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

Abstract

In this paper, we prove a sample-path comparison principle for the nonlinear stochastic fractional heat equation on R with measure-valued initial data. We give quantitative estimates about how close to zero the solution can be. 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. As an intermediate step, we prove the Hölder regularity of the solution starting from measure-valued initial data, which generalizes, in some sense, a recent work by Chen and Dalang [6].MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.

Comparison principle for stochastic heat equation on $\mathbb{R}^{d}$

“…Similar small-ball probabilities in various settings can be found in [8,9,13,20]. These nonnegativity statements can be translated into comparison statements by considering v = u 1 − u 2 .…”

“…Step 1: Let u ǫ,1 (t, x) and u ǫ,2 (t, x) be the solutions to (7.6) with initial data µ 1 and µ 2 , respectively. Following exactly the same lines as those in Step 2 of the proof in [8], we can prove that v ǫ (t, x) := u ǫ,2 (t, x) − u ǫ,1 (t, x) satisfies P v ǫ (t, x) ≥ 0, for every t > 0 and x ∈ R d = 1 . (7.8)…”

“…The proof consists of four steps. Both the setup and Steps 1 & 4 of the proof follow the same lines as those in the proof of Theorem 1.1 in [8] with some minor changes. The main difference lies in Step 2 and Step 3.…”

“…Proof. Since the d-dimensional heat kernel can be factored as a product of one-dimensional heat kernel, so the proof will be parallel with the proof of Lemma 4.1 in [8]. We will not repeat it here.…”

Dissipation in Parabolic SPDEs

“…We may apply the strong Markov property of u, with respect to F , at the stopping time T n in order to see that for all integers n 0, 1] solves (1.8) starting from the random initial profile v (n) 0 (x) := v (n) (0 , x) = u(T n , x ; λ). By the very definition of the stopping time T n , and since T n is finite a.s., v (n) 0 (x) = u(T n , x ; λ) e −1 inf y∈T u(T n−1 , y ; λ) · · · e −n inf y∈T u 0 (y) =: e −n u 0 , a.s. for every x ∈ T, and with identity for some x ∈ T a.s. Because u 0 > 0 [see (1.2)], it follows from a comparison theorem [3,21,25] that v (n) (t , x) w (n) (t , x) for all t 0 and x ∈ T a.s., where w (n) solves (1.8) [for a different Brownian sheet] starting from w (n) 0 (x) := w (n) (0 , x) = e −n u 0 . In particular, for all integers n 0 and reals τ ∈ (0 , 1),…”