Couplings via Comparison Principle and Exponential Ergodicity of SPDEs in the Hypoelliptic Setting

We develop a general framework for studying ergodicity of order-preserving Markov semigroups. We establish natural and in a certain sense optimal conditions for existence and uniqueness of the invariant measure and exponential convergence of transition probabilities of an order-preserving Markov process. As an application, we show exponential ergodicity and exponentially fast synchronization-by-noise of the stochastic reaction–diffusion equation in the hypoelliptic setting. This refines and complements corresponding results of Hairer and Mattingly (Electron J Probab 16:658–738, 2011). © 2020, The Author(s)

Statistical methods of SNP data analysis with applications

Various statistical methods important for genetic analysis are considered and developed. Namely, we concentrate on the multifactor dimensionality reduction, logic regression, random forests and stochastic gradient boosting. These methods and their new modifications, e.g., the MDR method with "independent rule", are used to study the risk of complex diseases such as cardiovascular ones. The roles of certain combinations of single nucleotide polymorphisms and external risk factors are examined. To perform the data analysis concerning the ischemic heart disease and myocardial infarction the supercomputer SKIF "Chebyshev" of the Lomonosov Moscow State University was employed

Stochastic equations with singular drift driven by fractional Brownian motion

We consider stochastic differential equation dXt=b(Xt)dt+dWHt, where the drift b is either a measure or an integrable function, and WH is a d-dimensional fractional Brownian motion with Hurst parameter H∈(0,1), d∈N. For the case where b∈Lp(Rd), p∈[1,∞] we show weak existence of solutions to this equation under the condition dp<1H−1, which is an extension of the Krylov-Röckner condition (2005) to the fractional case. We construct a counter-example showing optimality of this condition. If b is a Radon measure, particularly the delta measure, we prove weak existence of solutions to this equation under the optimal condition H<1d+1. We also show strong well-posedness of solutions to this equation under certain conditions. To establish these results, we utilize the stochastic sewing technique and develop a new version of the stochastic sewing lemma

Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation

We study stochastic reaction--diffusion equation $$\partial_tu_t(x)=\frac12 \partial^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in D$$ where $b$ is a generalized function in the Besov space $\mathcal{B}^\beta_{q,\infty}({\mathbb R})$, $D\subset{\mathbb R}$ and $\dot W$ is a space-time white noise on ${\mathbb R}_+\times D$. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever $\beta-1/q\ge-1$, $\beta>-1$ and $q\in[1,\infty]$. This class includes equations with $b$ being measures, in particular, $b=\delta_0$ which corresponds to the skewed stochastic heat equation. For $\beta-1/q > -3/2$, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalizes the results of Gy\"ongy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in L\^e~(2020)