Multivalued McKean-Vlasov SDEs

• Multivalued McKean-Vlasov SDEs are distribution-dependent stochastic equations driven by a maximal monotone, set-valued operator that captures nonsmooth dynamics.
• They extend classical SDE frameworks by incorporating path-dependent coefficients and non-Lipschitz effects via robust variational inequality formulations.
• The framework underpins analyses of stability, ergodicity, deviation principles, and averaging, with applications to reflected SDEs and backward equations.

Searching arXiv for papers on multivalued McKean–Vlasov stochastic differential equations and closely related path-dependent results. Multivalued McKean–Vlasov stochastic differential equations (MMVSDEs) are distribution-dependent stochastic differential inclusions in which the drift contains a maximal monotone set-valued operator and the coefficients depend on the current state, the law of the solution, and in some formulations its path segment. In canonical finite-dimensional form,

$$dX_t \in -A(X_t)\,dt + b(X_t,\mu_t)\,dt + \sigma(X_t,\mu_t)\,dW_t,\qquad \mu_t=\mathcal{L}(X_t),$$

while path-dependent versions replace $b(X_t,\mu_t)$ and $\sigma(X_t,\mu_t)$ by non-anticipative functionals of the segment $X_t$. The multivalued term is typically encoded through a finite-variation process $K$, so that $X_t$ solves an integral equation with $-K_t$ and the pair $(X,K)$ satisfies a Skorokhod-type variational inequality. This framework covers reflected McKean–Vlasov SDEs, stochastic variational inequalities, and mean-field systems with nonsmooth constraints, and recent work has developed well-posedness, path-dependent extensions, deviation theory, ergodicity, averaging, jump-noise variants, and backward analogues linked to nonlocal variational inequalities (Qiao et al., 2021, Ma et al., 21 Aug 2025).

1. Formal structure and operator framework

The defining structural feature of an MMVSDE is the presence of a multivalued maximal monotone operator $A:\mathbb{R}^d\to 2^{\mathbb{R}^d}$. Its graph is

$$\mathrm{Gr}(A)=\{(x,y)\in\mathbb{R}^{2d}:x\in D(A),\,y\in A(x)\},$$

and monotonicity means

$b(X_t,\mu_t)$0

Maximality requires that the graph be maximal among monotone graphs. Across the forward MMVSDE literature, the standing condition $b(X_t,\mu_t)$1 is central, because it yields coercivity estimates for the finite-variation term and stability of admissible solution pairs (Ma et al., 21 Aug 2025, Fang et al., 2022).

Rather than interpreting $b(X_t,\mu_t)$2 pointwise, the theory introduces an admissible pair $b(X_t,\mu_t)$3, with $b(X_t,\mu_t)$4 continuous or càdlàg depending on the noise model and $b(X_t,\mu_t)$5 a continuous finite-variation process with $b(X_t,\mu_t)$6. The pair belongs to a set $b(X_t,\mu_t)$7 defined by the variational condition

$b(X_t,\mu_t)$8

This formulation implies the monotonicity inequality

$b(X_t,\mu_t)$9

for any two admissible pairs, and supports coercivity estimates of the form

$\sigma(X_t,\mu_t)$0

or analogous bounds with squared norms, depending on the paper’s normalization (Ma et al., 21 Aug 2025, Fang et al., 2022, Qiao, 2022).

Two model classes recur throughout the literature. If $\sigma(X_t,\mu_t)$1 for a proper, lower semicontinuous convex function $\sigma(X_t,\mu_t)$2, the equation becomes a stochastic variational inequality with convex subdifferential drift. If $\sigma(X_t,\mu_t)$3 is the normal cone to a closed convex set $\sigma(X_t,\mu_t)$4, the term $\sigma(X_t,\mu_t)$5 is the minimal bounded-variation process that keeps the state in $\sigma(X_t,\mu_t)$6, recovering reflected McKean–Vlasov SDEs in convex domains (Ma et al., 21 Aug 2025, Qiao, 2022). Standard Yosida regularization,

$\sigma(X_t,\mu_t)$7

is used explicitly in several works on ergodicity, backward equations, and general deviation theory, whereas some path-dependent analyses proceed directly via variational formulations without using Yosida in the main proofs (Qiao, 2022, Gong et al., 2021, Cheng et al., 9 Jul 2025, Ma et al., 21 Aug 2025).

2. Solution concepts and well-posedness

For forward MMVSDEs, a strong solution is typically a pair $\sigma(X_t,\mu_t)$8 adapted to the Brownian filtration such that $\sigma(X_t,\mu_t)$9 almost surely, the coefficient integrability conditions hold, and

$X_t$0

Weak solutions allow the stochastic basis to vary, and martingale solutions are formulated on canonical path space through the associated martingale problem. These notions are all present in the modern MMVSDE literature, with the martingale formulation becoming especially useful in jump-driven settings (Fang et al., 2022, Cheng et al., 19 Jul 2025).

Under Lipschitz or monotonicity-type assumptions in the state and $X_t$1-Lipschitz dependence in the law, strong well-posedness is standard. One formulation assumes

$X_t$2

together with the corresponding bound for $X_t$3, and yields existence and uniqueness of strong solutions when $X_t$4 (Qiao et al., 2021). Another Lipschitz framework, used for small-noise deviation theory, assumes continuity and linear growth of $X_t$5 plus monotonicity/Lipschitz-type control in $X_t$6, again producing unique strong solutions (Fang et al., 2022).

A notable development is the treatment of non-Lipschitz coefficients. One class of results uses one-sided Lipschitz or Osgood-type assumptions in the state and global $X_t$7-control in the law, obtaining unique strong solutions through contraction on measure flows, monotonicity of $X_t$8, and Grönwall- or Bihari-type arguments (Qiao et al., 2021). A second line of work treats Lévy-noise MMVSDEs with non-Lipschitz coefficients and jump term

$X_t$9

proving strong existence and uniqueness under Osgood-type conditions, weak existence under linear growth, and existence of martingale solutions on the canonical càdlàg path space (Cheng et al., 19 Jul 2025). This shows that the multivalued mean-field framework is not restricted to continuous diffusions.

The proof technology varies with the model. Fixed-law decoupling and contraction in the space of measure flows appear in non-path-dependent Brownian models. Smoothing and approximation arguments are used in jump models with non-Lipschitz coefficients. Yosida penalization is prominent in backward and invariant-measure analyses. Across these settings, the indispensable ingredients are monotonicity of the multivalued operator, coercivity for $K$0, moment estimates from Itô-type formulas, and stability in Wasserstein distance (Qiao et al., 2021, Cheng et al., 19 Jul 2025, Qiao, 2022).

3. Path dependence and non-Lipschitz memory effects

The path-dependent theory replaces the current-state dependence by dependence on the recent history $K$1. Fixing $K$2, the path space is

$K$3

with uniform norm $K$4, and the segment process is

$K$5

The law variable then takes values in $K$6, equipped with the $K$7-Wasserstein metric

$K$8

and the path-dependent MMVSDE becomes

$K$9

(Ma et al., 21 Aug 2025).

The current well-posedness theory for this equation proceeds in three stages. First, for path-dependent multivalued SDEs without law dependence, strong well-posedness is proved under linear growth and global Lipschitz continuity in the path variable. The method is a Picard iteration on path space, where each iterate solves a multivalued SDE with frozen coefficients from the previous iterate, and the contraction estimate is closed on short intervals before patching to $X_t$0. Second, the path-dependent equation without law dependence is extended to non-Lipschitz coefficients by constructing bounded Lipschitz approximations, proving tightness of the approximate solutions in $X_t$1, and applying an Osgood uniqueness inequality of the form

$X_t$2

Third, the full path-dependent MMVSDE is obtained by iterating in distributions: one freezes $X_t$3, solves the path-dependent multivalued SDE for $X_t$4, sets $X_t$5, and proves convergence in $X_t$6 using coupled Osgood estimates in the path and law variables (Ma et al., 21 Aug 2025).

The non-Lipschitz assumptions are explicitly Osgood-type. In the path-only case, one assumes continuous $X_t$7 and a strictly increasing, continuous, concave modulus $X_t$8 with $X_t$9 and $-K_t$0. In the full MMVSDE, one replaces $-K_t$1 by two concave moduli $-K_t$2 governing the path and law variables, with

$-K_t$3

Under these hypotheses and $-K_t$4, the path-dependent MMVSDE admits a unique strong solution $-K_t$5 satisfying

$-K_t$6

(Ma et al., 21 Aug 2025).

The path-dependent framework broadens the admissible memory mechanisms. The coefficients may depend on delays such as $-K_t$7, on running suprema such as $-K_t$8, or on measure functionals of the path law, provided the corresponding continuity and Osgood-modulus estimates hold in $-K_t$9 and $(X,K)$0 (Ma et al., 21 Aug 2025). This suggests that MMVSDEs furnish a natural mean-field generalization of multivalued delay equations and reflecting equations with memory.

4. Deviation principles, fluctuation limits, and small-noise asymptotics

Small-noise asymptotics for MMVSDEs are developed through the weak convergence method of Budhiraja and Dupuis. In the state-dependent Brownian setting, one studies

$(X,K)$1

Under continuity, linear growth, and monotonicity/Lipschitz-type control in $(X,K)$2, the family $(X,K)$3 satisfies a large deviation principle in $(X,K)$4 with good rate function

$(X,K)$5

where $(X,K)$6 solves the deterministic controlled inclusion with law replaced by the Dirac mass at the path. The same work proves a central limit theorem for

$(X,K)$7

and a moderate deviation principle for

$(X,K)$8

The CLT limit contains the Lions derivative $(X,K)$9 of the drift, while no differentiability of $A:\mathbb{R}^d\to 2^{\mathbb{R}^d}$0 in $A:\mathbb{R}^d\to 2^{\mathbb{R}^d}$1 or $A:\mathbb{R}^d\to 2^{\mathbb{R}^d}$2 is assumed, so the diffusion term is evaluated directly along the deterministic limit path (Fang et al., 2022).

A related weak-convergence formulation proves large and moderate deviation principles for MMVSDEs under assumptions that isolate the deterministic limit law $A:\mathbb{R}^d\to 2^{\mathbb{R}^d}$3 in the skeleton. In that setting, the LDP skeleton solves

$A:\mathbb{R}^d\to 2^{\mathbb{R}^d}$4

and the MDP uses a linearized skeleton involving $A:\mathbb{R}^d\to 2^{\mathbb{R}^d}$5. The multivalued term persists through the compensator $A:\mathbb{R}^d\to 2^{\mathbb{R}^d}$6, and the main compactness input is the coercivity estimate for admissible pairs $A:\mathbb{R}^d\to 2^{\mathbb{R}^d}$7 (Zhu et al., 2022).

General large deviations and functional iterated logarithm laws have also been established under non-Lipschitz, concave-modulus assumptions on $A:\mathbb{R}^d\to 2^{\mathbb{R}^d}$8 and $A:\mathbb{R}^d\to 2^{\mathbb{R}^d}$9, allowing the operator $$\mathrm{Gr}(A)=\{(x,y)\in\mathbb{R}^{2d}:x\in D(A),\,y\in A(x)\},$$0 to vary with $$\mathrm{Gr}(A)=\{(x,y)\in\mathbb{R}^{2d}:x\in D(A),\,y\in A(x)\},$$1. The LDP and MDP are again derived from a weak convergence scheme, but the proofs combine Yosida approximation with Bihari’s inequality to handle the non-Lipschitz coefficients. The functional iterated logarithm law identifies the almost sure limit set with the unit sublevel set of the corresponding good rate function, thus extending classical Strassen-type behavior to multivalued mean-field diffusions (Cheng et al., 9 Jul 2025).

The path-dependent small-noise theory parallels the state-dependent one but introduces both path Fréchet derivatives and Lions derivatives. For

$$\mathrm{Gr}(A)=\{(x,y)\in\mathbb{R}^{2d}:x\in D(A),\,y\in A(x)\},$$2

the large deviation principle holds under non-Lipschitz path- and law-continuity conditions, the moderate deviation principle is formulated for

$$\mathrm{Gr}(A)=\{(x,y)\in\mathbb{R}^{2d}:x\in D(A),\,y\in A(x)\},$$3

and the central limit theorem is proved for

$$\mathrm{Gr}(A)=\{(x,y)\in\mathbb{R}^{2d}:x\in D(A),\,y\in A(x)\},$$4

The limiting CLT inclusion contains both

$$\mathrm{Gr}(A)=\{(x,y)\in\mathbb{R}^{2d}:x\in D(A),\,y\in A(x)\},$$5

from path differentiation and

$$\mathrm{Gr}(A)=\{(x,y)\in\mathbb{R}^{2d}:x\in D(A),\,y\in A(x)\},$$6

from the law derivative (Ma et al., 8 May 2026). This identifies path-dependent MMVSDE fluctuations as simultaneously controlled by memory linearization and measure linearization.

5. Stability, ergodicity, invariant measures, and averaging

A generalized Itô formula is available for functions $$\mathrm{Gr}(A)=\{(x,y)\in\mathbb{R}^{2d}:x\in D(A),\,y\in A(x)\},$$7 of the state and its law in the multivalued setting. Besides the standard drift, diffusion, and Lions-derivative terms, the formula contains the additional pairings with the finite-variation process $$\mathrm{Gr}(A)=\{(x,y)\in\mathbb{R}^{2d}:x\in D(A),\,y\in A(x)\},$$8,

$$\mathrm{Gr}(A)=\{(x,y)\in\mathbb{R}^{2d}:x\in D(A),\,y\in A(x)\},$$9

or equivalent formulations with $b(X_t,\mu_t)$00 when $b(X_t,\mu_t)$01. This extension underlies Lyapunov analyses for MMVSDEs with non-Lipschitz coefficients and yields criteria for asymptotic stability of second moments and almost sure asymptotic stability (Qiao et al., 2021).

Long-time behavior in Wasserstein distance has been analyzed under dissipativity assumptions of the form

$b(X_t,\mu_t)$02

When $b(X_t,\mu_t)$03, strong solutions satisfy an exponential contraction estimate

$b(X_t,\mu_t)$04

which implies the existence and uniqueness of an invariant probability measure and exponential ergodicity of the nonlinear semigroup $b(X_t,\mu_t)$05. The same framework supports convergence of invariant measures for sequences of MMVSDEs whose operators and coefficients converge via Yosida-resolvent convergence on compacts and coefficient convergence in $b(X_t,\mu_t)$06 (Qiao, 2022).

Jump-driven MMVSDEs admit both stability theory and averaging. For Brownian–Poisson systems with compensated jump measure $b(X_t,\mu_t)$07, a generalized Itô formula includes state-derivative, Lions-derivative, jump-compensator, and $b(X_t,\mu_t)$08-terms. With suitable Lyapunov inequalities, one obtains exponential stability of second moments, exponentially $b(X_t,\mu_t)$09-ultimate boundedness, and almost sure asymptotic stability. Under time-averaging assumptions on the coefficients, the solutions of fast-time-scaled MMVSDEs with jumps converge in mean square to the associated averaged MMVSDEs (Shen et al., 2023).

A multiscale theory has been developed for slow–fast multivalued McKean–Vlasov systems driven by two Brownian motions, with separate maximal monotone operators $b(X_t,\mu_t)$10 and $b(X_t,\mu_t)$11. Under non-Lipschitz well-posedness assumptions and stronger dissipativity for the frozen fast equation, the fast variable admits a unique invariant measure $b(X_t,\mu_t)$12, and the averaged slow drift is

$b(X_t,\mu_t)$13

Depending on the slow-noise scaling and whether the slow operator is general or reflection-type, the theory yields four averaging regimes, including deterministic and stochastic averaged limits, as well as a large deviation principle for the slow component in the small-noise case (Qiao, 2023). This suggests that MMVSDE averaging can accommodate both multivalued constraints and nonlinear law dependence without reducing to a classical smooth slow–fast system.

6. Variants, examples, and limitations

The class of examples covered by current MMVSDE theory is broad but structurally unified. Subdifferential operators $b(X_t,\mu_t)$14 model convex constraints and proximal-type dynamics. Normal cones $b(X_t,\mu_t)$15 or $b(X_t,\mu_t)$16 produce reflection in convex domains. Mean-field coefficients may depend on moments,

$b(X_t,\mu_t)$17

and path dependence may involve delays, running suprema, or other non-anticipative functionals of the segment (Ma et al., 21 Aug 2025). In jump models, an additional domain-invariance condition

$b(X_t,\mu_t)$18

ensures that jumps remain compatible with the constraint set (Cheng et al., 19 Jul 2025).

A significant variant is the backward multivalued McKean–Vlasov equation

$b(X_t,\mu_t)$19

where the multivalued constraint is again encoded by a finite-variation process $b(X_t,\mu_t)$20. Under Lipschitz assumptions in $b(X_t,\mu_t)$21 and in the law argument, this equation admits a unique solution $b(X_t,\mu_t)$22, depends continuously on the terminal value, and yields a probabilistic interpretation of viscosity solutions for nonlocal quasi-linear parabolic variational inequalities with a multivalued term $b(X_t,\mu_t)$23 (Gong et al., 2021). This backward theory connects the MMVSDE framework to nonlocal PDE and variational inequality methods.

The limitations of the present theory are also sharply defined. Many results require $b(X_t,\mu_t)$24 and maximal monotonicity, so nonconvex or nonmaximal multivalued drifts are خارج the established framework. Continuous-path path-dependent results are formulated on $b(X_t,\mu_t)$25 and do not treat discontinuous memory functionals. CLT and MDP results that linearize the measure dependence require Lions differentiability of the drift, and path-dependent CLT/MDP results additionally require Fréchet differentiability in the path variable (Ma et al., 21 Aug 2025, Ma et al., 8 May 2026). Several papers explicitly identify open directions including SPDEs with multivalued drifts, ergodicity and large deviations in path-dependent multivalued mean-field settings, numerical schemes such as Euler–Maruyama under multivalued path dependence, particle approximations and propagation of chaos for multivalued interactions, weakened dissipativity for invariant measures, and extensions to jumps or rougher noises beyond the specific frameworks already treated (Ma et al., 21 Aug 2025, Qiao, 2022, Cheng et al., 19 Jul 2025).

Taken together, these developments place MMVSDEs at the intersection of maximal monotone operator theory, mean-field stochastic analysis, Wasserstein calculus, stochastic variational inequalities, and asymptotic probability. The common analytic pattern is the replacement of a nonsmooth drift inclusion by an admissible pair $b(X_t,\mu_t)$26, followed by monotonicity, coercivity, and Wasserstein stability estimates; the diversity lies in how this pattern is adapted to memory, jumps, multiscale structure, backward equations, and small-noise asymptotics.