Stochastic differential equations and diffusion processes

In many applications it is important to be able to sample paths of SDEs conditional on observations of various kinds. This paper studies SPDEs which solve such sampling problems. The SPDE may be viewed as an infinite dimensional analogue of the Langevin SDE used in finite dimensional sampling. Here the theory is developed for conditioned Gaussian processes for which the resulting SPDE is linear. Applications include the Kalman-Bucy filter/smoother. A companion paper studies the nonlinear case, building on the linear analysis provided here.

Analysis of SPDEs Arising in Path Sampling Part I: The Gaussian Case

Introduction.- Weighted graphs and the associated Markov chains.- Heat kernel estimates - General theory.- Heat kernel estimates using effective resistance.- Heat kernel estimates for random weighted graphs.- Alexander-Orbach conjecture holds when two-point functions behave nicely.- Further results for random walk on IIC.- Random conductance model.

Random walks on disordered media and their scaling limits

Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field

We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in $\Z^d$. We show that for $d \geq 2$ and $p > p_c(\Z^d)$, the mixing time of simple random walk on the largest cluster inside $\{-n,...,n\}^d$ is $\Theta(n^2)$ - thus the mixing time is robust up to constant factor.

On the mixing time of simple random walk on the super critical percolation cluster

We study a continuous-time random walk, $X$, on $\mathbb{Z}^d$ in an environment of dynamic random conductances taking values in $(0, \infty)$. We assume that the law of the conductances is ergodic with respect to space-time shifts. We prove a quenched invariance principle for the Markov process $X$ under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser's iteration scheme.

Quenched invariance principle for random walks with time-dependent ergodic degenerate weights

We study a continuous-time random walk, XX, on ZdZd in an environment of dynamic random conductances taking values in (0,∞)(0,∞). We assume that the law of the conductances is ergodic with respect to space–time shifts. We prove a quenched invariance principle for the Markov process XX under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser’s iteration scheme.

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The large deviation principle in the small noise limit is derived for solutions of possibly degenerate Ito stochastic differential equations with predictable coefficients, which may depend also on the large deviation parameter. The result is established under mild assumptions using the Dupuis-Ellis weak convergence approach. Applications to certain systems with memory and to positive diffusions with square-root-like dispersion coefficient are included.

On large deviations for small noise It\^o processes

The large deviation principle in the small noise limit is derived for solutions of possibly degenerate Ito stochastic differential equations with predictable coefficients, which may also depend on the large deviation parameter. The result is established under mild assumptions using the Dupuis-Ellis weak convergence approach. Applications to certain systems with memory and to positive diffusions with square-root-like dispersion coefficient are included.

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On Large Deviations for Small Noise Itô Processes

Nous etudions une diffusion symetrique $X$ sur $R^{d}$ en forme de divergence dans un environnement aleatoire stationnaire et ergodique, dont les coefficients $a^{\omega}$ sont mesurables et degeneres. Cette diffusion qui est formellement engendree par l’operateur $L^{\omega}u=\nabla\cdot(a^{\omega}\nabla u)$, peut etre definie a l’aide de la theorie des formes de Dirichlet. Nous demontrons pour $X$ un principe d’invariance presque sur sous des conditions de moment de l’environnement; l’outil crucial est la sous-linearite du correcteur obtenu a l’aide de l’iteration introduite par J. Moser.

/pdf/invariance-principle-for-symmetric-diffusions-in-a-1gzbmcohhs.pdf

Alberto Chiarini

Papers

Quenched invariance principle for random walks with time-dependent ergodic degenerate weights

Quenched invariance principle for random walks with time-dependent ergodic degenerate weights

On large deviations for small noise It\^o processes

On Large Deviations for Small Noise Itô Processes

Invariance principle for symmetric diffusions in a degenerate and unbounded stationary and ergodic random medium