Controlled Branching McKean-Vlasov Diffusion

• Controlled branching McKean–Vlasov diffusion is a framework that models particle dynamics with branching rates influenced by individual states and the overall population distribution.
• The system uses closed-loop control policies under Lipschitz constraints, leading to optimization problems characterized by nonlinear Fokker–Planck PDEs and HJB master equations.
• Explicit Riccati equations in linear–quadratic settings demonstrate tractable solutions for optimal feedback controls in complex stochastic environments.

A controlled branching McKean–Vlasov diffusion is a population system in which each particle’s stochastic dynamics, branching (birth-death) rates, and offspring distribution may depend both on the particle’s current position and the distribution (law) of the entire population. The system is subject to a closed-loop control policy that prescribes feedback controls stable under Lipschitz constraints. The control objective is formalized as an optimization problem on the space of probability measures, seeking to minimize (or maximize) an expected sum of running and terminal costs aggregated across all particles through their evolving measure. This framework yields a nonlinear Fokker–Planck PDE for the marginal law of the system under control and leads to an infinite-dimensional Hamilton–Jacobi–Bellman (HJB) master equation, with explicit solutions in certain linear–quadratic settings (Claisse et al., 29 Nov 2025).

1. Model Formulation

Controlled branching McKean–Vlasov diffusions generalize classical branching processes and mean-field diffusions by combining interactive branching particle genealogies with controlled stochastic flows. The population’s genealogy is encoded using the Ulam–Harris tree $K$. At time $s$, the alive set is $K_s\subset K$; the state of the full system is given by

$$Z_s = \sum_{k\in K_s} \delta_{(k,X^k_s)} \in M(K\times\mathbb{R}^d),$$

where $X^k_s\in\mathbb{R}^d$ are particle positions. Mean–field interactions depend solely on the marginal empirical measure $\mu_s \in M_2(\mathbb{R}^d)$,

$$\mu_s(dx) = \mathbb{E}\left[\sum_{k\in K_s}\delta_{X^k_s}(dx)\right].$$

Each particle $k$ evolves via a controlled stochastic differential equation (SDE), \begin{align*} dX^k_s &= b\big(s,X^k_s,\mu_s,\alpha(s,X^k_s)\big)ds + \sigma\big(s,X^k_s,\mu_s,\alpha(s,X^k_s)\big)dW^k_s, \end{align*} where $W^k$ are independent Brownian motions. Branching occurs at state-dependent rate $\gamma(s,X^k_s,\mu_s,\alpha(s,X^k_s))$, and after branching, the particle is replaced by $\ell$ offspring with probability $p_\ell(s, X^k_s, \mu_s, \alpha(s, X^k_s))$, with all offspring locations coinciding with the parent.

The control $$\alpha : [0,T]\times\mathbb{R}^d \to A\subset\mathbb{R}^n$$ is Lipschitz in $x$ and linear growth-bounded. The optimization objective uses a running cost $$L:[0,T]\times\mathbb{R}^d\times M_2(\mathbb{R}^d)\times A \to\mathbb{R}$$ and terminal cost $$g:\mathbb{R}^d\times M_2(\mathbb{R}^d)\to\mathbb{R}$$. For initial measure $\xi$ at time $t$, the objective is \begin{align*} J(t,\xi,\alpha) &= \mathbb{E}\left[\int_t^T \sum_{k\in K_s} L(s,X^k_s,\mu_s,\alpha(s,X^k_s)) ds + \sum_{k\in K_T} g(X^k_T,\mu_T) \right] \ &= \int_t^T \langle L(s,\cdot,\mu_s,\alpha(s,\cdot)), \mu_s\rangle ds + \langle g(\cdot, \mu_T), \mu_T \rangle, \end{align*} reflecting the deterministic evolution of $\mu_s$.

2. Nonlinear Fokker–Planck Equation

For a given control $\alpha$, the marginal law $\mu_s$ evolves as a measure-valued solution to a nonlinear partial differential equation incorporating both diffusion and branching mechanisms. The branching increment is given by

$$\pi(s, x, \mu, a) = \gamma(s, x, \mu, a)\sum_{\ell\ge 0} (\ell-1)p_\ell(s, x, \mu, a).$$

For any test function $\varphi \in C^2_b(\mathbb{R}^d)$,

$$\frac{d}{ds}\langle\mu_s, \varphi\rangle = \langle \mu_s, L^\alpha_s\varphi + \pi(s,\cdot, \mu_s, \alpha(s,\cdot))\,\varphi \rangle,$$

where

$$L^\alpha_s\varphi(x) = b(s, x, \mu_s, \alpha(s, x)) \cdot \nabla\varphi(x) + \tfrac12 \mathrm{Tr}\left[ \sigma \sigma^T (s, x, \mu_s, \alpha(s, x)) \nabla^2\varphi(x) \right].$$

In density form, the evolution of $\mu_s$ satisfies: \begin{align*} \partial_s \mu_s + \nabla_x \cdot \Big(b(s,x,\mu_s,\alpha(s,x))\,\mu_s\Big) = \frac{1}{2} \sum_{i,j} \partial^2_{x_i x_j}\left((\sigma\sigma^T)_{ij}(s, x, \mu_s, \alpha(s, x))\,\mu_s\right) \ + \pi(s, x, \mu_s, \alpha(s, x))\,\mu_s. \end{align*} This deterministic nonlinear equation encodes the measure evolution for mean-field control of the branching system.

3. Dynamic Programming and the Value Function

The optimal control problem is formulated for the value function $v$ defined on $[0,T]\times M_2(\mathbb{R}^d)$: \begin{align*} v(t, \nu) = \inf_{\alpha} \inf_{\xi: \mathcal{L}(\xi)=\nu} J(t, \xi, \alpha) = \inf_{\alpha}\left{ \int_t^T \langle L(s, \cdot, \mu_s, \alpha(s, \cdot)), \mu_s \rangle ds + \langle g(\cdot, \mu_T), \mu_T \rangle \right}, \end{align*} where $\mu_s$ solves the Fokker–Planck equation from initial condition $\nu$.

A dynamic programming principle (DPP) holds:

$$v(t, \nu) = \inf_{\alpha} \left\{ \int_t^s \langle L(u, \cdot, \mu_u, \alpha(u, \cdot)), \mu_u \rangle du + v(s, \mu_s) \right\},$$

for any $t \leq s \leq T$. This leverages the flow property of the controlled measure evolution and control concatenation.

4. HJB Master Equation in the Measure Space

Assuming the value function $v$ is classically regular in $C^{1,2}([0,T]\times M_2(\mathbb{R}^d))$ (using the Lions’ linear derivative with respect to measure), the infinite-dimensional HJB equation takes form: \begin{align*} \begin{cases} \partial_t v(t, m) + \inf_{a(\cdot)} \left{ \langle L(t, \cdot, m, a(\cdot)), m \rangle + \langle G^a_t v(t, m)(\cdot), m \rangle \right} = 0 \ v(T, m) = \langle g(\cdot, m), m \rangle \end{cases} \end{align*} where, for feedback law $a: \mathbb{R}^d \to A$,

$$G^a_t v(t, m)(x) = b(t, x, m, a(x)) \cdot D_\mu v(t, m, x) + \frac12 \mathrm{Tr}\big(\sigma \sigma^T \partial_x D_\mu v\big)(t, m, x) + \gamma \sum_{\ell\ge 0} (\ell-1)p_\ell\,\delta_\mu v(t, m, x).$$

Viscosity and classical solution concepts are distinguished by the regularity and test-function admissibility in this infinite-dimensional setting.

5. Verification Theorem and Optimal Feedback

Under the existence of a classical solution $u \in C^{1,2}$ to the HJB master equation and for every $(t, m)$ a feedback $\hat a(t, \cdot)$ attaining the infimum, the optimal closed-loop control is

$\alpha^*_s(x) = \hat a(s, x, \mu^{t,m}_s).$

With this feedback, the equality $u(t, m) = v(t, m)$ holds, certifying both optimality and attainment by the value function.

6. Explicit Solution in the Linear–Quadratic Setting

As a concrete example, specialization to the linear–quadratic case in one dimension ($d=1$), with constant branching intensity $\gamma$ and fixed offspring law $p_\ell$, yields tractable Riccati equations:

• Drift and diffusion:

$$b(s, x, m, a) = b_1(s)x + b_2(s)m(\mathbb{R}) + b_3(s)a, \qquad \sigma(s, x, m, a) = \sigma.$$

• Cost functionals:

$$L(s,x,m,a) = L_1(s)x^2 + L_2(s)\bar m + L_3(s)m_1 + L_4(s)a^2, \qquad g(x,m) = g_1 x^2 + g_2 \bar m + g_3 m_1,$$

where $\bar m = m(\mathbb{R})$ and $m_1 = \int x\, m(dx)$.

• Value function Ansatz:

$$v(t,m) = \Lambda(t)m_2 + \Gamma_1(t)\bar m + \Gamma_2(t)\bar m m_1 + \Gamma_3(t)\bar m^2 + \Gamma_4(t)\bar m^3,$$

with $m_2 = \int x^2\, m(dx)$.

Inserting into the HJB yields a coupled Riccati ODE system for $(\Lambda, \Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4)$. The optimal feedback is pointwise:

$$\alpha^*(t, x, m) = -\frac{1}{2L_4(t)} \left[ 2b_3(t)\Lambda(t)x + b_3(t)\Gamma_2(t)\bar m \right].$$

Direct substitution verifies $(v, \alpha^*)$ as optimal and admissible, thereby solving the control problem in this regime and establishing explicit characterization of optimal closed-loop dynamics (Claisse et al., 29 Nov 2025).