In this paper, we establish a small-time annealed path large deviation principle for one-dimensional diffusions in a random environment associated with the generator ${\mathcal L}_W f(x)=e^{-\rho(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $\{\rho(x,\cdot):x\in\mathbb R\}$ and $\{a(x,\cdot):x\in\mathbb R\}$ are random. We assume that for each fixed realization of the environment, $\rho$ and $a$ are continuous and locally exponentially integrable, and that the support of the associated intrinsic coordinates is compact and non-collapsing. This framework includes the extensively studied Brox diffusion $dX_t=dB_t-\frac12\dot W(X_t)\,dt$, where $B$ is a standard Brownian motion and $W$ is an independent two-sided Brownian motion representing the environment. The It\^o--McKean representation of the diffusions and the estimates of the first exit probabilities derived via Moser iteration play a crucial role.
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.
Ping Chen, Tu-Sheng Zhang· 0 citations
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