Stochastic evolution equations driven by Liouville fractional Brownian motion

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of L(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in the interval (0,1). For Hurst parameters in (0,1/2) we show that a function F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fBm if and only if it is stochastically integrable with respect to an H-cylindrical fBm with the same Hurst parameter. As an application we show that second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by space-time noise which is white in space and Liouville fractional in time with Hurst parameter in (d/4,1) admit mild solution which are H\"older continuous both and space.Comment: To appear in Czech. Math.

Abstract Functional Stochastic Evolution Equations Driven by Fractional Brownian Motion

We investigate a class of abstract functional stochastic evolution equations driven by a fractional Brownianmotion in a real separable Hilbert space.Global existence results concerningmild solutions are formulated under various growth and compactness conditions. Continuous dependence estimates and convergence results are also established. Analysis of three stochastic partial differential equations, including a second-order stochastic evolution equation arising in the modeling of wave phenomena and a nonlinear diffusion equation, is provided to illustrate the applicability of the general theory

Measure-Dependent Stochastic Nonlinear Beam Equations Driven by Fractional Brownian Motion

We study a class of nonlinear stochastic partial differential equations arising in themathematicalmodeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separableHilbert space and is studied using the tools of cosine function theory, stochastic analysis, and fixed-point theory. Global existence and uniqueness results for mild solutions, continuous dependence estimates, and various approximation results are established and applied in the context of the model

Abstract Functional Stochastic Evolution Equations Driven by Fractional Brownian Motion

We investigate a class of abstract functional stochastic evolution equations driven by a fractional Brownian motion in a real separable Hilbert space. Global existence results concerning mild solutions are formulated under various growth and compactness conditions. Continuous dependence estimates and convergence results are also established. Analysis of three stochastic partial differential equations, including a second-order stochastic evolution equation arising in the modeling of wave phenomena and a nonlinear diffusion equation, is provided to illustrate the applicability of the general theory

Stochastic partial differential equations with fractal noise: two different approaches

This thesis deals with stochastic partial differential equations driven by fractional noises. In this work, problems related to this topics are tackled and solved from two fairly different points of view. On one side we prove existence, uniqueness and regularity for mild solutions to a parabolic transport diffusion type equation that involves a non-smooth coefficient. We investigate related Cauchy problems on bounded smooth domains with Dirichlet boundary conditions by means of semigroup theory and fixed point arguments. Main ingredients are the definition of a product of a function and a (not too irregular) distribution as well as a corresponding norm estimate. As an application, transport stochastic partial differential equations driven by fractional Brownian noises are considered in the pathwise sense. On the other side we deal with stochastic differential equations driven by fractal noises in Banach spaces. More precisely, we deal with abstract Cauchy problems driven by fractional Brownian processes in Banach spaces and look for weak and mild solutions. To this aim, a fractional Brownian motion in separable Banach spaces is introduced by means of cylindrical processes. The related stochastic integral is then defined as cylindrical stochastic process and its properties are investigated. When the Banach space is a function space then the equation becomes a stochastic partial differential equation driven by a fractional noise