Evelina Shamarova

Universidade Federal da Paraíba | UFPB · Departamento de Matemática

We study general problems modeling electrostatic microelectromechanical systems devices (Pλ )φ(r,−u′(r))=λ∫0rf(s)g(u(s))ds,r∈(0,1),0

We study general equations modeling electrostatic MEMS devices \begin{equation} \begin{cases} \label{P} \varphi\big(r,- u'(r)\big)=\lambda\int_0^r\frac{f(s)}{g(u(s))}\,\mathrm{d}s, & r\in(0,1), \\ 0 < u(r) < 1, & r\in(0,1), \\ u(1) = 0, \tag{$P_\lambda$} \end{cases} \end{equation} where $\varphi$, $g$, $f$ are some functions on $[0,1]$ and $\lambda...

We study a class of elliptic problems, involving a k-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. Furthermore, we study the multiplicity of regular solutions and bifurcation diagrams. An essential ingredien...

The aim of this paper is to study negative classical solutions to a $k$-Hessian equation involving a nonlinearity with a general weight \begin{equation} \label{Eq:Ma:0} \tag{$P$} \begin{cases} S_k(D^2u)= \lambda \rho(|x|) (1-u)^q &\mbox{in }\;\; B,\\ u=0 &\mbox{on }\partial B. \end{cases} \end{equation} Here, $B$ denotes the unit ball in $\mathbb R...

We obtain upper and lower Gaussian-type bounds on the density of each component Yti\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${Y^{i}_{t}}$\end{document} of the solu...

We establish the existence of smooth densities for solutions to a broad class of path-dependent SDEs under a Hörmander-type condition. The classical scheme based on the reduced Malliavin matrix turns out to be unavailable in the path-dependent context. We approach the problem by lifting the given n-dimensional path-dependent SDE into a suitable Lp-...

We study a class of elliptic problems, involving a $k$-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. Furthermore, we study the multiplicity of regular solutions and bifurcation diagrams. An important ingredi...

We establish the existence of smooth densities for solutions to a broad class of path-dependent SDEs under a H\"ormander-type condition. The classical scheme based on the reduced Malliavin matrix turns out to be unavailable in the path-dependent context. We approach the problem by lifting the given $n$-dimensional path-dependent SDE into a suitable...

We obtain upper and lower Gaussian density estimates for the laws of each component of the solution to a one‐dimensional fully coupled forward‐backward SDE. Our approach relies on the link between FBSDEs and quasilinear parabolic PDEs, and is fully based on the use of classical results on PDEs rather than on manipulation of FBSDEs, compared to othe...

We prove the existence and uniqueness of a classical solution to a multidimensional non-potential stochastic Burgers equation with Hölder continuous initial data. Our motivation is the adhesion model in the theory of formation of the large-scale structure of the universe. Importantly, we drop the assumption on the potentiality of the velocity flow...

We obtain upper and lower Gaussian-type bounds on the density of the law of each component $Y^i_t$ of the solution $Y_t$ to a multidimensional backward SDE. Our approach is based on the Nourdin-Viens formula and the analysis of the associated semilinear parabolic PDE. Furthermore, we apply our results to stochastic gene expression; namely, we estim...

We prove the existence and uniqueness of a classical solution to a multidimensional non-potential stochastic Burgers equation with H\"older continuous initial data. Our motivation is the adhesion model in the theory of formation of the large-scale structure of the universe. Importantly, we drop the assumption on the potentiality of the velocity flo...

We obtain an existence and uniqueness theorem for fully coupled forward–backward SDEs (FBSDEs) with jumps via the classical solution to the associated quasilinear parabolic partial integro-differential equation (PIDE), and provide the explicit form of the FBSDE solution. Moreover, we embed the associated PIDE into a suitable class of non-local quas...

We obtain upper and lower Gaussian density estimates for the law of each component of the solution to a one-dimensional fully coupled forward-backward SDE (FBSDE). Our approach relies on the link between FBSDEs and quasilinear parabolic PDEs, and is fully based on the use of classical results on PDEs rather than on manipulation of FBSDEs, compared...

In this article, we introduce a backward method to model stochastic gene expression and protein-level dynamics. The protein amount is regarded as a diffusion process and is described by a backward stochastic differential equation (BSDE). Unlike many other SDE techniques proposed in the literature, the BSDE method is backward in time; that is, inste...

We approximate the heat kernel h(x, y, t) on a compact connected Riemannian manifold M without boundary uniformly in \((x,y,t)\in M\times M\times [a,b]\), \(a>0\), by n-fold integrals over \(M^n\) of the densities of Brownian bridges. Moreover, we provide an estimate for the uniform convergence rate. As an immediate corollary, we get a uniform appr...

In this paper we prove a variant of the Stokes formula for differential forms of a finite codimension in a locally convex space (LCS)The main tool used by us for proving the mentioned formula is the surface layer theorem for surfaces of codimension 1 in a locally convex space which was proved earlier by the first authorMoreover, on some subspace of...

In this work, we prove the existence and uniqueness of the global classical solution to the multidimensional stochastically forced viscous Burgers equation without potential-type assumptions on forcing or terminal conditions. The main tool in our analysis is a novel probabilistic a priori gradient estimate which holds uniformly over subintervals of...

In this work, we establish pathwise functional Itô formulas for non-smooth functionals of real-valued continuous semimartingales. Under nite (p, q)- variation regularity assumptions in the sense of two-dimensional Young integration theory, we establish a pathwise local-time decomposition. Here, Xt = (X(s); 0 ≤ s ≤ t) is the continuous semimartingal...

In this work, we establish pathwise functional It\^o formulas for non-smooth functionals of real-valued continuous semimartingales. Under finite $(p,q)$-variation regularity assumptions in the sense of two-dimensional Young integration theory, we establish a pathwise local-time decomposition $$F_t(X_t) = F_0(X_0)+ \int_0^t\nabla^hF_s(X_s)ds + \int_...

We consider a stochastic evolution equation in a 2-smooth Banach space with a densely and continuously embedded Hilbert subspace. We prove that under Hörmander's bracket condition, the image measure of the solution law under any finite-rank bounded linear operator is absolutely continuous with respect to the Lebesgue measure. To obtain this result,...

We consider a stochastic evolution equation in a 2-smooth Banach space with a densely and continuously embedded Hilbert subspace. We prove that under H\"ormander's bracket condition, the image measure of the solution law under any finite-rank bounded linear operator is absolutely continuous with respect to the Lebesgue measure. To obtain this resul...

We construct a solution to the spatially periodic $d$-dimensional Navier-Stokes equations with a given distribution of the initial data. The solution takes values in the Sobolev space $H^\alpha$, where the index $\alpha\in R$ is fixed arbitrary. The distribution of the initial value is a Gaussian measure on $H^\alpha$ whose parameters depend on $\a...

We describe a probabilistic construction of $H^s$-regular solutions for the spatially periodic forced Burgers equation by using a characterization of this solution through a forward-backward stochastic system.

The classical Chernoff's theorem is a statement about discrete-time approximations of semigroups, where the approximations are consturcted as products of time-dependent contraction operators strongly differentiable at zero. We generalize the version of Chernoff's theorem for semigroups proved in a paper by Smolyanov et al., and obtain a theorem abo...

We establish a connection between the strong solution to the spatially periodic Navier–Stokes equations and a solution to a system of forward–backward stochastic differential equations (FBSDEs) on the group of volume-preserving diffeomorphisms of a flat torus. We construct representations of the strong solution to the Navier–Stokes equations in ter...

We develop a mathematical approach to the nonequilibrium work theorem which is traditionally referred to in statistical mechanics as Jarzynski's identity. We suggest a mathematically rigorous formulation and proof of the identity.

We prove a version of the Stokes formula for differential forms on locally convex spaces. The main tool used for proving this formula is the surface layer theorem proved in another paper by the author. Moreover, for differential forms of a Sobolev-type class relative to a differentiable measure, we compute the operator adjoint to the exterior diffe...

A generalized version of Chernoff's theorem has been obtained. Namely, the version of Chernoff's theorem for semigroups obtained in a paper by Smolyanov, Weizsaecker, and Wittich is generalized for a time-inhomogeneous case. The main theorem obtained in the current paper, Chernoff's theorem for evolution families, deals with a family of time-depend...

We develop a mathematical approach to the nonequilibrium work theorem which is traditionally referred to in statistical mechanics as Jarzynski's identity. We suggest a mathematically rigorous formulation and proof of the identity.

Without Abstract

The main result of the paper is an analog of the surface layer theorem for measures given on a locally convex space with a continuously and densely embedded Hilbert subspace (for a surface of finite codimension). Earlier, the surface layer theorem was proved only for Banach spaces: for surfaces of codimension 1 by Uglanov (1979) and for surfaces of...