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 Varadhan small-time asymptotics for the quenched and annealed heat kernels of one-dimensional diffusions in a random environment with generator $\mathcal L_W f(x)=e^{-\rho(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $a$ and $\rho$ are continuous in space and satisfy a local exponential moment condition....
We prove absolute continuity of the running maximum $X^{\ast}_T=\sup_{0\leq s\leq T}X_s$ of one-dimensional time-inhomogeneous L\'evy--It\^o diffusions driven by a Brownian motion and an independent non-truncated pure-jump L\'evy process. Using Bismut's directional Malliavin calculus on the Wiener--Poisson space togeth...
We consider the parabolic Anderson model $\partial_t u=\partial_x^2 u+\xi u$ on $\mathbb{R}_+\times\mathbb{R}$ with $u(0,\cdot)\equiv 1$, where $\xi$ is a spatial white noise. We study the long-time behavior of the spatial integral $U(t):=\int_{-L(t)/2}^{L(t)/2}u(t,x)\,dx$, where $L(t)=\exp(\alpha^3 t^3/24)$ with $\alp...
We study parabolic stochastic partial differential equations on metric measure spaces $(\mathbb{X}, d,m)$ of the form $$ \partial_t u(t,x) = \mathcal{L}^* u(t,x) + b(t,x,u(t,x)) + \sigma(t,x,u(t,x)) \dot{W}(t,x),\quad t>0,\, x \in \mathbb X, $$ where $\mathcal{L}$ is the generator of a Markov process which possesses tr...
Louis Wai-Tong Fan, Zhen-Yao Sun, Johnny Yang· 0 citations
Let $\pi(\mathrm{d} x)\propto e^{-U(x)}\, \mathrm{d} x$ on $\mathbb{R}^d$, where $U$ is continuously differentiable and $m$-strongly convex with a globally $L$-Lipschitz gradient, $0<m\leq L<\infty$, and $\kappa=L/m$. Fixed-step Metropolis-adjusted Langevin algorithm (MALA) has known warm-start mixing-time upper bounds...
We study large deviations for rare clusters in sparse systems of moving particles. In the regime \(nr_n^d\to0\) and \(\rho_{k,n}=n^kr_n^{d(k-1)}\to\infty\), we prove a large deviation principle for the empirical measure of isolated \(k\)-particle trajectory clusters. The speed is \(\rho_{k,n}\), and the rate function i...
Cai-Rui Duan, Paula de Dios Andres, Manish Pandey et al.· 0 citations
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