Lamperti-type Laws Overview

• Lamperti-type laws are representation theorems that relate self-similar processes to additive drivers through nonlinear time changes and logarithmic coordinate transformations.
• They offer rigorous frameworks for analyzing clock asymptotics, large deviations, and generalized arcsine laws in occupation-time limits.
• Extensions of these laws cover branching processes, Banach-valued and norm-dependent settings, as well as Gaussian random fields, enriching both theory and applications.

Searching arXiv for relevant papers on Lamperti-type laws, representations, and generalized arcsine laws. I’ll collect a compact set of arXiv papers spanning the main themes: classical pssMp clocks, real-valued Lamperti–Kiu representations, branching/CBI extensions, arcsine/Dynkin–Lamperti laws, random fields, and infinite-dimensional generalizations. Lamperti-type laws are representation theorems and asymptotic limit laws that relate self-similar objects to additive drivers through nonlinear time changes, logarithmic coordinate changes, or generalized arcsine limits. In the classical positive setting, they identify a positive self-similar Markov process with the exponential of a Lévy process run on an inverse exponential-functional clock; in later developments, analogous laws involve Markov additive processes, pairs of Lévy drivers, multitype compound Poisson systems, and multidimensional stable-ratio limits for occupation and passage variables (Rouault et al., 2014, Chaumont et al., 2011, Caballero et al., 2010, Sera et al., 2017).

1. Classical Lamperti correspondence for positive self-similar processes

A positive self-similar Markov process of index $\alpha>0$ is a $[0,\infty)$-valued càdlàg strong Markov process $X=(X_t)_{t\ge0}$ such that, for every $a,b>0$,

$$\left(\{bX_{b^{-\alpha}t},\, t\ge0\}, \mathbb Q_a\right)\overset{d}= \left(\{X_t,\, t\ge0\}, \mathbb Q_{ba}\right).$$

Lamperti’s correspondence identifies the two basic time-change objects attached to $X$: its clock

$T^{(X)}(t)=\int_0^t \frac{ds}{X_s^\alpha},$

with inverse

$A^{(X)}(t)=\inf\{u\ge0:\,T^{(X)}(u)\ge t\},$

and the associated Lévy process

$\xi_t=\log X_{A^{(X)}(t)}-\log X_0.$

Conversely, from a Lévy process $\xi$ started from $[0,\infty)$0, one forms the exponential functional

$[0,\infty)$1

and its inverse

$[0,\infty)$2

and then recovers the self-similar process by

$[0,\infty)$3

The two clocks are tied by the identity

$[0,\infty)$4

so large-time statements for $[0,\infty)$5 and $[0,\infty)$6 are equivalent (Rouault et al., 2014).

In the positive theory, the Lévy process is the complete Lamperti datum. For positive self-similar Markov processes absorbed at $[0,\infty)$7, the lifetime is represented by an exponential functional,

$[0,\infty)$8

with $[0,\infty)$9 an independent killing time when present. This is the prototype for later Lamperti-type laws: a multiplicative or branching object is reconstructed from additive independent-increment data through an inverse integral clock (Chaumont et al., 2011).

A standard simplification is that one may reduce to $X=(X_t)_{t\ge0}$0 by replacing $X=(X_t)_{t\ge0}$1 with $X=(X_t)_{t\ge0}$2. In that normalization, the clock is $X=(X_t)_{t\ge0}$3, and the asymptotic analysis becomes a problem about additive functionals of self-similar paths rather than about the original nonlinear time change (Rouault et al., 2014).

2. Clock asymptotics, large deviations, and logarithmic growth laws

For Lamperti clocks, the central asymptotic regime in the positive-drift case is logarithmic. If the Lévy process has positive mean,

$X=(X_t)_{t\ge0}$4

then the clock and inverse exponential functional satisfy the law of large numbers

$X=(X_t)_{t\ge0}$5

equivalently,

$X=(X_t)_{t\ge0}$6

Thus the physical clock grows only at logarithmic speed, with constant equal to the reciprocal drift of the underlying Lévy process (Rouault et al., 2014).

The proof uses the generalized Ornstein–Uhlenbeck transform

$X=(X_t)_{t\ge0}$7

Under suitable assumptions, $X=(X_t)_{t\ge0}$8 is stationary and ergodic under the entrance law $X=(X_t)_{t\ge0}$9, with invariant law expressed via the perpetuity

$a,b>0$0

Applying the ergodic theorem to $a,b>0$1 yields the clock LLN. This is a characteristic Lamperti argument: logarithmic time converts self-similar growth into stationarity (Rouault et al., 2014).

The same paper establishes a large deviation principle for

$a,b>0$2

under $a,b>0$3. Writing

$a,b>0$4

one obtains local asymptotics on the interval $a,b>0$5, exponential tightness of the right tail, and, under additional negligibility assumptions, a full LDP at speed $a,b>0$6. The normalized log-Laplace transform satisfies

$a,b>0$7

and the rate function admits the reciprocal form

$a,b>0$8

The LLN point

$a,b>0$9

is the unique minimizer of $$\left(\{bX_{b^{-\alpha}t},\, t\ge0\}, \mathbb Q_a\right)\overset{d}= \left(\{X_t,\, t\ge0\}, \mathbb Q_{ba}\right).$$0 (Rouault et al., 2014).

The proof combines self-similarity, Esscher tilting, entrance laws at $$\left(\{bX_{b^{-\alpha}t},\, t\ge0\}, \mathbb Q_a\right)\overset{d}= \left(\{X_t,\, t\ge0\}, \mathbb Q_{ba}\right).$$1, and the Gärtner–Ellis theorem. In the Brownian/Bessel case,

$$\left(\{bX_{b^{-\alpha}t},\, t\ge0\}, \mathbb Q_a\right)\overset{d}= \left(\{X_t,\, t\ge0\}, \mathbb Q_{ba}\right).$$2

recovering the Bessel-clock law of Yor and Zani. The paper also remarks on a functional LDP and conjectures a CLT with variance $$\left(\{bX_{b^{-\alpha}t},\, t\ge0\}, \mathbb Q_a\right)\overset{d}= \left(\{X_t,\, t\ge0\}, \mathbb Q_{ba}\right).$$3 when $$\left(\{bX_{b^{-\alpha}t},\, t\ge0\}, \mathbb Q_a\right)\overset{d}= \left(\{X_t,\, t\ge0\}, \mathbb Q_{ba}\right).$$4, but does not prove that CLT in the general Lévy-driven case (Rouault et al., 2014).

3. Real-valued self-similar processes and Lamperti–Kiu structure

For real-valued self-similar Markov processes, a single Lévy process no longer suffices because sign changes must be encoded. If $$\left(\{bX_{b^{-\alpha}t},\, t\ge0\}, \mathbb Q_a\right)\overset{d}= \left(\{X_t,\, t\ge0\}, \mathbb Q_{ba}\right).$$5 is a real-valued self-similar Markov process killed at

$$\left(\{bX_{b^{-\alpha}t},\, t\ge0\}, \mathbb Q_a\right)\overset{d}= \left(\{X_t,\, t\ge0\}, \mathbb Q_{ba}\right).$$6

the Kiu time change

$$\left(\{bX_{b^{-\alpha}t},\, t\ge0\}, \mathbb Q_a\right)\overset{d}= \left(\{X_t,\, t\ge0\}, \mathbb Q_{ba}\right).$$7

produces a multiplicatively invariant process $$\left(\{bX_{b^{-\alpha}t},\, t\ge0\}, \mathbb Q_a\right)\overset{d}= \left(\{X_t,\, t\ge0\}, \mathbb Q_{ba}\right).$$8. Chaumont, Pantí, and Rivero identify the precise object underlying this representation: a Lamperti–Kiu process, equivalently a two-state Markov additive structure with alternating sign regimes, two Lévy processes $$\left(\{bX_{b^{-\alpha}t},\, t\ge0\}, \mathbb Q_a\right)\overset{d}= \left(\{X_t,\, t\ge0\}, \mathbb Q_{ba}\right).$$9, two exponential holding times with rates $X$0, and sign-switch jumps $X$1. The reconstructed process has the form

$X$2

and its lifetime is

$X$3

This completes the real-valued analogue of Lamperti’s positive-state-space theorem (Chaumont et al., 2011).

A recurring misconception is that Lamperti-type laws always reduce self-similar dynamics to one Lévy process. The real-valued theory shows that the correct hidden additive object may be a Markov additive process. In the Lamperti–Kiu formulation, the pair $X$4 plays the role that $X$5 plays in the positive case, and the exponential functional remains central because it still controls the hitting time of $X$6 (Chaumont et al., 2011).

Döring’s jump-SDE approach gives a different Lamperti-type classification for a substantial symmetric subclass of real-valued self-similar Markov processes. Under symmetry, the assumption that the process only decreases its absolute value by jumps,

$X$7

and the requirement that it leave zero continuously, the process is classified by a quintuple

$X$8

where $X$9 is a spectrally negative Lévy triplet killed at rate $T^{(X)}(t)=\int_0^t \frac{ds}{X_s^\alpha},$0, and $T^{(X)}(t)=\int_0^t \frac{ds}{X_s^\alpha},$1 is a finite measure on $T^{(X)}(t)=\int_0^t \frac{ds}{X_s^\alpha},$2. The absorbed process solves the jump-type SDE

$T^{(X)}(t)=\int_0^t \frac{ds}{X_s^\alpha},$3

and there is a bijection between such processes and quintuples satisfying

$T^{(X)}(t)=\int_0^t \frac{ds}{X_s^\alpha},$4

The modulus $T^{(X)}(t)=\int_0^t \frac{ds}{X_s^\alpha},$5 is then a positive self-similar Markov process, so the real-valued extension criterion is controlled by the positive theory for the modulus (Doering, 2012).

The Lamperti–Kiu setting also supports asymptotic laws for exponential functionals. For a two-state MAP $T^{(X)}(t)=\int_0^t \frac{ds}{X_s^\alpha},$6, the functional

$T^{(X)}(t)=\int_0^t \frac{ds}{X_s^\alpha},$7

is finite exactly when the additive component drifts sufficiently to $T^{(X)}(t)=\int_0^t \frac{ds}{X_s^\alpha},$8: either $T^{(X)}(t)=\int_0^t \frac{ds}{X_s^\alpha},$9, when the drift is defined, or $A^{(X)}(t)=\inf\{u\ge0:\,T^{(X)}(u)\ge t\},$0 is undefined and $A^{(X)}(t)=\inf\{u\ge0:\,T^{(X)}(u)\ge t\},$1. Under a Cramér condition

$A^{(X)}(t)=\inf\{u\ge0:\,T^{(X)}(u)\ge t\},$2

where $A^{(X)}(t)=\inf\{u\ge0:\,T^{(X)}(u)\ge t\},$3 is the Perron–Frobenius eigenvalue of the matrix exponent $A^{(X)}(t)=\inf\{u\ge0:\,T^{(X)}(u)\ge t\},$4, one has

$A^{(X)}(t)=\inf\{u\ge0:\,T^{(X)}(u)\ge t\},$5

and in the strong subexponential regime,

$A^{(X)}(t)=\inf\{u\ge0:\,T^{(X)}(u)\ge t\},$6

These are Lamperti-type tail laws for real self-similar hitting times, with scalar Lévy exponents replaced by spectral data of a matrix exponent (Alili et al., 2018).

4. Branching, immigration, and multitype Lamperti representations

Lamperti-type laws extend from self-similar Markov processes to branching systems by replacing multiplicative scaling with population clocks. For continuous-state branching processes with immigration, the correct analogue of the classical Lamperti transform is not a one-driver representation but a two-input equation

$A^{(X)}(t)=\inf\{u\ge0:\,T^{(X)}(u)\ge t\},$7

where $A^{(X)}(t)=\inf\{u\ge0:\,T^{(X)}(u)\ge t\},$8 is a spectrally positive Lévy process with Laplace exponent $A^{(X)}(t)=\inf\{u\ge0:\,T^{(X)}(u)\ge t\},$9 and $\xi_t=\log X_{A^{(X)}(t)}-\log X_0.$0 is an independent subordinator with Laplace exponent $\xi_t=\log X_{A^{(X)}(t)}-\log X_0.$1. The unique càdlàg solution is a $\xi_t=\log X_{A^{(X)}(t)}-\log X_0.$2 process. The cumulative population

$\xi_t=\log X_{A^{(X)}(t)}-\log X_0.$3

remains the endogenous branching clock, while immigration enters in physical time. This is the paper’s central Lamperti-type law for affine branching models (Caballero et al., 2010).

The pathwise formulation is developed through deterministic admissible pairs $\xi_t=\log X_{A^{(X)}(t)}-\log X_0.$4 solving

$\xi_t=\log X_{A^{(X)}(t)}-\log X_0.$5

or, in primitive form,

$\xi_t=\log X_{A^{(X)}(t)}-\log X_0.$6

This deterministic/random ODE viewpoint yields existence, uniqueness, stability in Skorokhod $\xi_t=\log X_{A^{(X)}(t)}-\log X_0.$7 and uniform $\xi_t=\log X_{A^{(X)}(t)}-\log X_0.$8 topologies, weak continuity of CBI laws with respect to the mechanisms, scaling limits from Galton–Watson processes with immigration, and a simulation scheme. A basic conceptual point is that immigration forces a two-process Lamperti law: branching is clocked by accumulated population, immigration is exogenous and is not time-changed (Caballero et al., 2010).

In the multitype discrete setting, breadth-first search coding of $\xi_t=\log X_{A^{(X)}(t)}-\log X_0.$9-type forests produces a multidimensional Lamperti-type transformation. For continuous-time multitype branching processes with discrete state space, any such process can be obtained from a $\xi$0-dimensional compound Poisson process time changed by an integral functional. The representation takes the form

$\xi$1

so each source type $\xi$2 contributes through its own additive driver and its own occupation clock $\xi$3. This is a matrix-valued extension of the one-type Lamperti representation (Chaumont, 2014).

Taken together, these results show that Lamperti-type laws in branching theory are not merely formal analogies. They give exact pathwise constructions, identify the necessary driving processes, and explain when one driver, two drivers, or a full matrix of drivers is structurally unavoidable (Caballero et al., 2010, Chaumont, 2014).

5. Generalized arcsine laws, Dynkin–Lamperti laws, and occupation-time limits

A second major branch of Lamperti-type laws concerns generalized arcsine distributions. For infinite ergodic transformations with a multiray decomposition

$\xi$4

the normalized occupation-time vector

$\xi$5

converges, in the sense of strong distributional convergence, to the multidimensional generalized arcsine law

$\xi$6

where the $\xi$7 are independent one-sided $\xi$8-stable variables with Laplace transforms

$\xi$9

The parameters are determined by regular variation of the wandering rate,

$[0,\infty)$00

The same paper proves the inverse theorem: under the structural assumptions, any nontrivial weak limit of $[0,\infty)$01 must be of this form (Sera et al., 2017).

For intermittent maps with indifferent fixed points, there is also a functional multiray theorem. The joint process of occupations near the indifferent points, occupation of the finite junction, and waiting times for returns to the junction converges to the corresponding occupation times, local time, and last/next zero times of a skew Bessel diffusion on a multiray: $[0,\infty)$02 This unifies Darling–Kac local-time limits, Lamperti generalized arcsine occupation laws, and Dynkin–Lamperti waiting-time laws in a single path-level theorem (Sera, 2018).

For subordinators, the Dynkin–Lamperti theorem states that if the Laplace exponent is regularly varying with index $[0,\infty)$03, then

$[0,\infty)$04

either as $[0,\infty)$05 or $[0,\infty)$06, according to the long-range or short-range regime. Recent refinements quantify rare undershoots. If

$[0,\infty)$07

and

$[0,\infty)$08

then

$[0,\infty)$09

This is described as a large-deviation estimate related to the Dynkin–Lamperti theorem, but not as a full LDP (Sera, 10 Nov 2025).

Higher-order refinements go further by expanding the potential density of a killed subordinator and inserting that expansion into the exact identity

$[0,\infty)$10

Under detailed asymptotic assumptions on $[0,\infty)$11 and its derivatives, one obtains explicit higher-order approximations to the Beta/arcsine density for $[0,\infty)$12, both in the long-range and short-range regimes. In this branch of the theory, Lamperti-type laws are refined from weak convergence statements into full asymptotic expansions (Sera, 2024).

6. Banach-valued, norm-dependent, and measure-valued extensions

Lamperti-type correspondence persists in infinite-dimensional settings once the state space is viewed as a cone. For a conic subset $[0,\infty)$13 of a normed space, an $[0,\infty)$14-self-similar Markov process $[0,\infty)$15 admits a Lamperti transform through the additive functional

$[0,\infty)$16

and its inverse

$[0,\infty)$17

Then

$[0,\infty)$18

is a Markov additive process. Conversely, from a MAP $[0,\infty)$19 one recovers

$[0,\infty)$20

The resulting two-step equivalence

$[0,\infty)$21

extends Lamperti’s one-dimensional theorem to Banach-valued processes (Siri-Jégousse et al., 2024).

In the measure-valued population setting, this correspondence identifies the normalized state and the log-mass as a MAP. For the measure-valued self-similar processes constructed in that paper, the Lamperti time change implies that the frequency process

$[0,\infty)$22

is a $[0,\infty)$23-Fleming–Viot process, while the total mass evolves as a positive self-similar Markov process. This generalizes the stable-branching/Beta-Fleming–Viot correspondence to a setting in which the total size can be any positive self-similar Markov process with nonnegative jumps (Siri-Jégousse et al., 2024).

A related extension is norm-dependent rather than merely infinite-dimensional. On a Banach space $[0,\infty)$24, choosing a norm $[0,\infty)$25 determines the sphere

$[0,\infty)$26

and hence a norm-dependent correspondence between self-similar Markov processes on $[0,\infty)$27 and MAPs on $[0,\infty)$28. If $[0,\infty)$29 is such a MAP, then

$[0,\infty)$30

is self-similar, and conversely

$[0,\infty)$31

is a MAP. With the $[0,\infty)$32-norm on the orthant, this yields explicit MAP descriptions for stable processes killed at the boundary, reflected stable processes, and reflected Brownian motion in terms of generators, Lévy systems, and reflected SDEs (Kyprianou et al., 27 Jun 2025).

These developments show that Lamperti-type laws are sensitive to geometry. Outside the positive half-line, the hidden additive process is not canonical until a sign structure, angular decomposition, or norm has been fixed (Siri-Jégousse et al., 2024, Kyprianou et al., 27 Jun 2025).

7. Random fields, process families, and Gaussian Lamperti scaling

For random fields, Lamperti-type laws connect multiplicative scaling in the original parameter space to additive stationarity after logarithmic coordinates. If $[0,\infty)$33 is an $[0,\infty)$34-valued $[0,\infty)$35-multi-self-similar field on $[0,\infty)$36, then

$[0,\infty)$37

is stationary on $[0,\infty)$38, and conversely

$[0,\infty)$39

is multi-self-similar. In the same circle of ideas, domains of attraction under diagonal or operator scaling force the limit field to be self-similar, with normalization governed by multivariate regular variation (Davydov et al., 2017).

A discrete-field analogue uses

$[0,\infty)$40

so that

$[0,\infty)$41

That paper also derives an AR(1)-type characterization of stationary fields: $[0,\infty)$42 where $[0,\infty)$43 has stationary rectangular increments. Here the Lamperti transformation is not merely a correspondence between classes; it is the mechanism by which self-similarity produces the increment-noise field $[0,\infty)$44 (Voutilainen et al., 2023).

Another extension concerns one-parameter families of processes rather than single processes. A family $[0,\infty)$45 is called $[0,\infty)$46-self-similar if

$[0,\infty)$47

The corresponding Lamperti-type convergence theorem states that if

$[0,\infty)$48

then $[0,\infty)$49 is $[0,\infty)$50-self-similar in this generalized sense, and conversely any such family is a fixed point of the renormalization map. This framework covers Poisson processes, Brownian motion with drift, inverse Gaussian processes, Hougaard Lévy processes, and fractional Hougaard motions (Jørgensen et al., 2010).

In Gaussian random fields, Lamperti transformation also converts persistence problems. For tensor-product self-similar Gaussian fields $[0,\infty)$51 on $[0,\infty)$52, the homogeneous Gaussian field

$[0,\infty)$53

has persistence exponent on linearly growing domains, and for the self-similar field the logarithmic persistence exponent on $[0,\infty)$54 equals the homogeneous-field exponent on a simplex $[0,\infty)$55. In $[0,\infty)$56, that simplex can be replaced by a square after a geometric factor (Molchan, 2021).

A recent Gaussian development applies scaled Lamperti transforms to sub-fractional and bi-fractional Brownian motions and to Langevin-type integral processes driven by them. For a process that is $[0,\infty)$57-self-similar, the scaled Lamperti transform

$[0,\infty)$58

is stationary. In the sub-fractional case, the covariance decays as

$[0,\infty)$59

yielding an explicit exponential mixing rate $[0,\infty)$60. In the bi-fractional case, the decay rate is

$[0,\infty)$61

Through inverse Lamperti relations and Birkhoff’s theorem, the papers formulate single-trajectory reconstruction of ensemble quantities for the original non-stationary processes (Shokrollahi et al., 5 Jan 2026).

Taken together, these random-field and Gaussian results show that Lamperti-type laws are not confined to one-parameter Markov settings or to stationary increments. They apply to multi-parameter fields, generalized scaling families, and non-stationary Gaussian models once the correct logarithmic or scaled stationary image has been identified (Davydov et al., 2017, Voutilainen et al., 2023, Jørgensen et al., 2010, Molchan, 2021, Shokrollahi et al., 5 Jan 2026).