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

Abstract

We study stochastic reaction–diffusion equation where b is a generalized function in the Besov space , and is a space‐time white noise on . We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever , and . This class includes equations with b being measures, in particular, which corresponds to the skewed stochastic heat equation. For , 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 generalize the results of Gyöngy and Pardoux (1993) 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).

An extension of the stochastic sewing lemma and applications to fractional stochastic calculus

“…is 𝛿𝛼-Hölder continuous and 𝛿𝛼 + 𝛼 > 1 by our assumption of 𝛿. Therefore, 𝑥 (1) − 𝑥 (2) is a solution of the Young differential equation…”

“…The proof of Proposition 5.1 suggests that the pathwise uniqueness holds if, for any two (F 𝑡 )-adapted solutions 𝑋 (1) and 𝑋 (2) to (5.4), we can construct the integral…”

“…If 𝜃 𝑋 (1) 𝑠 + (1 − 𝜃) 𝑋 (2) 𝑠 is replaced by 𝐵 𝑠 , then the integral is constructed in Proposition 3.3. The difficulty here is that 𝑋 (𝑖) is not Gaussian and the Wiener chaos decomposition crucially used in the proof of Proposition 3.3 cannot be applied.…”

“…When ( 𝐴 𝑠,𝑡 ) 𝑠 ≤𝑡 is random, and when we want to prove the convergence of the sums (1.1), the above sewing lemma is often not sufficient. For instance, if 𝐴 𝑠,𝑡 := (𝑊 𝑡 − 𝑊 𝑠 ) 2 , the sums converge to the quadratic variation of the Brownian motion. However, we only have |𝛿𝐴 𝑠,𝑢,𝑡 (𝜔)| 𝜀, 𝜔 |𝑡 − 𝑠| 1−𝜀 almost surely for every 𝜀 > 0, and hence, we cannot apply the sewing lemma.…”

“…Finally, we remark that one of the most interesting applications of Lê's stochastic sewing lemma lies in the phenomenon of regularization by noise (see, e.g. [24], Athreya et al [2], [16], and Anzeletti et al [1]). In these works, they consider the stochastic differential equation (SDE)…”