Equations In Infinite Dimensions Research Articles

Backward stochastic partial differential equations driven by infinite-dimensional martingales and applications

This paper studies first a result of existence and uniqueness of the solution to a backward stochastic differential equation driven by an infinite-dimensional martingale. Then, we apply this result to find a unique solution to a backward stochastic partial differential equation in infinite dimensions. The filtration considered is an arbitrary right-continuous filtration, not necessarily the natural filtration of a Wiener process. This, in particular, allows us to study more applications, for example, the maximum principle for a controlled stochastic evolution system. Some examples are discussed in the paper as well.

Approximate Controllability for Linear Stochastic Differential Equations in Infinite Dimensions

The objective of the paper is to investigate the approximate controllability property of a linear stochastic control system with values in a separable real Hilbert space. In a first step we prove the existence and uniqueness for the solution of the dual linear backward stochastic differential equation. This equation has the particularity that in addition to an unbounded operator acting on the Y-component of the solution there is still another one acting on the Z-component. With the help of this dual equation we then deduce the duality between approximate controllability and observability. Finally, under the assumption that the unbounded operator acting on the state process of the forward equation is an infinitesimal generator of an exponentially stable semigroup, we show that the generalized Hautus test provides a necessary condition for the approximate controllability. The paper generalizes former results by Buckdahn, Quincampoix and Tessitore (2006) and Goreac (2007) from the finite dimensional to the infinite dimensional case.

Average and deviation for the stochastic FitzHugh--Nagumo system

An averaged system for the slow-fast stochastic FitzHugh--Nagumo system is derived in this paper. The rate of convergence in probability is obtained as a byproduct. Moreover the deviation between the original system and the averaged system is studied. A martingale approach proves that the deviation is described by a Gaussian process. The deviation gives a more accurate asymptotic approximation than previous work. References S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction--diffusion equations, to appear in Probab. Th. and Rel. Fields . doi:10.1007/s00440-008-0144-z R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane. Biophys J. , 1 (1961). H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Commun. Math. Phys. , 65 (1979), 79--128. doi:10.1007/3-540-08853-9 R. Z. Khasminskii, On the principle of averaging the Ito stochastic differential equations (Russian), Kibernetika , 4 (1968), 260--279. P. E. Kloeden and E. Platen. Numerical solution of stochastic differential equations , volume 23 of Applications of Mathematics . Springer--Verlag, 1992. M. Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces , Scuola Normale Superiore, Pisa, 1988. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions , Cambridge University Press, 1992. J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. , 146 (1987), 65--96. doi:10.1007/BF01762360 V. M. Volosov, Averaging in systems of ordinary differential equations. Russ. Math. Surv. , 17 (1962), 1--126. doi:10.1070/RM1962v017n06ABEH001130

Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions

Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions

Stochastic FitzHugh-Nagumo equations on networks with impulsive noise

We consider a system of nonlinear partial differential equations with stochastic dynamical boundary conditions that arises in models of neurophysiology for the diffusion of electrical potentials through a finite network of neurons. Motivated by the discussion in the biological literature, we impose a general diffusion equation on each edge through a generalized version of the FitzHugh-Nagumo model, while the noise acting on the boundary is described by a generalized stochastic Kirchhoff law on the nodes. In the abstract framework of matrix operators theory, we rewrite this stochastic boundary value problem as a stochastic evolution equation in infinite dimensions with a power-type nonlinearity, driven by an additive Lévy noise. We prove global well-posedness in the mild sense for such stochastic partial differential equation by monotonicity methods.

Backward Stochastic Differential Equations in Infinite Dimensions with Continuous Driver and Applications

In this paper we prove the existence of a solution to backward stochastic differential equations in infinite dimensions with continuous driver under various assumptions. We apply our results to a stochastic game problem with infinitely many players.

A new approach to Kolmogorov equations in infinite dimensions and applications to the stochastic 2D Navier–Stokes equation

In this Note we present a new approach to solve Kolmogorov equations in infinitely many variables in weighted spaces of weakly continuous functions, including the case of non-constant possibly degenerate diffusion coefficients. To cite this article: M. Röckner, Z. Sobol, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Backward stochastic partial differential equations in infinite dimensions

Backward stochastic partial differential equations in infinite dimensions

Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations

We develop a new method to uniquely solve a large class of heat equations, so-called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally Lipschitz drifts. Apart from general analytic interest, the main motivation is to apply this to uniquely solve martingale problems in the sense of Stroock–Varadhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic Navier–Stokes equations. In this paper this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.

Backward stochastic partial differential equations in infinite dimensions

Backward stochastic partial differential equations in infinite dimensions

Strong, mild and weak solutions of backward stochastic evolution equations

Four types of solutions ( mild, weak, weak mild and strong ) of the so-called backward stochastic evolution equations in infinite dimensions are proved to exist. A relationship between these solutions is established as well.

Microscopic derivations of several Hamilton–Jacobi equations in infinite dimensions, and large deviation of stochastic systems

We consider Hamilton–Jacobi equations which characterize optimal controlled partial differential equations of the following types: the Allen–Cahn equation, the Cahn–Hilliard equation, a nonlinear Fokker–Planck equation, and a Vlasov–Fokker–Planck equation. In each of the examples, the optimal control problem and its associated cost functional can be derived as limit from a microscopically defined stochastic system, using the probabilistic theory of large deviation. The physical context here makes it natural to derive a free energy inequality, which is very useful in proving the well-posedness of the Hamilton–Jacobi equation. The article is written using informal arguments. Rigorous results will appear elsewhere.

Uniform stability of autonomous linear stochastic functional differential equations in infinite dimensions

Existence, uniqueness and continuity of mild solutions are established for stochastic linear functional differential equations in an appropriate Hilbert space which is particularly suitable for stability analysis. An attempt is made to obtain some infinite dimensional stochastic extensions of the corresponding deterministic stability results. One of the most important results is to show that the uniformly asymptotic stability of the equations we try to handle is equivalent to their square integrability in some suitable sense. Subsequently, the stability results derived in retarded case are applied to coping with stability for a large class of neutral linear stochastic systems.

On the Continuity of Pathwise Solutions to Langevin Equations in Infinite Dimensions

We fix a rich probability space (Ω,F,P). Let (H,‖⋅‖) be a separable Hilbert space and let μ be the canonical cylindrical Gaussian measure μ on H. Given any abstract Wiener space (H,B,μ) over H, and for every Hilbert–Schmidt operator T: H⊂B→H which is (|{⋅}|,‖⋅‖)-continuous, where |{⋅}| stands for the (Gross-measurable) norm on B, we construct an Ornstein–Uhlenbeck process ξ: (Ω,F,P)×[0,1]→(B,|{⋅}|) as a pathwise solution of the following infinite-dimensional Langevin equation dξt=db t+T(ξt) dt with the initial data ξ0=0, where b is a B-valued Brownian motion based on the abstract Wiener space (H,B,μ). The richness of the probability space (Ω,F,P) then implies the following consequences: the probability space Ω is independent of the abstract Wiener space (H,B,μ) (in the sense that (Ω,F,P) does not depend on the choice of the Gross-measurable norm |{⋅}|) and the space C B consisting of all continuous B-valued functions on [0,1] is identical with the set of all paths of ξ. Finally, we present a way to obtain pathwise continuous solutions ξ: dξt= $$\sqrt {\left| \alpha \right|\beta }$$ db t+α⋅ξt dt with initial data ξ0=0, where α,β∈R,α≠0 and 0<β.

A new approach to Kolmogorov equations in infinite dimensions and applications to stochastic generalized Burgers equations

We develop a new method to uniquely solve a large class of heat equations, so called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. In this way, for the first time, the solutions are constructed everywhere without exceptional sets for equations with possibly non-locally Lipschitz drifts. Apart from general analytic interest, the main motivation is to apply this to uniquely solve martingale problems in the sense of Stroock–Varadhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic Navier–Stokes equations. In this Note this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows. To cite this article: M. Röckner, Z. Sobol, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Robustness of Pathwise Stability of Semilinear Perturbed Stochastic Evolution Equations

The purpose of this paper is to investigate pathwise stability for certain Hilbert space-valued stochastic evolution equations. We are especially interested in the robustness analysis of perturbed stochastic differential equations in infinite dimensions. Sufficient conditions are established to ensure the almost surely stable decay of the given stochastic systems. Lastly, a corollary and corresponding example are studied to illustrate our theory.

Existence of invariant manifolds for stochastic equations in infinite dimension

We provide a Frobenius type existence result for finite-dimensional invariant submanifolds for stochastic equations in infinite dimension, in the spirit of Da Prato and Zabczyk (Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992). We recapture and make use of the convenient calculus on Fréchet spaces, as developed by Kriegl and Michor (The Convenient Setting for Global Analysis, Surveys and Monographs, Vol. 53, Amer. Math. Soc., Providence, RI, 1997). Our main result is a weak version of the Frobenius theorem on Fréchet spaces. As an application, we characterize all finite-dimensional realizations for a stochastic equation which describes the evolution of the term structure of interest rates.

White Noise Limits for Inertial Particles in a Random Field

In this paper we present a rigorous analysis of a scaling limit related to the motion of an inertial particle in a Gaussian random field. The mathematical model comprises Stokes's law for the particle motion and an infinite dimensional Ornstein-Uhlenbeck process for the fluid velocity field. The scaling limit studied leads to a white noise limit for the fluid velocity, which balances particle inertia and the friction term. Strong convergence methods are used to justify the limiting equations. The rigorously derived limiting equations are of physical interest for the concrete problem under investigation and facilitate the study of two-point motions in the white noise limit. Furthermore, the methodology developed may also prove useful in the study of various other asymptotic problems for stochastic differential equations in infinite dimensions.

On exponential stability of Non-autonomous stochastic semilinear evolution equations

Sufficient conditions for the exponential stability of a class of nonlinear, non-autonomous stochastic differential equations in infinite dimensions are studied. The analysis consists of introducing a suitable approximating solution systems and usig a limiting argument to pass on stability of strong solutions to mild ones. Consequently, under these conditions the random attractors of given stochastic systems are reduced to zero with exponential decay. Lastly, two examples are investigated to illustrate the theory.

Measure transport on Wiener space and the Girsanov theorem

Let ( W, H, μ) be an abstract Wiener space, and assume that ν i , i=1,2, are two probability measures on (W, B(W)) which are absolutely continuous with respect to μ. Assume that the Wasserstein distance between ν 1 and ν 2 is finite. Then there exists a map T= I W + ξ of W into itself such that ξ: W→ H is measurable and 1-cyclically monotone such that the image of ν 1 under T is ν 2. Moreover T is invertible on the support of ν 2. We give also some applications of this result such as the existence of the solutions of the Monge–Ampère equation in infinite dimensions. To cite this article: D. Feyel, A.S. Üstünel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1025–1028.