Related papers: Operator-scaling Gaussian random fields via aggreg…
Continuous time random walks impose a random waiting time before each particle jump. Scaling limits of heavy tailed continuous time random walks are governed by fractional evolution equations. Space-fractional derivatives describe heavy…
Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The…
In this paper, we focus on multiple sampling problems for the estimation of the fractional Brownian motion when the maximum number of samples is limited, extending existing results in the literature in a non-Markovian framework. Two classes…
We study the level statistics for two classes of 1-dimensional random Schr\"odinger operators : (1) for operators whose coupling constants decay as the system size becomes large, and (2) for operators with critically decaying random…
We consider a scalar field governed by an advection-diffusion equation (or a more general evolution equation) with rapidly fluctuating, Gaussian distributed random coefficients. In the white noise limit, we derive the closed evolution…
The critical dynamics of conformal field theories on random surfaces is investigated beyond the previously studied dynamics of the overall area and the genus. It is found that the evolution of the order parameter in physical time performs a…
We consider anisotropic self-similar random fields, in particular, the fractional Brownian sheet. This Gaussian field is an extension of fractional Brownian motion. We prove some properties of covariance function for self-similar fields…
We consider metric graph Gaussian free field (GFF) defined on polygons of $\delta\mathbb{Z}^2$ with alternating boundary data. The crossing probabilities for level-set percolation of metric graph GFF have scaling limits. When the boundary…
In this paper, we analyze the scaling properties of a model that has as limiting cases the diffusion-limited aggregation (DLA) and the ballistic aggregation (BA) models. This model allows us to control the radial and angular scaling of the…
We consider continuous time random interlacements on $\mathbb{Z}^d$, $d \ge 3$, and characterize the distribution of the corresponding stationary random field of occupation times. When d = 3, we relate this random field to the…
Many growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties. Such structures can often be described effectively by space-time trajectories of interacting…
The main topic of these lecture notes is the continuum scaling limit of planar lattice models. One reason why this topic occupies an important place in the theory of probability and mathematical statistical physics is that scaling limits…
We consider quadrangulations with a boundary and derive explicit expressions for the generating functions of these maps with either a marked vertex at a prescribed distance from the boundary, or two boundary vertices at a prescribed mutual…
We consider the diffusion scaling limit of the vicious walkers and derive the time-dependent spatial-distribution function of walkers. The dependence on initial configurations of walkers is generally described by using the symmetric…
Two algorithms are proposed to simulate space-time Gaussian random fields with a covariance function belonging to an extended Gneiting class, the definition of which depends on a completely monotone function associated with the spatial…
Anomalous diffusion is frequently described by scaled Brownian motion (SBM), a Gaussian process with a power-law time dependent diffusion coefficient. Its mean squared displacement is $\langle x^2(t)\rangle\simeq\mathscr{K}(t)t$ with…
We derive the limiting distributions of exceedances point processes of randomly scaled weakly dependent stationary Gaussian sequences under some mild asymptotic conditions. In the literature analogous results are available only for…
We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of AR(1) process with random-coefficient $a \in [0,1)$ when $N$ and time scale $n$ increase at different rate. Assuming that $a$ has a density, regularly…
This paper presents a new model of textures, obtained as realizations of a new class of fractional Brownian fields. These fields, called weighted tensorized fractional Brownian fields, are obtained by a relaxation of the tensor-product…
Motivated by contemporary and rich applications of anomalous diffusion processes we propose a new statistical test for fractional Brownian motion, which is one of the most popular models for anomalous diffusion systems. The test is based on…