Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation

Abstract

We study stochastic reaction-diffusion equationwhere b is a generalized function in the Besov space B β q,∞ (R), D ⊂ R and Ẇ is a space-time white noise on R + × D. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever β −. This class includes equations with b being measures, in particular, b = δ 0 which corresponds to the skewed stochastic heat equation. For β − 1/q > −3/2, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalizes the results of to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in Lê (2020).

Regularisation by fractional noise for one-dimensional differential equations with nonnegative distributional drift

“…We first state a lemma which establishes various regularity estimates on the conditional expectation of fractional Brownian motion. It extends to the fBm the Lemma C.4 of [2] (which was for standard Brownian motion). It is used multiple times in the remainder of the paper and its proof is postponed to Appendix C. In particular, the proof of Lemma 5.1(b) relies on a new property of local nondeterminism of the fBm, see Lemma C.1.…”

“…In Section 5, we use some new regularity estimates on conditional expectations of the fBm (Lemma 5.1), and the stochastic sewing Lemma with random control (see Theorem 4.6 in [2]) to establish that weak solutions are in…”

“…In Section 6, in order to establish pathwise uniqueness of weak solutions to (1.1) (see Proposition 6.1), we adapt an approach developed recently for the stochastic heat equation with distributional drift, see [2]. This requires several regularity estimates on solutions which are derived from the crucial regularity Lemma D.2 ad the stochastic sewing Lemma with critical exponent (Theorem 4.4 in [2]). The proof of Lemma D.2 relies on the stochastic sewing Lemma and the aforementioned regularity estimates on conditional expectations of the fBm (Lemma 5.1).…”

“…We will apply the stochastic sewing Lemma with random controls as formulated in [2,Theorem 4.6]. In order to see that all conditions of this theorem are fulfilled (i.e.…”

“…To prove this theorem, we followed the recent approach of Athreya et al [2]. In this paper, the authors proved the strong well-posedness of the one-dimensional stochastic heat equation with Besov drift by a tightness-stability argument.…”