At the Mercy of the Common Noise: Blow-ups in a Conditional McKean--Vlasov Problem

Abstract

We extend a model of positive feedback and contagion in large mean-field systems, by introducing a common source of noise driven by Brownian motion. Although the driving dynamics are continuous, the positive feedback effect can lead to `blow-up' phenomena whereby solutions develop jump-discontinuities. Our main results are twofold and concern the conditional McKean--Vlasov formulation of the model. First and foremost, we show that there are global solutions to this McKean--Vlasov problem, which can be realised as limit points of a motivating particle system with common noise. Furthermore, we derive results on the occurrence of blow-ups, thereby showing how these events can be triggered or prevented by the pathwise realisations of the common noise.

Figures (3)

• Figure 1.1: The plots show two different realisations of (\ref{['MV']}) with $\rho = 0.5$ and the same initial condition. Each pixel represents the value of the density at that space-time point, with space on the vertical and time on the horizontal. On the right, the common noise, $B^0$, decreases sufficiently quickly to cause a blow-up. See Section \ref{['sec:numerics']} for the numerical algorithm.
• Figure 2.1: The figure displays the emergence of a blow-up. In the middle picture, the overall effect of the common noise is a prompt transportation of mass towards the origin, moving the system from $V_0$ to $V_s$. Since $V_s$ is concentrated near the origin, an adaptation of hambly_ledger_sojmark_2018 shows that there must be a blow-up, provided too much mass does not start escaping in the other direction. In this regard, the rightmost picture illustrates the nonlinear feedback becoming so strong that, after taking a limit as $s\uparrow t$, the resulting left-limit density $V_{t-}$ lies above $\alpha^{-1}$ near zero and hence $\Delta L_t \neq0$ in line with \ref{['eq:BU_PhysicalCondition']}.
• Figure 2.2: The figure displays an event where a blow-up is avoided, because the common noise keeps the mass away from the origin for an adequate amount of time. Specifically, the picture on the right illustrates how the common noise can allow sufficient elbow room for the diffusive effect to spread out the mass, so that the density $V_t$ ends up lying everywhere strictly below $\alpha^{-1}$. From here, a blow-up cannot occur at any later times, for this realisation of the common noise, since we can no longer get into a situation like the rightmost picture of Figure \ref{['fig:the2ndfigure']}.

Theorems & Definitions (27)

• Remark 1.1: Stochastic evolution equation
• Theorem 2.1: Blow-ups
• Lemma 2.2: Existence of densities
• Corollary 2.3: Initial curbs on blow-up
• Proposition 2.4: Linear spatial bounds for blow-up
• Proposition 2.5: Blow-up for transformed loss
• Example 2.6: Forcing blow-up
• Example 2.7: Averting blow-up
• Lemma 2.8: No-crossing lemma
• Proposition 2.9: Non-certain blow-up