Statistical inference for stochastic parabolic equations: a spectral approach.
Stochastic affine evolution equations with multiplicative fractional noise
A stochastic affine evolution equation with bilinear noise term is studied, where the driving process is a real-valued fractional Brownian motion with Hurst parameter greater than . Stochastic integration is understood in the Skorokhod sense. The existence and uniqueness of weak solution is proved and some results on the large time dynamics are obtained.
Stochastic averaging lemmas for kinetic equations
We develop a class of averaging lemmas for stochastic kinetic equations. The velocity is multiplied by a white noise which produces a remarkable change in time scale.Compared to the deterministic case and as far as we work in , the nature of regularity on averages is not changed in this stochastic kinetic equation and stays in the range of fractional Sobolev spaces at the price of an additional expectation. However all the exponents are changed; either time decay rates are slower (when the right...
Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem
Let H be a separable real Hilbert space and let E be a separable real Banach space. We develop a general theory of stochastic convolution of ℒ(H,E)-valued functions with respect to a cylindrical Wiener process with Cameron-Martin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP) (t∈ [0,T]), almost surely, where A is the generator of a -semigroup of bounded linear operators on...
Stochastic differential equations in Hilbert spaces
Stochastic diffrential equations on Banach spaces and their optimal feedback control
In this paper we consider stochastic differential equations on Banach spaces (not Hilbert). The system is semilinear and the principal operator generating a C₀-semigroup is perturbed by a class of bounded linear operators considered as feedback operators from an admissible set. We consider the corresponding family of measure valued functions and present sufficient conditions for weak compactness. Then we consider applications of this result to several interesting optimal feedback control problems....
Stochastic equation of population dynamics of prey-predator type with diffusion in a territory.
Stochastic evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality
In this paper we consider the question of optimal control for a class of stochastic evolution equations on infinite dimensional Hilbert spaces with controls appearing in both the drift and the diffusion operators. We consider relaxed controls (measure valued random processes) and briefly present some results on the question of existence of mild solutions including their regularity followed by a result on existence of partially observed optimal relaxed controls. Then we develop the necessary conditions...
Stochastic FitzHugh-Nagumo equations on networks with impulsive noise.
Stochastic fractional partial differential equations driven by Poisson white noise
We study a stochastic fractional partial differential equations of order driven by a compensated Poisson measure. We prove existence and uniqueness of the solution and we study the regularity of its trajectories.
Stochastic heat equation with white-noise drift
Stochastic integration of functions with values in a Banach space
Let H be a separable real Hilbert space and let E be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions Φ: (0,T) → ℒ(H,E) with respect to a cylindrical Wiener process . The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain E-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator which is...
Stochastic Modulation Equations on Unbounded Domains
We study the impact of small additive space-time white noise on nonlinear stochastic partial differential equations (SPDEs) on unbounded domains close to a bifurcation, where an infinite band of eigenvalues changes stability due to the unboundedness of the underlying domain. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, and we rely on the approximation via modulation or amplitude equations, which acts as a replacement for the lack of random...
Stochastic Navier-Stokes equations with artificial compressibility in random durations.
Stochastic nonlinear Schrödinger equations driven by a fractional noise, well-posedness, large deviations and support.
Stochastic partial differential equations for a class of interacting measure-valued diffusions
Stochastic partial differential equations with Dirichlet white-noise boundary conditions
Stochastic perturbation of the Allen-Cahn equation.
Stochastic Swift-Hohenberg equation near a change of stability
Stochastic Taylor expansions and heat kernel asymptotics
These notes focus on the applications of the stochastic Taylor expansion of solutions of stochastic differential equations to the study of heat kernels in small times. As an illustration of these methods we provide a new heat kernel proof of the Chern–Gauss–Bonnet theorem.