In this paper, we are concerned with the symmetric simple exclusion process on the regula tree $\mathbb{T}^d$ for $d\geq 2$. Our main result gives moderate deviation principles of occupation times of the process starting from an invariant product measure. Two replacement lemmas play key roles in the proof of our main result. To obtain these replacement lemmas, we utilize duality relationships between the symmetric exclusion process and two types of random walks on $\mathbb{T}^d$ and $\left(\mathbb{T}^d\right)^2$ respectively.
We derive the non-equilibrium fluctuations of one-dimensional symmetric simple exclusion processes in contact with slowed stochastic reservoirs which are regulated by a factor $n^{-θ}$. Depending on the range of $θ$ we obtain processes with various boundary conditions. Moreover, as a consequence of the previous result we deduce the non-equilibrium stationary fluctuations by using the matrix ansatz method which gives us information on the stationary measure for the model. The main ingredient to prove these results is the derivation of precise bounds on the two-point space-time correlation funct…
The drawdown process of an one-dimensional regular diffusion process $X$ is given by $X$ reflected at its running maximum. The drawup process is given by $X$ reflected at its running minimum. We calculate the probability that a drawdown proceeds a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we …
In this paper, we establish a universal variational characterization of the non-martingale components associated with weakly differentiable Wiener functionals in the sense of Leão, Ohashi and Simas. It is shown that any Dirichlet process (in particular semimartingales) is a differential form w.r.t Brownian motion driving noise. The drift components are characterized in terms of limits of integral functionals of horizontal-type perturbations and first-order variation driven by a two-parameter occupation time process. Applications to a class of path-dependent rough transformations of Brownian pa…
We consider the damped nonlinear wave (NLW) equation driven by a spatially regular white noise. Assuming that the noise is non-degenerate in all Fourier modes, we establish a large deviations principle (LDP) for the occupation measures of the trajectories. The lower bound in the LDP is of a local type, which is related to the weakly dissipative nature of the equation and seems to be new in the context of randomly forced PDE's. The proof is based on an extension of methods developed in \cite{JNPS-2012} and \cite{JNPS-2014} in the case of kick forced dissipative PDE's with parabolic regularisati…
The strong $L^2$-approximation of occupation time functionals is studied with respect to discrete observations of a $d$-dimensional càdlàg process. Upper bounds on the error are obtained under weak assumptions, generalizing previous results in the literature considerably. The approach relies on regularity for the marginals of the process and applies also to non-Markovian processes, such as fractional Brownian motion. The results are used to approximate occupation times and local times. For Brownian motion, the upper bounds are shown to be sharp up to a log-factor.
We show that the law of the overall supremum $\bar{X}_t=\sup_{s\le t}X_s$ of a Lévy process $X$ before the deterministic time $t$ is equivalent to the average occupation measure $μ_t(dx)=\int_0^t\p(X_s\in dx)\,ds$, whenever 0 is regular for both open halflines $(-\infty,0)$ and $(0,\infty)$. In this case, $\p(\bar{X}_t\in dx)$ is absolutely continuous for some (and hence for all) $t>0$, if and only if the resolvent measure of $X$ is absolutely continuous. We also study the cases where 0 is not regular for one of the halflines $(-\infty,0)$ or $(0,\infty)$. Then we give absolute continuity crit…
We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least theta occupied neighbors, occupied sites remain occupied forever. It is known that, when b>theta>1, the limiting density q=q(p) of occupied sites exhibits a jump at some p_t=p_t(b,theta) in (0,1) from q_t:=q(p_t)<1 to q(p)=1 when p>p_t. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p_t+h with h>0 and show that, as h decreases to 0, the system linge…
In this paper, we introduce an insurance ruin model with adaptive premium rate, thereafter refered to as restructuring/refraction, in which classical ruin and bankruptcy are distinguished. In this model, the premium rate is increased as soon as the wealth process falls into the red zone and is brought back to its regular level when the process recovers. The analysis is mainly focused on the time a refracted Lévy risk process spends in the red zone (analogous to the duration of the negative surplus). Building on results from Kyprianou and Loeffen (2010) and Loeffen et al. (2012), we identify th…
