Finite Explosion for One-dimensional L\'evy Driven SDEs and its Application to SPDEs

Abstract

We study finite-time explosion to $-\infty$ for the one-dimensional stochastic differential equation \[ X_t^x=x-\int_0^t b(X_s^x)\,\dd s+L_t,\quad x\in\R, \] where $b$ is locally Lipschitz and $(L_t)_{t\ge0}$ is a pure-jump L\'evy process. We give verifiable conditions involving Osgood-type integrability under which explosion occurs almost surely from every finite initial state. Under stronger tail conditions, we obtain a uniform bound on the mean explosion time. In particular, under suitable tail monotonicity, converse results make these criteria necessary and sufficient, and yield an explicit classification for Riccati drifts. As an application, we consider a semilinear parabolic stochastic partial differential equation (SPDE) with additive L\'evy space--time white noise and homogeneous Dirichlet boundary conditions. We give explicit conditions on the nonlinearity and the L\'evy measure under which every local weak solution defined up to explosion has an almost surely finite explosion time.