplace and inverse-Laplace transform partners (Chourasia et al., 2024).

A more abstract convex-duality generalization appears in the non-exponential extension of Sanov’s theorem. There, the classical pair

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$56

is replaced by a convex dual pair

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$57

and the main Laplace-principle limit is

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$58

The paper explicitly describes this as a vast extension of Sanov’s theorem “in Laplace principle form,” based on alternatives to the classical relative-entropy/exponential dual pair (Lacker, 2016).

Geometric transform theory supplies further examples. For the perturbed Bessel equation, Laplace–Borel methods convert a monodromic relation for $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$59 into a dual monodromic relation for a function $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$60, and the central theorem states that the Laplace transform

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$61

is a bijection (Gurarii et al., 2012). In quantum cohomology, the anticanonical case $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$62 identifies twisted quantum $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$63-modules with second structure connections arising as Fourier-Laplace transforms of the quantum $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$64-module of $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$65, and the duality pairing is identified with Dubrovin’s second metric (Iritani et al., 2014). In classical mechanics, Bohlin–Arnold–Vassiliev duality explains the relation between the isotropic harmonic oscillator and the Kepler problem, and the transformed conserved quantity is the Laplace–Runge–Lenz vector (Grandati et al., 2008). In statistical duality, the Laplace distribution appears among the statistically self-dual families for which the same density form serves both as a sampling law and as a confidence density after swapping the roles of observation and parameter (Bityukov et al., 2013).

7. Unifying themes, scope, and recurrent misconceptions

Several unifying themes recur across these mathematically disparate uses.

First, exponential kernels are the basic carriers of the duality. In stochastic-process duality, $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$66 intertwines semigroups and generators (Foucart et al., 13 Jul 2025, Foucart et al., 2022, Foucart et al., 7 May 2026). In log-concave analysis, exponential tilts are exchanged with translations (Li, 2023). In integration, the exponential weight $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$67 replaces the sublevel constraint $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$68 (Lasserre, 28 Feb 2025). In differential privacy, Laplace transforms of privacy-loss functionals connect two entire privacy notions (Chourasia et al., 2024).

Second, generator-level identities often precede semigroup-level identities. The relation

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$69

for CBCs (Foucart et al., 2022), the equality

$H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$70

for CBDI processes (Foucart et al., 7 May 2026), and the general identity on exponentials encoded by the Laplace symbol (Foucart et al., 13 Jul 2025) all exemplify this pattern. In the finite-dimensional representation theory of Markov processes, the same structural idea appears as diagonalization through spectral multipliers and a transform $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$71 (Kuznetsov et al., 2023).

Third, boundary behavior is not ancillary. In the general theory of positive Markov processes, the conventions for $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$72 and $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$73 determine weak continuity and absorbing behavior at the boundaries (Foucart et al., 13 Jul 2025). In CBC and CBDI models, extinction, explosion, entrance, exit, and regularity are analyzed through the dual diffusion or dual branching-interaction process (Foucart et al., 2022, Foucart et al., 7 May 2026).

Fourth, several papers distinguish transform duality from convex or operator duality. In the $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$74-Laplace equation, duality refers to the homeomorphism $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$75 induced by the derivative of the energy (Liu et al., 2012). In the $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$76-Laplace problem, it refers to Fenchel duality and the duality gap (Gazca-Orozco, 3 Jun 2026). In inverse obstacle problems, it refers to the equivalence between range and adjoint no-response tests (Lin et al., 2020). These usages are rigorous but not reducible to semigroup Laplace kernels.

A common misconception is therefore to treat “Laplace duality” as a single standardized doctrine. The literature instead shows a cluster of related structures. The common denominator is that the Laplace transform, Laplace kernel, or an exponential penalty converts one description into another in a way that preserves enough analytic structure to make classification, inversion, or well-posedness possible.

A second misconception is to regard Laplace duality as inherently probabilistic. The characterization of the Laplace transform on log-concave functions (Li, 2023), the integration counterpart of Lagrangian duality (Lasserre, 28 Feb 2025), and Fourier-Laplace realizations in quantum $H(x,y)=e^{-xy},\qquad (x,y)\in [0,\infty]^2,$77-modules (Iritani et al., 2014) show that the idea is equally geometric and functional-analytic.

A plausible implication is that the modern research program on Laplace duality is increasingly organized around three questions. The first is structural: which generators, symbols, or valuations are forced by exponential intertwining? The second is analytic: when do boundary conditions, complete monotonicity, or convexity suffice to produce a dual object? The third is computational: when does the dual representation convert a difficult constrained, nonlocal, or interacting problem into a tractable one? The papers surveyed here answer these questions differently, but they do so within a recognizable common language of exponentials, transforms, and dual representations.