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.
We consider the existence and pathwise uniqueness of the stochastic heat equation with a multiplicative colored noise term on R d for d ≥ 1. We focus on the case of non-Lipschitz noise coefficients and singular spatial noise correlations. In the course of the proof a new result on Hölder continuity of the solutions near zero is established.
We show that the Hausdorff dimension of the boundary of d-dimensional super-Brownian motion is 0, if d = 1, 4 − 2 √ 2, if d = 2, and (9 − √ 17)/2, if d = 3. September 13, 2018 AMS 2000 subject classifications. Primary 60H15, 60G57. Secondary 28A78, 35J65, 60J55, 60H40, 60J80.
We consider reaction-diffusion equations of KPP type in one spatial dimension, perturbed by a Fisher-Wright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed Fisher-KPP equationswhereẆ =Ẇ (t, x) is a space-time white noise.
General stochastic equations with jumps are studied. We provide criteria for the uniqueness and existence of strong solutions under non-Lipschitz conditions of Yamada-Watanabe type. The results are applied to stochastic equations driven by spectrally positive Lévy processes.Mathematics subject classifications (2000). Primary: 60H10, 60H20; Secondary: 60J80.
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