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S. V. Lototsky

Publications and source records attributed to S. V. Lototsky.

16 recordsLinked to original sources
arXiv2022

A Sharp Rate of Convergence in the Functional Central Limit Theorem with Gaussian Input

When the underlying random variables are Gaussian, the classical Central Limit Theorem (CLT) is trivial, but the functional CLT is not. The objective of the paper is to investigate the functional CLT for stationary Gaussian processes in the Wasserstein-1 metric on the space of continuous functions. Matching upper and lower bounds are established, indicating that the convergence rate is slightly faster than in the Lévy-Prokhorov metric.

math.PR↗
arXiv2010

A Note on Generalized Malliavin Calculus

The Malliavin derivative, divergence operator, and the Ornstein-Uhlenbeck operator are extended from the traditional Gaussian setting to generalized processes from the higher-order chaos spaces.

math.PR↗
arXiv2009

Parameter Estimation in Diagonalizable Stochastic Hyperbolic Equations

A parameter estimation problem is considered for a linear stochastic hyperbolic equation driven by additive space-time Gaussian white noise. The damping/amplification operator is allowed to be unbounded. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimator are studied as the number of the Fourier coefficients increases, while the observation time and the noise intensity are fixed.

math.PR↗
arXiv2008

Stochastic Differential Equations Driven by Purely Spatial Noise

We study stochastic parabolic and elliptic PDEs driven by purely spatial white noise. Even the simplest equations driven by this noise often do not have a square-integrable solution and must be solved in special weighted spaces. We demonstrate that the Cameron-Martin version of the Wiener chaos decomposition is an effective tool to study both stationary and evolution equations driven by space-only noise. The paper presents results about solvability of such equations in weighted Wiener chaos spaces and studies the long-time behavior of the solutions of evolution equations with space-only noise.

math.PR↗
arXiv2008

Estimating Speed and Damping in the Stochastic Wave Equation

A parameter estimation problem is considered for a one-dimensional stochastic wave equation driven by additive space-time Gaussian white noise. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimator are studied as the number of the Fourier coefficients increases, while the observation time and the noise intensity are fixed.

math.PR↗
arXiv2007

Stochastic Integrals and Evolution Equations with Gaussian Random Fields

The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic integrands. The problem is then to extend the definition to random integrands. An orthogonal decomposition of the chaos space of the random field, combined with the Wick product, leads to the \Ito-Skorokhod integral, and provides an efficient tool to study the integral, both analytically and numerically. For a Gaussian process, a natural definition of the integral follows from a canonical correspondence between random processes and a special class of random fields. Some examples of the corresponding stochastic differential equations are also considered.

math.PR↗
arXiv2007

A Unified Approach to Stochastic Evolution Equations Using the Skorokhod Integral

We study stochastic evolution equations driven by Gaussian noise. The key features of the model are that the operators in the deterministic and stochastic parts can have the same order and the noise can be time-only, space-only, or space-time. Even the simplest equations of this kind do not have a square-integrable solution and must be solved in special weighted spaces. We demonstrate that the Cameron-Martin version of the Wiener chaos decomposition leads to natural weights and a natural replacement of the square integrability condition.

math.PR↗
arXiv2007

Solving SPDEs driven by colored noise: a chaos approach

An Ito-Skorokhod bi-linear equation driven by infinitely many independent colored noises is considered in a normal triple of Hilbert spaces. The special feature of the equation is the appearance of the Wick product in the definition of the Ito-Skorokhod integral, requiring innovative approaches to computing the solution. A chaos expansion of the solution is derived and several truncations of this expansion are studied. A recursive approximation of the solution is suggested and the corresponding approximation error bound is computed.

math.PR↗
arXiv2007

Stochastic Parabolic Equations of Full Second Order

A procedure is described for defining a generalized solution for stochastic differential equations using the Cameron-Martin version of the Wiener Chaos expansion. Existence and uniqueness of this Wiener Chaos solution is established for parabolic stochastic PDEs such that both the drift and the diffusion operators are of the second order.

math.PR↗
arXiv2007

From Random Processes to Generalized Fields: A Unified Approach to Stochastic Integration

The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic integrands. The problem is then to extend the definition to random integrands. An orthogonal decomposition of chaos space of the random field leads to two such extensions, corresponding to the \Ito-Skorokhod and the Stratononovich integrals, and provides an efficient tool to study these integrals, both analytically and numerically. For a Gaussian process, a natural definition of the integral follows from a canonical correspondence between random processes and a special class of random fields.

math.PR↗
arXiv2006

Wiener chaos solutions of linear stochastic evolution equations

A new method is described for constructing a generalized solution of a stochastic evolution equation. Existence, uniqueness, regularity and a probabilistic representation of this Wiener Chaos solution are established for a large class of equations. As an application of the general theory, new results are obtained for several types of the passive scalar equation.

math.PR↗
arXiv2005

Stochastic Differential Equations: A Wiener Chaos Approach

A new method is described for constructing a generalized solution for stochastic differential equations. The method is based on the Cameron-Martin version of the Wiener Chaos expansion and provides a unified framework for the study of ordinary and partial differential equations driven by finite- or infinite-dimensional noise with either adapted or anticipating input. Existence, uniqueness, regularity, and probabilistic representation of this Wiener Chaos solution is established for a large class of equations. A number of examples are presented to illustrate the general constructions. A detailed analysis is presented for the various forms of the passive scalar equation and for the first-order Itô stochastic partial differential equation. Applications to nonlinear filtering if diffusion processes and to the stochastic Navier-Stokes equation are also discussed.

math.PR↗
arXiv2003

Time Evolution of a Passive Scalar in a Turbulent Incompressible Gaussian Velocity Field

Passive scalar equation is considered in a turbulent homogeneous incompressible Gaussian velocity field. The turbulent nature of the field results in non-smooth coefficients in the equation. A strong, in the stochastic sense, solution of the equation is constructed using the Wiener Chaos, and the properties of the solution are studied. The results apply to both viscous and conservative motions.

math-ph↗