Long-time existence for signed solutions of the heat equation with a noise term

Abstract

Let I be the circle 0Y t with the ends identi®ed. We prove longtime existence for the following equation.Here, tY x is 2-parameter white noise, and we assume that u 0 x is a continuous function on I. We show that if gu grows no faster than g 0 1 juj c for some c`3a2, g 0 b 0, then this equation has a unique solution utY x valid for all times t b 0.

Higher-order stochastic partial differential equations with branching noises

“…However, the tightness is still useful in obtaining weak convergence, which is similar as in Refs. [10,12,[15][16][17]. As to the tightness arguments here, we will need some fundamental estimates on the Green kernel corresponding to the operator ∂ ∂t + ∆ 2 .…”

“…Refs. [16][17][18]). Therein a powerful tool is the comparison theorem, and the (mild) solution of the equation is positive if the initial u 0 is positive.…”

Can the stochastic wave equation with strong drift hit zero?

“…As for SPDE analogues of the Bessel process, the only results known to the authors are in Mueller [Mue98] and Mueller and Pardoux [MP99]. Here we assume that u(t, x) is scalar valued, and as before t > 0.…”

“…Let τ be the first time at which u hits 0, and let τ = ∞ if u does not hit 0. Then P(τ < ∞) > 0 if α < 3, see [Mue98] Corollary 1.1. Also, P(τ < ∞) = 0 if α > 3, see Theorem 1 of [MP99].…”

The critical parameter for the heat equation with a noise term to blow up in finite time

“…Krylov [Kry94] gave another proof of this fact for a more general class of equations. The papers [Mue97a] and [Mue97b] are also relevant. We refer the reader to Pardoux [Par93] for this and other questions about parabolic SPDE.…”