Stochastic explosion and non-uniqueness for α-Riccati equation

Abstract We consider the problem of global in time existence and uniqueness for the initial value problems u ′ ( t ) = − u ( t ) + u 2 ( α t ) , u ( 0 ) = u 0 ≥ 0 for scaling parameters α ≥ 0 and a large class of initial data u 0 . Much of the focus is on the “super-critical” case α > 1 where by using the multiplicative stochastic cascade techniques we prove global existence for small initial data and finite-time blow-up for large initial data. However, while uniqueness holds for solutions satisfying a growth condition, in general it fails even for arbitrary small initial data. We demonstrate that this lack of uniqueness is directly connected to the stochastic explosion of an associated multiplicative stochastic cascade process. The key tool in establishing the above-mentioned results is an iterative algorithm that allows one to exploit the stochastic explosion of the underlying multiplicative cascade to establish both existence and lack of uniqueness of solutions.