Related papers: Assouad dimension of random processes
We consider the Assouad spectrum, introduced by Fraser and Yu, along with a natural variant that we call the `upper Assouad spectrum'. These spectra are designed to interpolate between the upper box-counting and Assouad dimensions. It is…
We consider a set of one-dimensional transformations of Gaussian random functions. Under natural assumptions we obtain a connection between $L_2$-small ball asymptotics of the transformed function and of the original one. Also the explicit…
We study various classes of random processes defined on the regular tree $T_d$ that are invariant under the automorphism group of $T_d$. Most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov…
We consider high frequency samples from ergodic L\'evy driven stochastic differential equation (SDE) with drift coefficient $a(x,\alpha)$ and scale coefficient $c(x,\gamma)$ involving unknown parameters $\alpha$ and $\gamma$. We suppose…
We study the packing dimension of Borel measures under orthogonal projections. We give a necessary and sufficient condition such that typical projections of Borel probability measures have full packing dimension and derive general lower…
In this note we prove the well-posedness for stochastic 2D Navier-Stokes equation driven by general L\'evy processes (in particular, $\alpha$-stable processes), and obtain the existence of invariant measures.
We calculate the Assouad dimension of a planar self-affine set $X$ satisfying the strong separation condition and the projection condition and show that $X$ is minimal for the conformal Assouad dimension. Furthermore, we see that such a…
We study random graphs with latent geometric structure, where the probability of each edge depends on the underlying random positions corresponding to the two endpoints. We focus on the setting where this conditional probability is a…
We formulate and analyze a graphical model selection method for inferring the conditional independence graph of a high-dimensional nonstationary Gaussian random process (time series) from a finite-length observation. The observed process…
We provide a detailed derivation of the Karhunen-Lo\`eve expansion of a stochastic process. We also discuss briefly Gaussian processes, and provide a simple numerical study for the purpose of illustration.
It is proved that the random integral mappings (some type of functionals of L\'evy processes) are always isomorphisms between convolution semigroups of infinitely divisible measures. However, the inverse mappings are no longer of the random…
For a series of Markov processes we prove stochastic duality relations with duality functions given by orthogonal polynomials. This means that expectations with respect to the original process (which evolves the variable of the orthogonal…
In this article we consider the Levy processes and the corresponding semigroup. We represent the generator of this semigroup in a convolution form. Using the obtained convolution form and the theory of integral equations we investigate the…
For a self-similar set in $\mathbb{R}^d$ that is the attractor of an iterated function system that does not verify the weak separation property, Fraser, Henderson, Olson and Robinson showed that its Assouad dimension is at least $1$. In…
Generative diffusion models and many stochastic models in science and engineering naturally live in infinite dimensions before discretisation. To incorporate observed data for statistical and learning tasks, one needs to condition on…
In this paper, we determine the almost sure values of the $\Phi $-dimensions of random measures supported on random Moran sets that satisfy a uniform separation condition. The $\Phi $-dimensions are intermediate Assouad-like dimensions, the…
We present an alternative construction of the infinite dimensional It\^{o} integral with respect to a Hilbert space valued L\'{e}vy process. This approach is based on the well-known theory of real-valued stochastic integration, and the…
The study of multidimensional stochastic processes involves complex computations in intricate functional spaces. In particular, the diffusion processes, which include the practically important Gauss-Markov processes, are ordinarily defined…
We study point processes that consist of certain centers of point tuples of an underlying Poisson process. Such processes arise in stochastic geometry in the study of exceedances of various functionals describing geometric properties of the…
We explore the limit of stochastic differential equations driven by some random processes satisfying singularly perturbed second order stochastic differential equations. The main tool we employ is the universal limit theorem in rough path…