# Borel Function

The Experts below are selected from a list of 210 Experts worldwide ranked by ideXlab platform

### David Nualart – 1st expert on this subject based on the ideXlab platform

• ##### Hyperbolic Stochastic Partial Differential Equations with Additive Fractional Brownian Sheet
Stochastics and Dynamics, 2020
Co-Authors: Mohamed Erraoui, Youssef Ouknine, David Nualart

Abstract:

Let [Formula: see text] be a fractional Brownian sheet with Hurst parameters H, H′ ≤ 1/2. We prove the existence and uniqueness of a strong solution for a class of hyperbolic stochastic partial differential equations with additive fractional Brownian sheet of the form [Formula: see text], where b(ζ, x) is a Borel Function satisfying some growth and monotonicity assumptions. We also prove the convergence of Euler’s approximation scheme for this equation.

• ##### Hyperbolic Stochastic Partial Differential Equations with Additive Fractional Brownian Sheet
Stochastics and Dynamics, 2003
Co-Authors: Mohamed Erraoui, Youssef Ouknine, David Nualart

Abstract:

Let be a fractional Brownian sheet with Hurst parameters H, H′ ≤ 1/2. We prove the existence and uniqueness of a strong solution for a class of hyperbolic stochastic partial differential equations with additive fractional Brownian sheet of the form , where b(ζ, x) is a Borel Function satisfying some growth and monotonicity assumptions. We also prove the convergence of Euler’s approximation scheme for this equation.

• ##### Regularization of differential equations by fractional noise
Stochastic Processes and their Applications, 2002
Co-Authors: David Nualart, Youssef Ouknine

Abstract:

Let {BtH,t[set membership, variant][0,T]} be a fractional Brownian motion with Hurst parameter H. We prove the existence and uniqueness of a strong solution for a stochastic differential equation of the form , where b(s,x) is a bounded Borel Function with linear growth in x (case ) or a Holder continuous Function of order strictly larger than 1-1/2H in x and than in time (case ).

### Paweł Sztonyk – 2nd expert on this subject based on the ideXlab platform

• ##### Strong Feller Property for SDEs Driven by Multiplicative Cylindrical Stable Noise
Potential Analysis, 2020
Co-Authors: Tadeusz Kulczycki, Michał Ryznar, Paweł Sztonyk

Abstract:

We consider the stochastic differential equation d X _ t = A ( X _ t −) d Z _ t , X _0 = x , driven by cylindrical α -stable process Z _ t in , where α ∈ (0,1) and d ≥ 2. We assume that the determinant of A ( x ) = ( a _ i j ( x )) is bounded away from zero, and a _ i j ( x ) are bounded and Lipschitz continuous. We show that for any fixed γ ∈ (0, α ) the semigroup P _ t of the process X _ t satisfies | P t f ( x ) − P t f ( y ) | ≤ c t − γ / α | x − y | γ | | f | | ∞ $|P_{t} f(x) – P_{t} f(y)| \le c t^{-\gamma /\alpha } |x – y|^{\gamma } ||f||_{\infty }$ for arbitrary bounded Borel Function f . Our approach is based on Levi’s method.

### Hisashi Kikuchi – 3rd expert on this subject based on the ideXlab platform

• ##### valley approximation for the BorelFunction
Physical Review D, 1992
Co-Authors: Hisashi Kikuchi

Abstract:

A simple one-dimensional integral is investigated as a model for large-order estimation of the perturbative expansion in quantum mechanics with degenerate vacua. A Borel Function analysis allows us to separate nonperturbative contributions from perturbative ones. Issues such as the cancellation between the perturbative and nonperturbative contributions of ambiguity due to non-Borel summability and the large-order estimation in terms of a dispersion integral are discussed. A stationary-point approximation for the Borel Function is proposed to connect the simple integral to the quantum-mechanical case based on the new valley trajectory, which was recently formulated.