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.
Stochastic two-scale convergence in the mean and applications.
Stochastic weak attractor for a dissipative Euler equation.
Strong Feller solutions to SPDE's are strong Feller in the weak topology
For a wide class of Markov processes on a Hilbert space H, defined by semilinear stochastic partial differential equations, we show that their transition semigroups map bounded Borel functions to functions weakly continuous on bounded sets, provided they map bounded Borel functions into functions continuous in the norm topology. In particular, an Ornstein-Uhlenbeck process in H is strong Feller in the norm topology if and only if it is strong Feller in the bounded weak topology. As a consequence,...
Symmetrized solutions for nonlinear stochastic differential equations.
Taylor expansion of the density in a stochastic heat equation.
The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs.
The Hausdorff measure of the closed support of super-brownian motion
The internal stabilization by noise of the linearized Navier-Stokes equation
One shows that the linearized Navier-Stokes equation in , around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller , , where are independent Brownian motions in a probability space and is a system of functions on with support in an arbitrary open subset . The stochastic control input is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium...