We establish the well-posedness of stochastic scalar conservation laws on moving hypersurfaces driven by Brownian motion. To handle the interaction between stochastic forcing and evolving geometry, we derive an It\^{o} formula on moving surfaces and introduce the notion of generalized entropy solutions incorporating the relevant stochastic interaction terms. A martingale entropy solution is constructed via the vanishing-viscosity method, based on a uniform $L^\infty$-bound in space and time, an $L^1$-estimate for the spatial gradient, an $L^1$-continuity estimate in time, and a suitable tightness argument. Pathwise uniqueness is established by adapting Kruzhkov's doubling-of-variables method to moving hypersurfaces, yielding an $L^1$-contraction property. Finally, together with the Yamada-Watanabe theorem, these results yield the well-posedness of the problem.
We develop an intrinsic well-posedness theory for nonlinear backward stochastic differential equations (BSDEs) driven simultaneously by Brownian motion and a (level-$2$) rough path of finite $p$-variation. Unlike earlier approaches based on smooth approximation or transformation methods, we formulate the equation directly by viewing its solution as a rough semimartingale (in the sense of \cite{friz2023rough}). This framework is particularly suited to BSDEs, whose Brownian martingale component is only implicitly defined and lacks the \emph{a priori} time regularity required by stochastic-sewing-based controlled rough path methods \cite{fhl21,allan2024rough}. We establish comparison, existence, uniqueness, and stability under the natural regularity condition $H\in C_b^\gamma$, $\gamma>p$. The main analytical difficulty, namely, the loss of integrability arising from nonlinear composition, is overcome through conditional $p$-variation norms with BMO-type properties. Finally, by randomizing the rough driver as a Brownian rough path, we establish a direct correspondence between rough BSDEs and backward doubly stochastic differential equations (BDSDEs).
P. Friz, Jian Song, Hui-Lin Zhang et al.· 0 citations
Consider the three-dimensional primitive equations subject to Neumann wind stress driven by finitely many Brownian motions. For arbitrary smooth spatial wind profiles, global pathwise existence and uniqueness without assuming a vertical derivative of the initial datum is proved, extending the local theory of Binz, Hieber, Hussein, and Saal~\cite{BHHS-24} for this class of data and wind profiles. The proof combines a local maximal $\rL^2$ regularity construction with weighted estimates for the boundary stochastic convolution.
This work investigates the mean-field stochastic PDEs involving a class of pseudo-monotone kernels. We first study the well-posedness -- in both the strong and weak sense -- within the variational framework by introducing a notion of measure-dependent pseudo-monotone operators, which generalizes the classical framework due to Br\'{e}zis. Furthermore, we establish the quantitative dimension-free propagation of chaos within a $p$-uniformly convex Banach space for general infinite-dimensional weakly interacting systems, obtaining convergence rates that are near-optimal in a suitable sense. Our results reveal a new insight: the convergence rate of the mean-field limit is intrinsically governed by the geometry of the underlying solution space, specifically its modulus of convexity. As applications, we study several finite- and infinite-dimensional interacting particle systems arising in machine learning and fluid mechanics, including stochastic Stein variational gradient descent, mean-field Allen-Cahn equations, and Lagrangian-averaged Burgers equations.
We study a first-order stochastic transport equation driven by Brownian transport noise and a nonlinear state-dependent Poisson jump term. The drift vector field is merely integrable and satisfies the subcritical Krylov--R\"ockner condition. The dependence of the jump coefficient on the solution creates a nontrivial coupling between the solution value and its spatial gradient. To handle this coupling, we construct a stochastic characteristic system for the position, the solution value, and the gradient, and derive a characteristic representation by means of an It\^o--Wentzell formula with jumps. For singular drifts, we combine smooth approximation, the Zvonkin transformation, estimates for stochastic flows, and stochastic Gronwall inequalities to pass to a weakly differentiable limit. Uniqueness is established directly in the weakly differentiable class through a renormalized energy estimate for the difference of two solutions, including the contribution of the Poisson compensator. Under suitable integrability and differentiability assumptions on the jump coefficient, we prove the existence and uniqueness of weakly differentiable solutions. We further show that the first-order spatial Sobolev regularity of the initial datum is preserved by the evolution.
This paper studies doubly nonlinear parabolic obstacle problems. We introduce a renormalized entropy formulation in which the reaction measure is encoded by evaluating the entropy multiplier at the obstacle. For diffusion depending only on the gradient and continuous time-independent obstacles, we prove a global $L^1$ comparison estimate, uniqueness of the solution, and a minimality principle among a measure-free entropy supersolution class. For diffusion depending on both the solution and its gradient, we prove existence for continuous time-dependent obstacles with a boundary-controlled decomposition. Using two ordered penalization procedures, we obtain convergence of the approximate solutions and of the corresponding reaction terms. The limiting reaction consists of an absolutely continuous part with bounded density and a finite Radon measure concentrated on a prescribed compact spatial region. A one-sided time regularization yields strong convergence of the gradients and identifies the nonlinear flux. Together, the results yield existence, uniqueness and $L^1$ stability under their common assumptions.
This paper studies the obstacle problem for first-order scalar conservation laws driven by multiplicative noise. By adapting a barrier-substitution strategy to the kinetic formulation, we permit the reflection measure to be a general Radon measure and eliminate the usual obstacle-noise compatibility condition. We establish the existence of kinetic solutions for continuous obstacles and further derive $L^1$-contraction and uniqueness under stronger spatial regularity. Moreover, we prove a Freidlin--Wentzell large deviation principle in $L^1(0,T;L^1(\mathbb{T}^N))$. Unlike existing large-deviation arguments based on control-uniform penalization of the obstacle, we work directly with the reflected skeleton equation and retain the possibly singular Radon reflection measure throughout, while a viscous approximation provides the spatial $H^1$-regularity required for compactness before passing to the inviscid limit. Our results provide a hyperbolic counterpart to the large deviation theory for parabolic reflected SPDEs developed by Matoussi, Sabbagh, and Zhang (2021).
Ruoyang Liu, Rangrang Zhang· 0 citations
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