Regularity of the Value Function in Discounted Infinite-Time Mean Field Games

Abstract: In [17], we introduced the discounted infinite-time mean field games. Subsequently, in [18], we studied the connection between infinite-time mean field FBSDEs and elliptic master equations. In this paper, we further investigate the regularity of the representative player's value function. Specifically, we first prove the strong existence and uniqueness, as well as the uniqueness in law, for an extended class of infinite-time FBSDEs. We then establish the Lions-differentiability for the derivative of the representative player's value function with respect to the measure argument, and provide an explicit characterization for it using solutions to FBSDEs.

• The paper proves strong existence, uniqueness, and uniqueness in law for a broad class of discounted infinite-time McKean–Vlasov FBSDEs under monotonicity and Lipschitz conditions.
• The paper constructs the Lions derivative of the value function with respect to the population measure through auxiliary linear infinite-time FBSDEs, overcoming the lack of uniform directional continuity in the infinite-horizon setting.
• The paper establishes a bounded, jointly continuous measure derivative and upgrades directional differentiability to a Fréchet expansion, yielding C¹ regularity in the measure variable for the elliptic master-equation analysis.

This paper develops the regularity theory for the value function of a representative player in discounted infinite-time mean field games, extending the program begun in the authors' earlier works [(2603.15141) context: arXiv (Yang et al., 21 May 2025) and (Yang et al., 4 Oct 2025)]. Two main contributions are established: first, strong existence and uniqueness together with uniqueness in law for an extended class of infinite-time (McKean–Vlasov) FBSDEs; second, Lions-differentiability of the value function $V(x,\mu)$ with respect to its measure argument, with an explicit representation of $\partial_\mu V$ via solutions of auxiliary infinite-time FBSDEs.

Setting and motivation

The framework is the discounted infinite-time mean field game introduced previously by the authors. The equilibrium state process $X^\xi$ and the representative player's process $X^{x,\xi}$ are governed by two coupled infinite-time FBSDEs driven by the Hamiltonian $$H(x,\mu,y)=\min_{a\in\mathbb{R}}[b(x,\mu,a)\cdot y+f(x,\mu,a)]$$ with discount factor $r>0$. The value function

$$V(x,\mu)\triangleq Y_0^{x,\xi},\qquad \mathcal{L}_\xi=\mu,$$

was shown in prior work to be a viscosity solution of the elliptic master equation on the Wasserstein space $\mathscr{P}_2$. In finite-horizon mean field games, Lions-differentiability of $V(t,x,\mu)$ follows directly from differentiability of the associated finite-time FBSDEs, and explicit formulas exist via the works of Gangbo–Mészáros–Mou–Zhang and Mou–Zhang. The paper's central obstacle is that this route fails in the infinite-time setting: the continuity of the directional-derivative functional in $\xi$ obtained from linearized FBSDEs is merely pointwise and not uniform over perturbation directions $\partial_\mu V$0, which precludes the classical argument. The paper therefore constructs the Lions-derivative by an alternative route.

Strong solvability and uniqueness in law for infinite-time FBSDEs

The paper first considers the general infinite-time McKean–Vlasov FBSDE

$\partial_\mu V$1

with $\partial_\mu V$2 a given adapted square-integrable input process, under Assumption (monotonicity-type condition): a Lipschitz estimate in $\partial_\mu V$3 and the conditional law, plus a coercivity inequality

$\partial_\mu V$4

for some $\partial_\mu V$5. Under these conditions, existence and uniqueness hold in the exponentially weighted space $\partial_\mu V$6, with norm $\partial_\mu V$7. Uniqueness follows from Itô's formula applied to $\partial_\mu V$8 along a sequence $\partial_\mu V$9; existence uses the continuity method, interpolating between the decoupled system at $X^\xi$0 (solvable by a lemma from Bayraktar–Zhang) and the target system at $X^\xi$1. The contraction constant is explicit, $X^\xi$2, and the procedure iterates finitely many steps to reach $X^\xi$3. This result generalizes Shi–Zhao's elliptic PDE connection and Bayraktar–Zhang's McKean–Vlasov extension to a broader class admitting external adapted inputs $X^\xi$4 and pathwise dependence through $X^\xi$5.

The paper then adapts the Yamada–Watanabe scheme to this class: if the FBSDE has a unique strong solution on any set-up with inputs $X^\xi$6, then the law of $X^\xi$7 depends only on $X^\xi$8, and there exists a measurable "solution map" $X^\xi$9 such that $X^{x,\xi}$0 a.s. The proof glues two solutions via regular conditional laws on an enlarged space and verifies that the glued driving noise remains a Brownian motion relative to the canonical filtration; the argument requires the absolute continuity of $X^{x,\xi}$1 and recovers $X^{x,\xi}$2 as a derivative of $X^{x,\xi}$3. Uniqueness in law is essential later: it licenses identifying quantities computed on different probability spaces, which is precisely what makes $X^{x,\xi}$4 a function of the law of $X^{x,\xi}$5 alone.

Existence of the directional derivative

Linearizing the equilibrium and representative-player FBSDEs in the initial condition $X^{x,\xi}$6 along direction $X^{x,\xi}$7 yields two linear infinite-time FBSDEs whose solutions $X^{x,\xi}$8 and $X^{x,\xi}$9 satisfy

$$H(x,\mu,y)=\min_{a\in\mathbb{R}}[b(x,\mu,a)\cdot y+f(x,\mu,a)]$$0

Under the standing Assumption (at most quadratic growth, Lipschitz derivatives up to mixed measure derivatives, and monotonicity constants satisfying $$H(x,\mu,y)=\min_{a\in\mathbb{R}}[b(x,\mu,a)\cdot y+f(x,\mu,a)]$$1), these systems admit unique $$H(x,\mu,y)=\min_{a\in\mathbb{R}}[b(x,\mu,a)\cdot y+f(x,\mu,a)]$$2 solutions. A key lemma shows that, for fixed $$H(x,\mu,y)=\min_{a\in\mathbb{R}}[b(x,\mu,a)\cdot y+f(x,\mu,a)]$$3, the map $$H(x,\mu,y)=\min_{a\in\mathbb{R}}[b(x,\mu,a)\cdot y+f(x,\mu,a)]$$4 is a bounded linear functional on the Hilbert space $$H(x,\mu,y)=\min_{a\in\mathbb{R}}[b(x,\mu,a)\cdot y+f(x,\mu,a)]$$5; the bound is obtained by energy estimates from Itô's formula applied to $$H(x,\mu,y)=\min_{a\in\mathbb{R}}[b(x,\mu,a)\cdot y+f(x,\mu,a)]$$6 and $$H(x,\mu,y)=\min_{a\in\mathbb{R}}[b(x,\mu,a)\cdot y+f(x,\mu,a)]$$7, using the sign conditions $$H(x,\mu,y)=\min_{a\in\mathbb{R}}[b(x,\mu,a)\cdot y+f(x,\mu,a)]$$8 and $$H(x,\mu,y)=\min_{a\in\mathbb{R}}[b(x,\mu,a)\cdot y+f(x,\mu,a)]$$9. By Riesz representation there exists $r>0$0 with $r>0$1.

Two structural facts then identify $r>0$2 with a version of $r>0$3:

• Measurability reduction: a contradiction argument using an i.i.d. copy $r>0$4 of any hypothetical residual component shows $r>0$5 a.s., so $r>0$6 for a Borel $r>0$7.
• Law-dependence: weak uniqueness implies that for $r>0$8 with common law $r>0$9, the resulting functions agree $$V(x,\mu)\triangleq Y_0^{x,\xi},\qquad \mathcal{L}_\xi=\mu,$$0-a.e., since $$V(x,\mu)\triangleq Y_0^{x,\xi},\qquad \mathcal{L}_\xi=\mu,$$1 for all bounded continuous $$V(x,\mu)\triangleq Y_0^{x,\xi},\qquad \mathcal{L}_\xi=\mu,$$2.

Consequently there exists a Borel function $$V(x,\mu)\triangleq Y_0^{x,\xi},\qquad \mathcal{L}_\xi=\mu,$$3 satisfying

$$V(x,\mu)\triangleq Y_0^{x,\xi},\qquad \mathcal{L}_\xi=\mu,$$4

The same construction applies when $$V(x,\mu)\triangleq Y_0^{x,\xi},\qquad \mathcal{L}_\xi=\mu,$$5 is written in its integral form via $$V(x,\mu)\triangleq Y_0^{x,\xi},\qquad \mathcal{L}_\xi=\mu,$$6. At this stage the derivative exists only as a Gâteaux (directional) derivative; upgrading it to a genuine Lions-derivative requires a uniformly bounded, jointly continuous version, which is the subject of the next section. It is worth noting that the authors explicitly acknowledge that the pointwise continuity available here does not suffice for the standard finite-time argument — the entire alternative construction below exists because of this gap.

Representation and continuity of the Lions-derivative

For discrete $$V(x,\mu)\triangleq Y_0^{x,\xi},\qquad \mathcal{L}_\xi=\mu,$$7 taking values $$V(x,\mu)\triangleq Y_0^{x,\xi},\qquad \mathcal{L}_\xi=\mu,$$8 with probabilities $$V(x,\mu)\triangleq Y_0^{x,\xi},\qquad \mathcal{L}_\xi=\mu,$$9, the directional derivative in direction $\mathscr{P}_2$0 satisfies

$\mathscr{P}_2$1

The paper introduces two coupled FBSDEs: one solved jointly by pairs $\mathscr{P}_2$2 tracking perturbations localized at $\mathscr{P}_2$3 both within the equilibrium system and within the tagged component $\mathscr{P}_2$4 of the population, and a "total" system computing $\mathscr{P}_2$5. Because all equations are linear and their solutions are explicit, the relation

$\mathscr{P}_2$6

is verified by direct substitution, exploiting the decomposition $\mathscr{P}_2$7 and independence of the localized processes from $\mathscr{P}_2$8. Hence $\mathscr{P}_2$9 on the support of discrete measures.

For absolutely continuous $V(t,x,\mu)$0, the strategy approximates $V(t,x,\mu)$1 by quantizations $V(t,x,\mu)$2 on grids of mesh $V(t,x,\mu)$3 truncated at $V(t,x,\mu)$4. Three auxiliary FBSDEs are introduced: the spatial-derivative system with unit initial condition ($V(t,x,\mu)$5, $V(t,x,\mu)$6), a system driven only through the equilibrium pair, and a total system combining both channels. Lemma (boundedness and continuity) establishes that these admit unique $V(t,x,\mu)$7 solutions whose maps into $V(t,x,\mu)$8 are uniformly bounded and jointly continuous; define $V(t,x,\mu)$9, well-defined independently of the choice of $\xi$0 by law-uniqueness. The critical limiting lemma then shows, for absolutely continuous $\xi$1, that the discrete-system solutions converge in $\xi$2:

$\xi$3

and analogously for the star-perturbed and total systems. The proof hinges on the uniform $\xi$4 boundedness of all solutions across initial values, which permits dominated convergence; the mass term $\xi$5 vanishes exactly because $\xi$6 has no atoms, killing the coupling terms involving $\xi$7.

Combining the pieces, the main theorem states: defining $\xi$8, the function $\xi$9 is a version of $\partial_\mu V$00 for every $\partial_\mu V$01, including singular $\partial_\mu V$02 via approximation by absolutely continuous measures. Since $\partial_\mu V$03 is bounded and jointly continuous on $\partial_\mu V$04, the Gâteaux derivative strengthens to a Fréchet expansion

$\partial_\mu V$05

which is precisely the definition of the Lions-derivative. An immediate consequence is that $\partial_\mu V$06 in the measure variable, complementing the known viscosity-solution property of $\partial_\mu V$07 for the elliptic master equation and providing the probabilistic ingredient typically needed for classical (rather than viscosity) wellposedness results on the master equation.

Limitations and open questions

Several qualifications attach to these results. All regularity conclusions require the strong Assumption on $\partial_\mu V$08: Lipschitz continuity of derivatives through order two including the mixed measure derivatives $\partial_\mu V$09, uniform bounds $\partial_\mu V$10, and the spectral-type condition $\partial_\mu V$11 tying the discount rate to the monotonicity constants. Whether Lions-differentiability persists under weaker displacement-monotonicity or nonseparable Hamiltonians, as handled in finite time by Gangbo et al., is not addressed. The solvability theorem treats scalar equations started at $\partial_\mu V$12, with multidimensional and arbitrary-start-time extensions asserted but not proved in detail. Finally, the analysis yields $\partial_\mu V$13 regularity in $\partial_\mu V$14 only; whether $\partial_\mu V$15 possesses higher-order Lions-differentiability sufficient for a classical solution of the elliptic master equation remains open, as does the corresponding convergence problem for the $\partial_\mu V$16-player game.

Conclusion

The paper completes a three-part development of discounted infinite-time mean field games: it supplies the wellposedness infrastructure (strong existence/uniqueness and Yamada–Watanabe uniqueness in law for a broad class of infinite-time McKean–Vlasov FBSDEs) and proves that the representative player's value function is Lions-differentiable in the measure argument, with the derivative explicitly represented as the initial backward component of a computable linear infinite-time FBSDE and shown to be bounded and continuous. The methodological departure from the finite-time literature — bypassing uniform-in-$\partial_\mu V$17 continuity by combining Riesz representation, weak uniqueness, and discrete-to-absolutely-continuous quantization limits — is the technical core, and the resulting $\partial_\mu V$18 regularity places the elliptic master equation analysis on a footing comparable to its parabolic counterpart.