On the (strict) positivity of solutions of the stochastic heat equation

Abstract

We give a new proof of the fact that the solutions of the stochastic heat equation, started with non-negative initial conditions, are strictly positive at positive times. The proof uses concentration of measure arguments for discrete directed polymers in Gaussian environments, originated in M. Talagrand's work on spin glasses and brought to directed polymers by Ph. Carmona and Y. Hu. We also get slightly improved bounds on the lower tail of the solutions of the stochastic heat equation started with a delta initial condition.

On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations

“…These results extend Mueller's comparison principle on the stochastic heat equation to allow more general initial data such as the (Dirac) delta measure and measures with heavier tails than linear exponential growth at ±∞. These results generalize a recent work by Moreno Flores [25], who proves the strict positivity of the solution to the stochastic heat equation with the delta initial data. As one application, we establish the full intermittency for the equation.…”

“…The authors also thank Carl Mueller for many helpful suggestions. The authors thank Tom Alberts for some interesting discussions and for his pointing out the reference [25]. The first author thanks Robert Dalang and Roger Tribe for many interesting discussions.…”

Universality for the pinning model in the weak coupling regime

“…In this section we prove Proposition 2.8. Taking inspiration from [17], we first prove (2.27), using concentration results, and later we prove (2.26). We start with some preliminary results.…”

Renormalizing the Kardar–Parisi–Zhang Equation in $$d\ge 3$$ in Weak Disorder

“…We invoke a second moment method combined with the Talagrand's concentration inequality as in [5] (see also [22,Sect. 2.2]).…”