Stieltjes Derivative: A Unified Measure Differential Calculus

• The Stieltjes derivative is a generalized derivative defined relative to a derivator that replaces standard increments with Lebesgue–Stieltjes measure increments.
• It unifies continuous, discrete, and impulsive dynamics, accurately capturing jump points and intervals of constancy in various systems.
• The framework supports specific algebraic rules, Stieltjes-analytic expansions, and numerical schemes, extending classical calculus to more complex scenarios.

The Stieltjes derivative is a generalized derivative taken with respect to a derivator $g$ or $\alpha$: classically a left-continuous nondecreasing function, and in more recent work a left-continuous function of bounded variation. It replaces increments of the independent variable by increments of the derivator, so that rates of change are measured relative to the Lebesgue–Stieltjes measure induced by that derivator. In this form it unifies continuous, discrete and impulsive dynamics, accommodates jump points and intervals of constancy, and provides the differential side of Stieltjes ordinary and partial differential equations (Fernández et al., 2020, Maia et al., 5 Sep 2025, Fernández et al., 2020).

1. Classical definition and the geometry of derivators

Let $g:\mathbb R\to\mathbb R$ be nondecreasing and left-continuous. The basic sets attached to $g$ are

$$\Delta g(t)=g(t^+)-g(t),\qquad D_g=\{t:\Delta g(t)>0\},\qquad C_g=\{t:g\text{ is constant on some neighborhood of }t\}.$$

If $C_g=\bigcup_n(a_n,b_n)$ is the decomposition into maximal flat intervals, one defines

$$t^*= \begin{cases} t,&t\notin C_g,\ b_n,&t\in(a_n,b_n). \end{cases}$$

For $f:[a,b]\to\mathbb F$ ($\mathbb F=\mathbb R$ or $\mathbb C$), the Stieltjes derivative at $\alpha$0 is

$\alpha$1

with the corresponding two-sided limit away from $\alpha$2, and the corresponding right-sided limit at jumps or plateaux. At a jump $\alpha$3,

$\alpha$4

Equivalent formulations appear throughout the literature, including the notation $\alpha$5 for differentiation with respect to a nondecreasing $\alpha$6 (Fernández et al., 2022, Fernández et al., 2021, Senín et al., 18 May 2026, Fernández et al., 2020).

This definition is tailored to the geometry of $\alpha$7. On open intervals where $\alpha$8 is strictly increasing, it behaves like an ordinary derivative after reparametrization by $\alpha$9. On $g:\mathbb R\to\mathbb R$0, it records impulsive increments by dividing the jump of $g:\mathbb R\to\mathbb R$1 by the jump of $g:\mathbb R\to\mathbb R$2. On $g:\mathbb R\to\mathbb R$3, where the derivator is locally constant, the derivative is transferred to the plateau endpoint $g:\mathbb R\to\mathbb R$4. In the classical monotone setting $g:\mathbb R\to\mathbb R$5 is at most countable, and $g:\mathbb R\to\mathbb R$6 admits a decomposition into continuous and jump parts, written $g:\mathbb R\to\mathbb R$7 or $g:\mathbb R\to\mathbb R$8 (Fernández et al., 2020, Cora et al., 2023).

A closely related notion is $g:\mathbb R\to\mathbb R$9-continuity: $g$0 is $g$1-continuous at $g$2 if

$g$3

This is continuity in the pseudometric generated by $g$4, rather than ordinary Euclidean continuity. In the monotone theory, the pair $g$5 determines the induced $g$6-topology up to equivalence (Fernández et al., 2022, Albés et al., 11 Jan 2025).

2. Algebraic rules, higher-order derivatives, and Stieltjes-analytic expansions

The Stieltjes derivative is linear: $g$7 Its product rule differs from the classical one by an additional jump term: $g$8 When denominators avoid zeros, there is also a quotient rule, and whenever $g$9 is $$\Delta g(t)=g(t^+)-g(t),\qquad D_g=\{t:\Delta g(t)>0\},\qquad C_g=\{t:g\text{ is constant on some neighborhood of }t\}.$$0 in the ordinary sense one recovers a chain rule of the form

$$\Delta g(t)=g(t^+)-g(t),\qquad D_g=\{t:\Delta g(t)>0\},\qquad C_g=\{t:g\text{ is constant on some neighborhood of }t\}.$$1

subject to the appropriate one-sided interpretation at jumps (Fernández et al., 2022, Fernández et al., 2022, Fernández et al., 2021).

The extra factor involving $$\Delta g(t)=g(t^+)-g(t),\qquad D_g=\{t:\Delta g(t)>0\},\qquad C_g=\{t:g\text{ is constant on some neighborhood of }t\}.$$2 is the characteristic algebraic feature of the theory. It controls the behavior of products at jump points and is the mechanism by which ordinary differential calculus is modified in impulsive regimes. Fernández, Márquez Albés, and Tojo showed that the higher-order calculus inherits the same phenomenon: for an $$\Delta g(t)=g(t^+)-g(t),\qquad D_g=\{t:\Delta g(t)>0\},\qquad C_g=\{t:g\text{ is constant on some neighborhood of }t\}.$$3-fold product $$\Delta g(t)=g(t^+)-g(t),\qquad D_g=\{t:\Delta g(t)>0\},\qquad C_g=\{t:g\text{ is constant on some neighborhood of }t\}.$$4, the first derivative contains a sum of terms weighted by powers of $$\Delta g(t)=g(t^+)-g(t),\qquad D_g=\{t:\Delta g(t)>0\},\qquad C_g=\{t:g\text{ is constant on some neighborhood of }t\}.$$5, and explicit second- and third-order formulas require differentiability of $$\Delta g(t)=g(t^+)-g(t),\qquad D_g=\{t:\Delta g(t)>0\},\qquad C_g=\{t:g\text{ is constant on some neighborhood of }t\}.$$6 and of $$\Delta g(t)=g(t^+)-g(t),\qquad D_g=\{t:\Delta g(t)>0\},\qquad C_g=\{t:g\text{ is constant on some neighborhood of }t\}.$$7. They also proved a regularity criterion: $$\Delta g(t)=g(t^+)-g(t),\qquad D_g=\{t:\Delta g(t)>0\},\qquad C_g=\{t:g\text{ is constant on some neighborhood of }t\}.$$8 at every point where $$\Delta g(t)=g(t^+)-g(t),\qquad D_g=\{t:\Delta g(t)>0\},\qquad C_g=\{t:g\text{ is constant on some neighborhood of }t\}.$$9 jumps (Fernández et al., 2022).

A parallel development is the theory of Stieltjes-analytic functions. For a fixed center $C_g=\bigcup_n(a_n,b_n)$0, the Stieltjes monomials are defined recursively by

$C_g=\bigcup_n(a_n,b_n)$1

They satisfy

$C_g=\bigcup_n(a_n,b_n)$2

and lead to local expansions

$C_g=\bigcup_n(a_n,b_n)$3

with term-by-term Stieltjes differentiation wherever the series converges and $C_g=\bigcup_n(a_n,b_n)$4. The coefficients are recovered from the Stieltjes jet at the center: $C_g=\bigcup_n(a_n,b_n)$5 This is the basis for the Stieltjes-exponential series and for the treatment of higher-order linear equations with constant coefficients (Cora et al., 2023).

3. Fundamental theorem of calculus and absolute continuity

The integral side of the theory is expressed through the Lebesgue–Stieltjes measure $C_g=\bigcup_n(a_n,b_n)$6, determined in the monotone case by

$C_g=\bigcup_n(a_n,b_n)$7

For nondecreasing derivators, the fundamental theorem takes the expected measure-theoretic form. If

$C_g=\bigcup_n(a_n,b_n)$8

then $C_g=\bigcup_n(a_n,b_n)$9 is $$t^*= \begin{cases} t,&t\notin C_g,\ b_n,&t\in(a_n,b_n). \end{cases}$$0-differentiable and $$t^*= \begin{cases} t,&t\notin C_g,\ b_n,&t\in(a_n,b_n). \end{cases}$$1 for $$t^*= \begin{cases} t,&t\notin C_g,\ b_n,&t\in(a_n,b_n). \end{cases}$$2-almost every $$t^*= \begin{cases} t,&t\notin C_g,\ b_n,&t\in(a_n,b_n). \end{cases}$$3. Conversely, $$t^*= \begin{cases} t,&t\notin C_g,\ b_n,&t\in(a_n,b_n). \end{cases}$$4 if and only if $$t^*= \begin{cases} t,&t\notin C_g,\ b_n,&t\in(a_n,b_n). \end{cases}$$5 exists for $$t^*= \begin{cases} t,&t\notin C_g,\ b_n,&t\in(a_n,b_n). \end{cases}$$6-almost every $$t^*= \begin{cases} t,&t\notin C_g,\ b_n,&t\in(a_n,b_n). \end{cases}$$7, $$t^*= \begin{cases} t,&t\notin C_g,\ b_n,&t\in(a_n,b_n). \end{cases}$$8, and

$$t^*= \begin{cases} t,&t\notin C_g,\ b_n,&t\in(a_n,b_n). \end{cases}$$9

Equivalent forms of this statement appear in the monotone theory of López–Rodríguez and in later treatments of $f:[a,b]\to\mathbb F$0 and $f:[a,b]\to\mathbb F$1 spaces (Albés et al., 11 Jan 2025, Fernández et al., 2021, Maia et al., 5 Sep 2025).

Recent work by Maia and Tojo extends this fundamental theorem to left-continuous derivators of locally bounded variation, relaxing the monotonicity constraint. The key device is the variation function

$f:[a,b]\to\mathbb F$2

together with a refined definition of $f:[a,b]\to\mathbb F$3-differentiability valid across the whole domain. In this broader setting there is still an almost-everywhere theorem, but an everywhere version requires a nondegeneracy condition: $f:[a,b]\to\mathbb F$4 If $f:[a,b]\to\mathbb F$5 and $f:[a,b]\to\mathbb F$6 for all $f:[a,b]\to\mathbb F$7, then

$f:[a,b]\to\mathbb F$8

Moreover, this positivity condition is necessary: if $f:[a,b]\to\mathbb F$9 at some $\mathbb F=\mathbb R$0, one can construct a continuous $\mathbb F=\mathbb R$1 whose primitive fails to admit a $\mathbb F=\mathbb R$2-derivative at that point (Maia et al., 5 Sep 2025, Maia et al., 8 Dec 2025).

This framework makes precise the relation between Stieltjes absolute continuity and measure-theoretic differentiation. In the bounded-variation setting, the $\mathbb F=\mathbb R$3-derivative coincides with the Radon–Nikodým derivative $\mathbb F=\mathbb R$4 almost everywhere, and under the nondegeneracy condition even everywhere. A plausible implication is that many arguments from measure differential equations can be transported into the Stieltjes language with only the additional bookkeeping required by jump points and plateaux (Maia et al., 5 Sep 2025).

4. Linear equations, integrating factors, and the $\mathbb F=\mathbb R$5-exponential

A first-order linear Stieltjes ordinary differential equation can be written as

$\mathbb F=\mathbb R$6

or equivalently

$\mathbb F=\mathbb R$7

In the monotone framework of Pouso and coauthors, an integrating factor is introduced by

$\mathbb F=\mathbb R$8

and the product rule yields

$\mathbb F=\mathbb R$9

Hence

$\mathbb C$0

This is the direct Stieltjes analogue of the classical integrating-factor formula (Fernández et al., 2020).

In the $\mathbb C$1 and $\mathbb C$2 formulations, the integrating factor is often written as a $\mathbb C$3-exponential. For $\mathbb C$4, $\mathbb C$5, one has

$\mathbb C$6

and the unique solution of

$\mathbb C$7

is

$\mathbb C$8

Under stronger regularity $\mathbb C$9, the solution belongs to $\alpha$00 (Fernández et al., 2021).

For nonmonotone derivators of controlled variation, the $\alpha$01-exponential is defined by

$\alpha$02

where

$\alpha$03

Theorem 7.3 in the controlled-variation theory shows that this function solves $\alpha$04 and satisfies the usual variation-of-constants formula (Frigon et al., 11 Jan 2025).

The same structure extends to second-order equations. For

$\alpha$05

with constant coefficients, if the characteristic polynomial $\alpha$06 has distinct roots $\alpha$07, then the solutions are $\alpha$08 and $\alpha$09; repeated roots introduce the same kind of generalized second solution as in the classical theory. Fernández, Márquez Albés, and Tojo further introduced full and simplified $\alpha$10-Wronskians and derived a variation-of-parameters method for nonhomogeneous second-order equations with $\alpha$11-continuous coefficients (Fernández et al., 2021, Fernández et al., 2022).

5. Beyond monotonicity: bounded variation, controlled variation, and several derivators

The main limitation of the classical definition is monotonicity. If $\alpha$12 is not nondecreasing, then on intervals where $\alpha$13 falls, the quantity $\alpha$14 may vanish or change sign, and the difference quotient may oscillate or be undefined. Two closely related 2025 developments address this point (Maia et al., 5 Sep 2025, Frigon et al., 11 Jan 2025).

One approach introduces derivators of controlled variation. Let $\alpha$15 and $\alpha$16 be left-continuous and of bounded variation. If there is a closed set $\alpha$17 such that $\alpha$18 is monotonic on each connected component of $\alpha$19, and $\alpha$20, then $\alpha$21 is of controlled variation. Proposition 3.2 shows that such $\alpha$22 form a dense subclass of all left-continuous BV-functions. On this class, the $\alpha$23-derivative coincides with the Radon–Nikodým derivative of $\alpha$24 with respect to $\alpha$25, and one obtains linearity, chain-rule properties, a notion of $\alpha$26-absolute continuity, an explicit $\alpha$27-exponential, and a Peano-type existence theorem for

$\alpha$28

under Stieltjes–Carathéodory hypotheses (Frigon et al., 11 Jan 2025).

A second approach works directly with $\alpha$29, using the variation function $\alpha$30 and the refined remainder-based definition of differentiability. This formulation is designed to be valid across the entire domain, including constancy intervals and one-sided singular points. It produces both almost-everywhere and everywhere versions of the fundamental theorem and shows that Stieltjes differential equations

$\alpha$31

and measure-differential equations

$\alpha$32

lead to the same integral equation

$\alpha$33

In this sense, the framework bridges the gap between Stieltjes differential equations and measure differential equations (Maia et al., 5 Sep 2025).

Another extension concerns systems with several derivators. If $\alpha$34 and each $\alpha$35 is a derivator, then for $\alpha$36 one has

$\alpha$37

Solutions of

$\alpha$38

are defined componentwise by $\alpha$39 for $\alpha$40-almost every $\alpha$41. Under Osgood or Montel–Tonelli conditions the system has at most one solution on a short interval, and under $\alpha$42-Carathéodory conditions a local solution exists by a Schauder fixed-point argument (Albés et al., 11 Jan 2025).

The bounded-variation theory has also been extended to vector-valued derivators $\alpha$43, together with notions of $\alpha$44-continuity and $\alpha$45-continuity. A characterization in terms of the single nondecreasing derivator $\alpha$46 yields compactness tools and everywhere differentiability statements for scalar $\alpha$47-absolutely continuous primitives under a nondegeneracy condition analogous to $\alpha$48 (Maia et al., 8 Dec 2025).

6. Kernel, function spaces, and nonuniqueness phenomena

Unlike the classical derivative, the Stieltjes derivative may have a nontrivial kernel. This issue is studied in detail in the analysis of the operator $\alpha$49 (Fernández et al., 31 Mar 2025).

Three regimes are distinguished. If $\alpha$50 is $\alpha$51-absolutely continuous, then

$\alpha$52

If $\alpha$53 is $\alpha$54-continuous and $\alpha$55-differentiable everywhere, then again

$\alpha$56

But if $\alpha$57 is only $\alpha$58-differentiable everywhere, without $\alpha$59-continuity, then $\alpha$60 forces $\alpha$61 to be constant only on each connected component of

$\alpha$62

where $\alpha$63 are right-endpoints of $\alpha$64-flat intervals. Conversely, any function constant off $\alpha$65 has zero $\alpha$66-derivative (Fernández et al., 31 Mar 2025).

This phenomenon motivates the introduction of spaces of bounded Stieltjes-differentiable functions. The space $\alpha$67 consists of bounded functions with the appropriate $\alpha$68-continuity off the singular sets and one-sided $\alpha$69-continuity at plateau endpoints. Recursively, $\alpha$70 is defined by requiring $\alpha$71. In general $\alpha$72 is not norm-complete under the sup norm, so a metric

$\alpha$73

is introduced, where $\alpha$74 is built from the chordal distance between difference quotients. The resulting space $\alpha$75 is complete, and differentiation is continuous into $\alpha$76 (Fernández et al., 31 Mar 2025).

The nontrivial kernel has direct consequences for Cauchy problems. For the linear homogeneous equation

$\alpha$77

with $\alpha$78 $\alpha$79-regressive, the $\alpha$80-exponential is the unique solution in the absolutely continuous class $\alpha$81. But in the larger space $\alpha$82, every element $\alpha$83 with $\alpha$84 generates another solution by multiplication: $\alpha$85 Hence first-order Stieltjes differential equations are, in general, not unique unless extra continuity or $\alpha$86-hypotheses are imposed (Fernández et al., 31 Mar 2025).

A common misconception is therefore that the Stieltjes derivative behaves exactly like an ordinary derivative once $\alpha$87 is fixed. The kernel results show that this is false at the level of function spaces, even though uniqueness is recovered in more restrictive classes such as $\alpha$88, $\alpha$89, or under the Osgood and Montel–Tonelli conditions for systems (Fernández et al., 31 Mar 2025, Albés et al., 11 Jan 2025).

7. Numerical analysis, parabolic equations, and representative applications

A substantial part of the theory concerns constructive solution methods. For Stieltjes ordinary differential equations

$\alpha$90

Pouso and collaborators proposed a quadrature-based predictor–corrector scheme on a uniform partition of step $\alpha$91: $\alpha$92

$\alpha$93

Under hypotheses $\alpha$94–$\alpha$95, the local errors satisfy

$\alpha$96

the global error satisfies

$\alpha$97

and under perturbations of size $\alpha$98 the method is zero-stable with global error $\alpha$99. In a silkworm population model with $g:\mathbb R\to\mathbb R$00, $g:\mathbb R\to\mathbb R$01, $g:\mathbb R\to\mathbb R$02 on $g:\mathbb R\to\mathbb R$03, numerical experiments show convergence $g:\mathbb R\to\mathbb R$04 in the maximum norm (Fernández et al., 2020).

The same impulsive-and-plateau structure appears in partial differential equations. In the Stieltjes–Bochner framework, a Banach-space-valued curve $g:\mathbb R\to\mathbb R$05 is $g:\mathbb R\to\mathbb R$06-differentiable if

$g:\mathbb R\to\mathbb R$07

and one builds spaces $g:\mathbb R\to\mathbb R$08 and $g:\mathbb R\to\mathbb R$09. This formulation allows the study of parabolic equations that present impulses at certain times or lapses where the system does not evolve at all and presents an elliptic behavior. For the semilinear problem

$g:\mathbb R\to\mathbb R$10

with $g:\mathbb R\to\mathbb R$11, existence, uniqueness, a priori bounds, and a full Galerkin/FEM algorithm are obtained (Fernández et al., 2020).

Senín and Tojo developed a constructive treatment of the one-dimensional heat equation with Stieltjes derivatives. With derivators $g:\mathbb R\to\mathbb R$12 in time and $g:\mathbb R\to\mathbb R$13 in space, the equation is written

$g:\mathbb R\to\mathbb R$14

or, in the single-derivator case $g:\mathbb R\to\mathbb R$15,

$g:\mathbb R\to\mathbb R$16

They proved existence for finite-mode data, existence by series under interchange conditions in Stieltjes–Sobolev spaces, and explicit separated solutions

$g:\mathbb R\to\mathbb R$17

leading for Dirichlet data to expansions of the form

$g:\mathbb R\to\mathbb R$18

They also introduced multivariable derivators $g:\mathbb R\to\mathbb R$19 and corresponding partial Stieltjes derivatives (Senín et al., 18 May 2026).

Representative applied models use the derivative to encode mixed continuous/jump behavior directly in the derivator. In the silkworm life-cycle model, the derivator has jumps at $g:\mathbb R\to\mathbb R$20, while the right-hand side alternates between continuous mortality, a mortality impulse at $g:\mathbb R\to\mathbb R$21, and a birth impulse at $g:\mathbb R\to\mathbb R$22 (Fernández et al., 2020). In fluid stratification on buoyant miscible jets and plumes, setting $g:\mathbb R\to\mathbb R$23 yields

$g:\mathbb R\to\mathbb R$24

so that non-monotonicity of the ambient density encodes thermoclines or density jumps directly via the Stieltjes derivative (Frigon et al., 11 Jan 2025). In a PV–battery model, the derivators

$g:\mathbb R\to\mathbb R$25

govern energy, state of health, and cumulative thermal stress, and the Carathéodory–Picard theorem yields a unique locally defined solution (Maia et al., 8 Dec 2025).

Taken together, these developments show that the Stieltjes derivative is not merely a formal quotient but a full differential calculus adapted to Lebesgue–Stieltjes measures. In the monotone case it supports a theory of $g:\mathbb R\to\mathbb R$26 spaces, $g:\mathbb R\to\mathbb R$27-exponentials, Wronskians, Green’s functions, and convergent numerical schemes; in the bounded-variation case it extends to non-monotonic derivators, measure-differential equations, several derivators, and vector-valued continuity structures. The resulting framework treats continuous evolution, waiting periods, and instantaneous impulses on equal footing (Fernández et al., 2021, Fernández et al., 2022, Maia et al., 5 Sep 2025).