Distribution-Dependent SDEs

• Distribution-dependent SDEs are stochastic equations whose drift and diffusion depend on both the current state and its probability law, linking them to nonlinear Fokker–Planck dynamics.
• They employ fixed-point methods and measure differentiation techniques to achieve well-posedness under various noise regimes including Brownian, Lévy, and fractional models.
• These models reveal unique nonlinear phenomena—such as multiple equilibria, phase transitions, and critical non-uniqueness—that enrich the understanding of mean-field interactions.

Distribution-dependent stochastic differential equations (DDSDEs), also called McKean–Vlasov SDEs or mean-field SDEs, are stochastic equations whose coefficients depend not only on the current state of the process but also on its current law. A standard model is

$$\mathrm dX_t=b(t,X_t,\mathcal L_{X_t})\,\mathrm dt+\sigma(t,X_t,\mathcal L_{X_t})\,\mathrm dW_t,$$

so the dynamics are nonlinear at the level of probability distributions. This law dependence gives DDSDEs an intrinsic link with nonlinear Fokker–Planck equations, nonlinear semigroups on spaces of probability measures, and mean-field limits, while recent work has pushed the theory toward singular coefficients, non-Brownian noises, measure differentiation, long-time dynamics, and even critical-regime non-uniqueness (Huang et al., 2020, Ren, 2024, Ye, 14 Apr 2026).

1. Canonical formulation and nonlinear law flow

The defining feature of a DDSDE is that the drift and diffusion depend on the unknown marginal law $\mu_t:=\mathcal L(X_t)$. In the notation used in recent work, one often writes

$$P_t^*\mu:=\mathcal L_{X_t^\mu},\qquad P_t f(\mu):=\mathbb E[f(X_t^\mu)],$$

where $X_t^\mu$ denotes a solution with initial law $\mu$. This produces a nonlinear evolution on measures and a nonlinear Markov semigroup on observables (Ren, 2024).

The associated nonlinear Fokker–Planck equation is

$\partial_t\mu_t=L_{t,\mu_t}^*\mu_t,$

with

$$L_{t,\mu} = \frac12 \sum_{i,j=1}^d (\sigma\sigma^*)_{ij}(t,\cdot,\mu)\,\partial_i\partial_j +\sum_{i=1}^d b_i(t,\cdot,\mu)\,\partial_i.$$

A central structural fact is the superposition principle: weak solutions of the DDSDE correspond to solutions of the nonlinear Fokker–Planck equation, and conversely (Huang et al., 2020). This correspondence is one of the main reasons DDSDEs are studied simultaneously by probabilistic and PDE methods.

The semigroup is generally not linear in the initial law. An explicit example is

$$\mathrm dX_t = \operatorname{Var}(X_t)\,\mathrm dt + \mathrm dB_t,$$

for which

$$P_t^*\delta_x \sim N\!\left(x+\frac{t^2}{2},\,t\right), \qquad P_t^*\nu_0 \sim N\!\left(t+\frac{t^2}{2},\,1+t\right)$$

when $\nu_0$ is standard normal, and

$\mu_t:=\mathcal L(X_t)$0

Thus the law evolution is not affine in the initial distribution (Sun et al., 15 Jan 2025). This nonlinearity underlies many phenomena that are absent in classical distribution-free SDEs, including multiple stationary laws, phase transitions, and nonlinear sensitivity with respect to the initial measure.

2. Well-posedness theory and singular-coefficient regimes

A basic route to well-posedness is to freeze the law path $\mu_t:=\mathcal L(X_t)$1 and solve the decoupled SDE

$\mu_t:=\mathcal L(X_t)$2

then seek a fixed point in the space of law flows. In monotone or Wasserstein-Lipschitz settings, this program yields strong and weak well-posedness, moment bounds, and stability estimates in Wasserstein distance (Huang et al., 2020).

The contemporary theory, however, is no longer confined to globally Lipschitz coefficients. One direction treats singular drifts through probabilistic regularization and fixed-point arguments in law. In the extrinsic-derivative framework, a Brownian DDSDE on $\mu_t:=\mathcal L(X_t)$3,

$\mu_t:=\mathcal L(X_t)$4

is studied under the factorization

$\mu_t:=\mathcal L(X_t)$5

which makes Girsanov transforms available. The abstract theory applies in particular to nondegenerate DDSDEs with space-time singular drift and to degenerate DDSDEs with weakly monotone coefficients (Ren, 2024). The point is not merely existence of solutions, but existence in a form compatible with differentiating the nonlinear semigroup in the initial law.

Another major regime is additive Lévy noise. For

$\mu_t:=\mathcal L(X_t)$6

weak existence has been proved under low regularity assumptions by combining a Krylov-type estimate for Lévy-driven semimartingales with approximation, tightness, Prokhorov compactness, and Skorokhod representation. The drift is controlled through an approximating sequence $\mu_t:=\mathcal L(X_t)$7 satisfying pointwise convergence along converging laws, uniform $\mu_t:=\mathcal L(X_t)$8 domination, and Lipschitz continuity in $\mu_t:=\mathcal L(X_t)$9 (Ye, 14 Apr 2026). This extends Brownian compactness methods to a nonlocal setting that includes Brownian motion, general nondegenerate $$P_t^*\mu:=\mathcal L_{X_t^\mu},\qquad P_t f(\mu):=\mathbb E[f(X_t^\mu)],$$0-stable processes, cylindrical $$P_t^*\mu:=\mathcal L_{X_t^\mu},\qquad P_t f(\mu):=\mathbb E[f(X_t^\mu)],$$1-stable processes, stable-type, tempered stable, truncated stable, and mixed Lévy noises (Ye, 14 Apr 2026).

Non-Markovian noises also admit strong well-posedness results. For additive fractional Brownian motion, one can solve DDSDEs with irregular, even distributional drifts,

$$P_t^*\mu:=\mathcal L_{X_t^\mu},\qquad P_t f(\mu):=\mathbb E[f(X_t^\mu)],$$2

by combining regularization by fBm with a Wasserstein fixed-point argument. A model case is

$$P_t^*\mu:=\mathcal L_{X_t^\mu},\qquad P_t f(\mu):=\mathbb E[f(X_t^\mu)],$$3

and for $$P_t^*\mu:=\mathcal L_{X_t^\mu},\qquad P_t f(\mu):=\mathbb E[f(X_t^\mu)],$$4 the threshold may be negative, so distributional drifts are allowed (Galeati et al., 2021).

Delay and infinite-dimensional variants are also now available. Distribution-dependent stochastic differential delay equations,

$$P_t^*\mu:=\mathcal L_{X_t^\mu},\qquad P_t f(\mu):=\mathbb E[f(X_t^\mu)],$$5

have unique strong solutions in finite-dimensional path space and in infinite-dimensional variational settings under continuity, coercivity, monotonicity, and growth assumptions on the segment process and its law (Heinemann, 2020). This places memory effects and McKean–Vlasov interactions in a common monotonicity-based framework.

3. Differentiation with respect to the initial law

A major recent theme is differentiation of $$P_t^*\mu:=\mathcal L_{X_t^\mu},\qquad P_t f(\mu):=\mathbb E[f(X_t^\mu)],$$6. Several non-equivalent notions of measure derivative are now in active use, and the distinction matters technically.

One notion is the extrinsic derivative, defined by the birth–death perturbation

$$P_t^*\mu:=\mathcal L_{X_t^\mu},\qquad P_t f(\mu):=\mathbb E[f(X_t^\mu)],$$7

This is closely related to the linear functional derivative through

$$P_t^*\mu:=\mathcal L_{X_t^\mu},\qquad P_t f(\mu):=\mathbb E[f(X_t^\mu)],$$8

It is explicitly distinguished from the Lions derivative: the Lions derivative differentiates along $$P_t^*\mu:=\mathcal L_{X_t^\mu},\qquad P_t f(\mu):=\mathbb E[f(X_t^\mu)],$$9-lifts and is intrinsic to moving mass, whereas the extrinsic derivative captures insertion of a Dirac mass (Ren, 2024).

For DDSDEs with

$X_t^\mu$0

the paper on extrinsic derivatives proves a Bismut-type formula for the nonlinear semigroup,

$X_t^\mu$1

where the first term is the direct contribution of changing the initial distribution while freezing the law flow, and the second term is the genuinely nonlinear correction generated by McKean–Vlasov feedback (Ren, 2024). The same work derives a gradient-type bound

$X_t^\mu$2

which regularizes merely measurable test functions.

A different notion is the intrinsic derivative, defined at $X_t^\mu$3 by perturbing the identity map: $X_t^\mu$4 For singular convolution interactions

$X_t^\mu$5

recent work proves a Bismut formula for the intrinsic derivative of $X_t^\mu$6 without assuming Lions differentiability of the drift in the measure variable. The resulting formula decomposes into a decoupled-SDE Bismut term and a law-interaction correction involving the singular kernel $X_t^\mu$7 (Ren, 10 Apr 2026). This is specifically designed for singular interactions where the usual differentiability of $X_t^\mu$8 in $X_t^\mu$9 fails or becomes singular.

The Lions derivative remains central in smoother regimes. For DDSDEs driven by fractional Brownian motion with Hurst parameter $\mu$0, one has a Bismut formula of the form

$\mu$1

together with short-time estimates for $\mu$2 and total variation bounds between solution laws (Fan et al., 2021). In that setting, Malliavin calculus, the Volterra representation of fractional Brownian motion, and the law derivative are combined in a genuinely non-semimartingale framework.

Taken together, these results show that “differentiation with respect to the law” is not a single construction. Birth–death perturbations, pushforward perturbations, and Lions’ $\mu$3-lift calculus lead to different objects, and current DDSDE theory uses each of them where it is structurally natural.

4. Noise models and generalized probabilistic frameworks

The Brownian McKean–Vlasov setting is only one part of the present landscape. Several recent directions replace the classical noise or even the underlying notion of distribution.

For additive Lévy noise,

$\mu$4

the analytical input is a semigroup estimate

$\mu$5

supplemented by moment and density assumptions on the Lévy process (Ye, 14 Apr 2026). The corresponding generator is nonlocal,

$\mu$6

so the Krylov theory becomes integro-differential rather than parabolic (Ye, 14 Apr 2026).

Under $\mu$7-expectation, the very meaning of distribution dependence changes. De Sun, Jiang-Lun Wu, and Panyu Wu formulate a distribution-dependent $\mu$8-SDE by replacing the classical law $\mu$9 with the sublinear distribution functional

$\partial_t\mu_t=L_{t,\mu_t}^*\mu_t,$0

They introduce the metric

$\partial_t\mu_t=L_{t,\mu_t}^*\mu_t,$1

prove that in the classical linear-expectation case it coincides with $\partial_t\mu_t=L_{t,\mu_t}^*\mu_t,$2 by Kantorovich–Rubinstein duality, and establish existence and uniqueness for

$\partial_t\mu_t=L_{t,\mu_t}^*\mu_t,$3

under global Lipschitz assumptions in state and distribution variables (Sun et al., 2023).

Fractional Brownian motion yields a different extension. For $\partial_t\mu_t=L_{t,\mu_t}^*\mu_t,$4, DDSDEs driven by fBm are well posed under Wasserstein-Lipschitz hypotheses, and Bismut formulas for the Lions derivative can be established in both non-degenerate and degenerate cases (Fan et al., 2021). For additive fBm with singular or distributional drift, Catellier–Gubinelli-type regularization leads to strong existence, pathwise uniqueness, and uniqueness in law in Besov scales (Galeati et al., 2021). A key technical insight there is that solvability of the frozen-law singular SDE requires drift regularity $\partial_t\mu_t=L_{t,\mu_t}^*\mu_t,$5, but stability only needs differences of drifts in the weaker norm $\partial_t\mu_t=L_{t,\mu_t}^*\mu_t,$6, which matches Wasserstein control of the law dependence (Galeati et al., 2021).

Pathwise regularization has also been used for degenerate multiplicative noise. For equations of the form

$\partial_t\mu_t=L_{t,\mu_t}^*\mu_t,$7

weak existence is proved when $\partial_t\mu_t=L_{t,\mu_t}^*\mu_t,$8, $\partial_t\mu_t=L_{t,\mu_t}^*\mu_t,$9, and $$L_{t,\mu} = \frac12 \sum_{i,j=1}^d (\sigma\sigma^*)_{ij}(t,\cdot,\mu)\,\partial_i\partial_j +\sum_{i=1}^d b_i(t,\cdot,\mu)\,\partial_i.$$0 lie in $$L_{t,\mu} = \frac12 \sum_{i,j=1}^d (\sigma\sigma^*)_{ij}(t,\cdot,\mu)\,\partial_i\partial_j +\sum_{i=1}^d b_i(t,\cdot,\mu)\,\partial_i.$$1, the map $$L_{t,\mu} = \frac12 \sum_{i,j=1}^d (\sigma\sigma^*)_{ij}(t,\cdot,\mu)\,\partial_i\partial_j +\sum_{i=1}^d b_i(t,\cdot,\mu)\,\partial_i.$$2 is Lipschitz in $$L_{t,\mu} = \frac12 \sum_{i,j=1}^d (\sigma\sigma^*)_{ij}(t,\cdot,\mu)\,\partial_i\partial_j +\sum_{i=1}^d b_i(t,\cdot,\mu)\,\partial_i.$$3, and the external path $$L_{t,\mu} = \frac12 \sum_{i,j=1}^d (\sigma\sigma^*)_{ij}(t,\cdot,\mu)\,\partial_i\partial_j +\sum_{i=1}^d b_i(t,\cdot,\mu)\,\partial_i.$$4 is sufficiently locally non-deterministic (Harang et al., 3 Sep 2025). The regularization mechanism is the averaging operator

$$L_{t,\mu} = \frac12 \sum_{i,j=1}^d (\sigma\sigma^*)_{ij}(t,\cdot,\mu)\,\partial_i\partial_j +\sum_{i=1}^d b_i(t,\cdot,\mu)\,\partial_i.$$5

which converts time integration along a rough path into spatial convolution with local time (Harang et al., 3 Sep 2025). This suggests a distinct route to DDSDE existence: irregular noise can regularize a law-dependent equation even when the multiplicative diffusion is degenerate and the coefficients are singular.

5. Long-time behavior, periodicity, and equilibrium selection

Long-time analysis for DDSDEs differs sharply from the classical Markov case because the law flow is nonlinear and stationary distributions need not be unique. One recent line studies existence of periodic and stationary solutions directly at the nonlinear semigroup level. For

$$L_{t,\mu} = \frac12 \sum_{i,j=1}^d (\sigma\sigma^*)_{ij}(t,\cdot,\mu)\,\partial_i\partial_j +\sum_{i=1}^d b_i(t,\cdot,\mu)\,\partial_i.$$6

with $$L_{t,\mu} = \frac12 \sum_{i,j=1}^d (\sigma\sigma^*)_{ij}(t,\cdot,\mu)\,\partial_i\partial_j +\sum_{i=1}^d b_i(t,\cdot,\mu)\,\partial_i.$$7-periodic coefficients, a $$L_{t,\mu} = \frac12 \sum_{i,j=1}^d (\sigma\sigma^*)_{ij}(t,\cdot,\mu)\,\partial_i\partial_j +\sum_{i=1}^d b_i(t,\cdot,\mu)\,\partial_i.$$8-periodic solution is equivalent to a $$L_{t,\mu} = \frac12 \sum_{i,j=1}^d (\sigma\sigma^*)_{ij}(t,\cdot,\mu)\,\partial_i\partial_j +\sum_{i=1}^d b_i(t,\cdot,\mu)\,\partial_i.$$9-periodic measure $$\mathrm dX_t = \operatorname{Var}(X_t)\,\mathrm dt + \mathrm dB_t,$$0 satisfying

$$\mathrm dX_t = \operatorname{Var}(X_t)\,\mathrm dt + \mathrm dB_t,$$1

Existence is obtained by combining weak convergence or Krylov–Bogoliubov-type averaging with Lyapunov estimates, or alternatively by applying Schauder’s theorem on a compact convex set of measures (Sun et al., 15 Jan 2025). In the autonomous case, the same framework yields stationary solutions.

When a stationary distribution $$\mathrm dX_t = \operatorname{Var}(X_t)\,\mathrm dt + \mathrm dB_t,$$2 is fixed, one can linearize the nonlinear semigroup around it. For

$$\mathrm dX_t = \operatorname{Var}(X_t)\,\mathrm dt + \mathrm dB_t,$$3

the frozen generator is

$$\mathrm dX_t = \operatorname{Var}(X_t)\,\mathrm dt + \mathrm dB_t,$$4

and the law derivative of the drift defines a mean-field correction

$$\mathrm dX_t = \operatorname{Var}(X_t)\,\mathrm dt + \mathrm dB_t,$$5

The linearized semigroup $$\mathrm dX_t = \operatorname{Var}(X_t)\,\mathrm dt + \mathrm dB_t,$$6 generated by $$\mathrm dX_t = \operatorname{Var}(X_t)\,\mathrm dt + \mathrm dB_t,$$7 governs local behavior near equilibrium (Zhang, 8 Jan 2025). If the frozen semigroup is contractive in a weighted Lipschitz metric and $$\mathrm dX_t = \operatorname{Var}(X_t)\,\mathrm dt + \mathrm dB_t,$$8 is exponentially stable on mean-zero functions in $$\mathrm dX_t = \operatorname{Var}(X_t)\,\mathrm dt + \mathrm dB_t,$$9, then the full nonlinear DDSDE converges locally exponentially fast back to $$P_t^*\delta_x \sim N\!\left(x+\frac{t^2}{2},\,t\right), \qquad P_t^*\nu_0 \sim N\!\left(t+\frac{t^2}{2},\,1+t\right)$$0 (Zhang, 8 Jan 2025).

The complementary instability theory shows that multiple stationary laws are not merely a static phenomenon. Under suitable Lyapunov, differentiability, and quasi-compactness hypotheses, if the spectrum of the adjoint linearized generator satisfies

$$P_t^*\delta_x \sim N\!\left(x+\frac{t^2}{2},\,t\right), \qquad P_t^*\nu_0 \sim N\!\left(t+\frac{t^2}{2},\,1+t\right)$$1

then the stationary distribution is unstable in a weighted Kantorovich-type metric (Zhang, 5 Oct 2025). Concrete examples include Dawson’s model

$$P_t^*\delta_x \sim N\!\left(x+\frac{t^2}{2},\,t\right), \qquad P_t^*\nu_0 \sim N\!\left(t+\frac{t^2}{2},\,1+t\right)$$2

where the symmetric stationary law becomes unstable in a low-noise regime, as well as other granular-media and transformed double-well models (Zhang, 5 Oct 2025). This connects McKean–Vlasov phase-transition heuristics to a precise spectral criterion.

The broader ergodic picture is summarized in the survey literature: under dissipativity, one can obtain unique invariant measures together with exponential convergence in Wasserstein distance and entropy, and in some cases log-Harnack inequalities and Donsker–Varadhan large deviations for empirical measures (Huang et al., 2020). The important point is that DDSDEs support both regimes: uniqueness with exponential ergodicity under strong contraction, and multiple equilibria with local stability or instability near selected stationary laws.

6. Critical non-uniqueness, PDE ill-posedness, and scope of the theory

Recent work has shown that DDSDE well-posedness can fail dramatically at critical roughness. On $$P_t^*\delta_x \sim N\!\left(x+\frac{t^2}{2},\,t\right), \qquad P_t^*\nu_0 \sim N\!\left(t+\frac{t^2}{2},\,1+t\right)$$3 or $$P_t^*\delta_x \sim N\!\left(x+\frac{t^2}{2},\,t\right), \qquad P_t^*\nu_0 \sim N\!\left(t+\frac{t^2}{2},\,1+t\right)$$4, $$P_t^*\delta_x \sim N\!\left(x+\frac{t^2}{2},\,t\right), \qquad P_t^*\nu_0 \sim N\!\left(t+\frac{t^2}{2},\,1+t\right)$$5, one studies DDSDEs whose coefficients depend on the density $$P_t^*\delta_x \sim N\!\left(x+\frac{t^2}{2},\,t\right), \qquad P_t^*\nu_0 \sim N\!\left(t+\frac{t^2}{2},\,1+t\right)$$6,

$$P_t^*\delta_x \sim N\!\left(x+\frac{t^2}{2},\,t\right), \qquad P_t^*\nu_0 \sim N\!\left(t+\frac{t^2}{2},\,1+t\right)$$7

The construction is carried out first at the level of the nonlinear Fokker–Planck equation, and then lifted to the DDSDE by the superposition principle (Lü, 30 Jun 2026).

At the time-dependent level, one can construct a divergence-free drift

$$P_t^*\delta_x \sim N\!\left(x+\frac{t^2}{2},\,t\right), \qquad P_t^*\nu_0 \sim N\!\left(t+\frac{t^2}{2},\,1+t\right)$$8

arbitrarily close to a smooth divergence-free field, such that the nonlinear Fokker–Planck equation admits infinitely many distinct probability density solutions from the stationary initial density $$P_t^*\delta_x \sim N\!\left(x+\frac{t^2}{2},\,t\right), \qquad P_t^*\nu_0 \sim N\!\left(t+\frac{t^2}{2},\,1+t\right)$$9, and the corresponding DDSDE admits infinitely many distinct martingale solutions starting from volume measure (Lü, 30 Jun 2026). The paper identifies this as a critical-threshold phenomenon: in several models, well-posedness is expected for drifts in $\nu_0$0, whereas the construction sits at $\nu_0$1 (Lü, 30 Jun 2026).

At the stationary level, for $\nu_0$2 and any prescribed $\nu_0$3, one can construct divergence-free drifts for which the stationary nonlinear Fokker–Planck equation has at least $\nu_0$4 distinct non-constant probability density solutions, and the associated DDSDE has at least $\nu_0$5 distinct stationary martingale solutions (Lü, 30 Jun 2026). This gives a rigorous multistability mechanism that is not based on smooth double-well dynamics but on rough critical drifts and convex-integration-based non-uniqueness.

These results also clarify the current scope of DDSDE theory. Many of the strongest derivative formulas are finite-dimensional and rely on specific structures such as the factorization

$\nu_0$6

and continuity of the measure derivative in weighted total variation rather than Wasserstein distance (Ren, 2024). Lévy-noise weak existence presently treats additive noise only, not multiplicative law-dependent jump coefficients (Ye, 14 Apr 2026). Pathwise-regularization results with degenerate multiplicative noise encode the law through a finite-dimensional map $\nu_0$7, not a fully general infinite-dimensional dependence on $\nu_0$8 (Harang et al., 3 Sep 2025). Stability and instability results near equilibria are local rather than global, and they do not classify all basins of attraction (Zhang, 8 Jan 2025, Zhang, 5 Oct 2025).

The cumulative picture is therefore two-sided. On one side, DDSDEs now admit robust well-posedness, differentiation, and long-time theories across Brownian, Lévy, fractional, $\nu_0$9-Brownian, delayed, and singular-interaction settings. On the other, the same class of equations exhibits genuinely nonlinear phenomena—multiple invariant laws, phase transitions, and critical-regime non-uniqueness—that have no analogue in the classical linear Markov framework.