Finite Explosion for One-dimensional Lévy Driven SDEs and its Application to SPDEs

This paper addresses finite-time explosion of one-dimensional stochastic differential equations (SDEs) driven by pure-jump Lévy processes $$X_t^x=x-{\int }_0^{t}b(X_s^x)\text{dd}s+L_t,$$ where the drift coefficient $b$ is locally Lipschitz continuous, and $(L_t{)}_{t\ge 0}$ is a pure jump Lévy process. We formulate three sets of conditions: a right-tail return condition; a nondegeneracy condition together with a left-tail Osgood bound; and a right-tail Osgood bound. The first two imply almost-sure explosion to $-\mathrm{\infty }$ from every finite initial state, whereas the latter two imply a uniform bound on the mean explosion time. As an application, we give explicit conditions on the nonlinearity and the Lévy measure under which every local weak solution of a semilinear parabolic stochastic partial differential equation (SPDE) with additive Lévy space–time white noise and homogeneous Dirichlet boundary conditions has an almost surely finite lifetime.