Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case

Abstract

Abstract. We prove pathwise uniqueness for solutions of parabolic stochastic pde's with multiplicative white noise if the coefficient is Hölder continuous of index γ > 3/4. The method of proof is an infinite-dimensional version of the Yamada-Watanabe argument for ordinary stochastic differential equations.

Stochastic Analysis: A Series of Lectures

“…Using superprocess methods, it was shown that if γ < 3/4 then there exists a second solutionũ t (x) which is nonzero with positive probability. This result complements a result of Mytnik and Perkins [MP11], who showed that strong uniqueness holds for the equation…”

On the boundary of the zero set of super-Brownian motion and its local time

“…(1.2) BZ t = ∂{x : X(t, x) = 0} = ∂{x : X(t, x) > 0}, was studied in [10]. The increased regularity of X on and near this set has played an important role in the study of SPDE's such as (1.1) (see [12] and [11]). Mytnik and Perkins (unpublished) had obtained side conditions on X which would give pathwise uniqueness in (1.1) but which would imply that dim(BZ t ), the Hausdorff dimension of BZ t , is zero.…”

“…In fact numerical estimates of λ 0 due to Peiyuan Zhu suggest that (1.4) implies (1.5) dim(BZ t ) ≈ .224 a.s. on {X t (1) > 0}, perhaps larger than one may think given that X(t, ·) is Hölder 1−η in space near its zero set for any η > 0 (see Theorem 2.3 in [11]). We briefly discuss this approximation below and give some evidence for the accuracy of the estimate to the digits given.…”

Stochastic equations of super-Lévy processes with general branching mechanism

“…The main difficulty comes from the unbounded drift coefficient and the non-Lipschitz diffusion coefficient. See, e.g., [10,11] for some important progresses in the subject. A different approach for pathwise uniqueness of the super-Brownian motion was suggested in the recent work of Xiong [17], where a stochastic equation for the distribution function process of {X t : t ≥ 0} was formulated and the strong existence and uniqueness for the equation were established.…”