A McKean--Vlasov equation with positive feedback and blow-ups

Abstract

We study a McKean--Vlasov equation arising from a mean-field model of a particle system with positive feedback. As particles hit a barrier they cause the other particles to jump in the direction of the barrier and this feedback mechanism leads to the possibility that the system can exhibit contagious blow-ups. Using a fixed-point argument we construct a differentiable solution up to a first explosion time. Our main contribution is a proof of uniqueness in the class of càdlàg functions, which confirms the validity of related propagation-of-chaos results in the literature. We extend the allowed initial conditions to include densities with any power law decay at the boundary, and connect the exponent of decay with the growth exponent of the solution in small time in a precise way. This takes us asymptotically close to the control on initial conditions required for a global solution theory. A novel minimality result and trapping technique are introduced to prove uniqueness.

Figures (5)

• Figure 1.1: Example of a solution to (\ref{['eq:Intro_PDEproblem']}, \ref{['eq:Intro_DensityPDE']}) showing two blow-up times. Pixel intensity represents the value of the solution density at that space-time coordinate. The initial condition is a linear combination of indicator functions of three disjoint sets.
• Figure 1.2: On the left, $V_{t-}$ is the density just before a jump of size $\Delta L_t$. This density is then translated by $\alpha \Delta L_t$ and the mass falling below the boundary at zero equals the change in the loss, which gives (\ref{['eq:Intro_JumpCondition']}). After the jump, the system is restarted from the density on the right. Notice that, in general, this new initial condition will not vanish at the origin.
• Figure 2.1: The function from Example \ref{['MinimalJumps_Ex_SomeV0']} is on the left. The candidate jumps --- that is, solutions to (\ref{['eq:Intro_JumpCondition']}) --- are the points on the right where the graphs intersect. The point $(1 ,1)$ gives the minimal allowed jump size of $1$, since $x = 1$ is the value given by (\ref{['eq:Intro_PhysicalJumpCondition']}).
• Figure 5.1: Given an initial density, $V_0$, the $\varepsilon$-deleted initial condition constructed in (\ref{['eq:Boot_Contruction_F_solution']}) is obtained by killing the mass on $(0,\varepsilon)$ and shifting the density towards the origin by the amount $\alpha\nu_0(0,\varepsilon) + \tfrac{1}{4} \varepsilon$. The proof of Lemma \ref{['Boot_Lem_epsilonSolution']} shows us that $V^\varepsilon_0$ vanishes in a neighbourhood of zero.
• Figure 5.2: On a small time interval, the unique differentiable solution, $L$, from Corollary \ref{['Boot_Cor_Bootstrap']} and the $\varepsilon$-deleted solutions from Definition \ref{['Boot_Def_DeletedSolution']} trap any candidate càdlàg solution, $\bar{L}$, by Lemma \ref{['Boot_Lem_Monotonicity']}. Here $\eta > \varepsilon$. Lemma \ref{['Boot_Lem_Convergence']} shows that $L^\varepsilon \to L$ uniformly on a small time interval as $\varepsilon \to 0$, and so we see that $L = \bar{L}$ is forced since the area between the curves above shrinks to zero.

Theorems & Definitions (36)

• Theorem 1.1: Blow-up for large $\alpha$
• Proposition 1.2: Physical solutions have minimal jumps
• Conjecture 1.3: Global uniqueness
• Conjecture 1.4: Uniqueness for non-vanishing initial laws
• Definition 1.5
• Theorem 1.6: Contraction and minimality
• Theorem 1.7: Stability and fixed point
• Theorem 1.8: Uniqueness up to explosion
• Remark 1.9: Propagation of chaos
• Proposition 2.1: Existence of a density process