On strong solution of a multidimensional SDE: extension of Yamada -- Watanabe's theorem

TL;DR

This work addresses strong well-posedness for a multidimensional SDE with diagonal, non-degenerate diffusion and a drift that is partially irregular. It develops a localized Yamada--Watanabe framework and a constructive PDE-based transform: solving decoupled elliptic equations $L^i u^i = 0$ yields a coordinate change $u$ with $u^i'(x^i)>0$, and Ito--Krylov is used to transfer the problem to transformed coefficients $\hat b, \hat\sigma$ that preserve divergence-type continuity. Under assumptions $b_1$ has modulus $\rho_b$ with $\int_0^1 \rho_b^{-1}(s) ds = \infty$, $\sigma$ has modulus $\rho_\sigma$ with $\int_0^1 \rho_\sigma^{-2}(s) ds = \infty$, and $\sigma^2$ is uniformly non-degenerate, the paper proves pathwise uniqueness via a localized-to-global argument. By combining Yamada--Watanabe, Zvonkin-type transformations, and localization, the results extend strong uniqueness to higher dimensions with partially irregular drift, offering a constructive method for ensuring well-posedness in this class of SDEs.

Abstract

A new strong uniqueness result for a multidimensional SDE with a non-degenerate diffusion and partially irregular drift is established. It may be regarded as a combined variation on the themes of Yamada \& Watanabe (1971), of Zvonkin (1974), and of the second author of the present paper (1980).

Theorems & Definitions (8)

• Proposition 1: YW71a
• Proposition 2: Localized Yamada -- Wanatabe theorem
• Remark 1
• Remark 2
• Theorem 1
• Remark 3
• Remark 4
• Lemma 1: simplified Bihari -- LaSalle lemma