Stochastic Differential Equations with Discontinuous Drift in Hilbert Space with Applications to Interacting Particle Systems (original) (raw)

ON THE EXISTENCE OF WEAK VARIATIONAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS

Communications on Stochastic Analysis, 2010

We study the existence of weak variational solutions in a Gelfand triplet of real separable Hilbert spaces, under continuity, growth, and coercivity conditions on the coefficients of the stochastic differential equation. The laws of finite dimensional approximations are proved to weakly converge to the limit which is identified as a weak solution. The solution is an Hvalued continuous process in

Strong uniqueness for SDEs in Hilbert spaces with nonregular drift

The Annals of Probability, 2016

We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear drift parts are sums of the sub-differential of a convex function and a bounded part. This generalizes a classical result by one of the authors to infinite dimensions. Our results also generalize and improve recent results by N. Champagnat and P. E. Jabin, proved in finite dimensions, in the case where their diffusion matrix is constant and nondegenerate and their weakly differentiable drift is the (weak) gradient of a convex function. We also prove weak existence, hence obtain unique strong solutions by the Yamada-Watanabe theorem. The proofs are based in part on a recent maximal regularity result in infinite dimensions, the theory of quasi-regular Dirichlet forms and an infinite dimensional version of a Zvonkin-type transformation. As a main application, we show pathwise uniqueness for stochastic reaction diffusion equations perturbed by a Borel measurable bounded drift. Hence, such SDE have a unique strong solution.

Weak Solutions of Stochastic Differential Equations with Discontinuous Coefficients

2001

In the present paper, we consider the stochastic differential equation dx(t) = f (t, x(t))dt + g(t, x(t))dW (t) (1) with Borel measurable functions f : R + × R d → R d and g : R + × R d → R d×d , introduce various definitions of weak solutions, and prove the corresponding existence theorems. We establish a theorem on the dependence of solutions on the initial conditions and right-hand sides and a theorem on the compactness of the set of probability laws of weak solutions. The existence theorem is proved under the only assumption that the mappings f and g are Borel measurable and locally bounded and without any additional conditions, in contrast with other well-known existence theorems [1-5]. Weak solutions of Eq. (1) are defined as weak solutions of some stochastic differential inclusion corresponding to Eq. (1). We use the following notation: R d×r is the space of real d × r matrices equipped with the Euclidean norm • ; R d×1 = R d ; R + = [0, +∞[, conv R d×r is the metric space of nonempty convex compact subsets of R d×r with metric κ(A, B) = max{β(A, B), β(B, A)}, where β(A, B) = sup a∈A inf b∈B a − b ; [B] α = x ∈ R d×r | inf y∈B x − y ≤ α is the α-neighborhood of a set B ⊂ R d×r ; µ is the Lebesgue measure on R + ; C R + , R d×r is the space of continuous functions a : R + → R d×r equipped with the metric (a 1 , a 2) = ∞ k=1 2 −k max 0≤t≤k a 1 (t) − a 2 (t) ∧ 1 , b 1 ∧ b 2 = min {b 1 , b 2 } ,

On Interacting Systems of Hilbert-Space-Valued Diffusions

Applied Mathematics and Optimization, 1998

A nonlinear Hilbert-space-valued stochastic differential equation where L −1 (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L −1 , the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L −1 is not nuclear, a solution in most cases lies not in H but in a larger Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove under suitable conditions that a unique strong solution can still be found to lie in the space H itself. Uniqueness of the weak solution is proved without moment assumptions on the initial random variable.

Stochastic differential equations with coefficients in Sobolev spaces

Journal of Functional Analysis, 2010

We consider Itô SDE dX t = m j=1 A j (X t) dw j t + A 0 (X t) dt on R d. The diffusion coefficients A 1 , • • • , A m are supposed to be in the Sobolev space W 1,p loc (R d) with p > d, and to have linear growth; for the drift coefficient A 0 , we consider two cases: (i) A 0 is continuous whose distributional divergence δ(A 0) w.r.t. the Gaussian measure γ d exists, (ii) A 0 has the Sobolev regularity W 1,p ′ loc for some p ′ > 1. Assume R d exp λ 0 |δ(A 0)| + m j=1 (|δ(A j)| 2 + |∇A j | 2) dγ d < +∞ for some λ 0 > 0, in the case (i), if the pathwise uniqueness of solutions holds, then the push-forward (X t) # γ d admits a density with respect to γ d. In particular, if the coefficients are bounded Lipschitz continuous, then X t leaves the Lebesgue measure Leb d quasi-invariant. In the case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish the existence and uniqueness of stochastic flow of maps.

Linear Stochastic Differential Equations in The Dual Of A Multi-Hilbertian Space

2008

We prove the existence and uniqueness of strong solutions for linear stochastic differential equations in the space dual to a multi–Hilbertian space driven by a finite dimensional Brownian motion under relaxed assumptions on the coefficients. As an application, we consider equtions in S � with coefficients which are differential operators violating the typical growth and monotonicity conditions. 1. Assumptions We consider a countably Hilbertian space (Φ ,τ ), whose topology τ is determined by a family of separable Hilbertian seminorms �·� p, p ∈ R (for a detailed exposition, see [4]). For any p ∈ R+ ,w e identifyφ ∈ Φw ith [φ]p ∈ Φ/ ker �·� p and denote the completion of Φ in �·� p by Hp .T henHp is a real separable Hilbert space containing Φ as its dense subspace, and the embedding (Φ ,τ ) � → (Hp, �·� p) is continuous. Assume that, for q ≤ p, the canonical embedding (Hp, �·� p) � → (Hq, �· � q) is continuous, i.e., �· � p dominates �·� q, denoted by �·� q ≺� ·� p. In applications,...

Stochastic Differential Equations with Discontinuous Drift

The study of interacting particle systems arise in physics in case of spin systems and behavior of systems following Glauber dynamics. They can be modeled with stochastic differential equations in ℝ∞ or in mathbbRmathbbZd{\mathbb{R}}^{{\mathbb{Z}}^{d}}mathbbRmathbbZd , d>1. One wishes to know if a solution exists and determine its state space. For example, it may be interesting if the solution is in l 2. However, we consider models where the drift coefficient is not continuous on l 2. This complication, together with the fact that the Peano theorem fails in infinite dimensions, can be overcome by noting that the embedding from l 2 to ℝ∞ is continuous and compact, and by applying the “method of compact embedding” from Chap. 3. We construct weak solutions, generally in a Hilbert space H, which are not continuous as functions into the state space with its natural topology. However, by identifying H with l 2↪ℝ∞, they turn out to be continuous in the topology induced on H from ℝ∞. This part is related to the work of Leha and Ritter (Math. Ann. 270:109–123, 1985), where equations in general form are studied and the motivation comes from modeling of unbounded spin systems. We show the existence of solutions under weaker condition on the drift and provide Galerkin approximation in the case studied by Leha and Ritter. In the second part we prove existence of solutions for quantum lattice systems in mathbbRmathbbZd{\mathbb{R}}^{{\mathbb{Z}}^{d}}mathbbRmathbbZd under weaker assumptions than those considered by Albeverio et al. (Rev. Math. Phys. 13(1):51–124, 2001) in justifying Glauber dynamics. To that end, we change the set-up to equations in a dual to a nuclear space and obtain weak solutions using compact embeddings.

Dissipative stochastic equations in Hilbert space with time dependent coefficients

Rendiconti Lincei - Matematica e Applicazioni, 2000

Mathematical analysis.-Dissipative stochastic equations in Hilbert space with time dependent coefficients, by GIUSEPPE DA PRATO and MICHAEL RÖCKNER, communicated on 23 June 2006. ABSTRACT.-We prove existence and, under an additional assumption, uniqueness of an evolution system of measures (ν t) t∈R for a stochastic differential equation with time dependent dissipative coefficients. We prove that if P s,t denotes the corresponding transition evolution operator, then P s,t ϕ behaves asymptotically as t → +∞ like a limit curve (which is independent of s) for any continuous and bounded "observable" ϕ.