Uniform Lipschitz estimates in stochastic homogenization

Scott Armstrong

Displaying similar documents to “Uniform Lipschitz estimates in stochastic homogenization”

On Stochastic Differential Equations with Reflecting Boundary Condition in Convex Domains

Weronika Łaukajtys (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let D be an open convex set in $ℝ^{d}$ and let F be a Lipschitz operator defined on the space of adapted càdlàg processes. We show that for any adapted process H and any semimartingale Z there exists a unique strong solution of the following stochastic differential equation (SDE) with reflection on the boundary of D: $X_{t} = H_{t} + \int_{0}^{t} ⟨ F {(X)}_{s -}, d Z_{s} ⟩ + K_{t}$ , t ∈ ℝ⁺. Our proofs are based on new a priori estimates for solutions of the deterministic Skorokhod problem.

Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings

Marc Peigné, Wolfgang Woess (2011)

Colloquium Mathematicae

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In this continuation of the preceding paper (Part I), we consider a sequence ${(F ₙ)}_{n \geq 0}$ of i.i.d. random Lipschitz mappings → , where is a proper metric space. We investigate existence and uniqueness of invariant measures, as well as recurrence and ergodicity of the induced stochastic dynamical system (SDS) $X ₙ^{x} = F ₙ \circ . . . \circ F ₁ (x)$ starting at x ∈ . The main results concern the case when the associated Lipschitz constants are log-centered. Principal tools are local contractivity, as considered in detail in Part I,...

Some Results on Stochastic Porous Media Equations

Viorel Barbu, Giuseppe Da Prato, Michael Röckner (2008)

Bollettino dell'Unione Matematica Italiana

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Some recent results about nonnegative solutions of stochastic porous media equations in bounded open subsets of $\mathbb{R}^{3}$ are considered. The existence of an invariant measure is proved.

Viability theorems for stochastic inclusions

Michał Kisielewicz (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Sufficient conditions for the existence of solutions to stochastic inclusions $x_{t} - x_{s} \in \int_{s}^{t} F_{τ} (x_{τ}) d τ + \int_{s}^{t} G_{τ} (x_{τ}) d w_{τ} + \int_{s}^{t} \int_{I R ⁿ} H_{τ, z} (x_{τ}) ν ̃ (d τ, d z)$ beloning to a given set K of n-dimensional cádlág processes are given.

Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity

Marc Peigné, Wolfgang Woess (2011)

Colloquium Mathematicae

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Consider a proper metric space and a sequence ${(F ₙ)}_{n \geq 0}$ of i.i.d. random continuous mappings → . It induces the stochastic dynamical system (SDS) $X ₙ^{x} = F ₙ \circ . . . \circ F ₁ (x)$ starting at x ∈ . In this and the subsequent paper, we study existence and uniqueness of invariant measures, as well as recurrence and ergodicity of this process. In the present first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying...

Spatial Besov regularity for stochastic partial differential equations on Lipschitz domains

Petru A. Cioica, Stephan Dahlke, Stefan Kinzel, Felix Lindner, Thorsten Raasch, Klaus Ritter, René L. Schilling (2011)

Studia Mathematica

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We use the scale of Besov spaces $B_{τ, τ}^{α} ()$ , 1/τ = α/d + 1/p, α > 0, p fixed, to study the spatial regularity of solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains ⊂ ℝ. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.

Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem

Zdzisław Brzeźniak, Jan van Neerven (2000)

Studia Mathematica

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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 ${W_{t}^{H}}_{t \in [0, T]}$ 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) $d X_{t} = A X_{t} d t + B d W_{t}^{H}$ (t∈ [0,T]), $X_{0} = 0$ almost surely, where A is the generator of a $C_{0}$ -semigroup ${S (t)}_{t \geq 0}$ of bounded linear...

Limit theorems for stochastic recursions with Markov dependent coefficients

Dariusz Buraczewski, Małgorzata Letachowicz (2012)

Colloquium Mathematicae

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We consider the stochastic recursion $X ₙ = A ₙ X_{n - 1} + B ₙ$ for Markov dependent coefficients (Aₙ,Bₙ) ∈ ℝ⁺ × ℝ. We prove the central limit theorem, the local limit theorem and the renewal theorem for the partial sums Sₙ = X₁+ ⋯ + Xₙ.

A free stochastic partial differential equation

Yoann Dabrowski (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We get stationary solutions of a free stochastic partial differential equation. As an application, we prove equality of non-microstate and microstate free entropy dimensions under a Lipschitz like condition on conjugate variables, assuming also the von Neumann algebra $R^{ω}$ embeddable. This includes an $N$ -tuple of $q$ -Gaussian random variables e.g. for $| q | N \leq 0.13$ .

Euler's Approximations of Solutions of Reflecting SDEs with Discontinuous Coefficients

Alina Semrau-Giłka (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let D be either a convex domain in $ℝ^{d}$ or a domain satisfying the conditions (A) and (B) considered by Lions and Sznitman (1984) and Saisho (1987). We investigate convergence in law as well as in $L^{p}$ for the Euler and Euler-Peano schemes for stochastic differential equations in D with normal reflection at the boundary. The coefficients are measurable, continuous almost everywhere with respect to the Lebesgue measure, and the diffusion coefficient may degenerate on some subsets of the domain. ...

Non-autonomous stochastic Cauchy problems in Banach spaces

Mark Veraar, Jan Zimmerschied (2008)

Studia Mathematica

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We study the non-autonomous stochastic Cauchy problem on a real Banach space E, $d U (t) = A (t) U (t) d t + B (t) d W_{H} (t)$ , t ∈ [0,T], U(0) = u₀. Here, $W_{H}$ is a cylindrical Brownian motion on a real separable Hilbert space H, ${(B (t))}_{t \in [0, T]}$ are closed and densely defined operators from a constant domain (B) ⊂ H into E, ${(A (t))}_{t \in [0, T]}$ denotes the generator of an evolution family on E, and u₀ ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then we use a well-known factorisation method in the setting...

Set-valued stochastic integrals and stochastic inclusions in a plane

Władysław Sosulski (2001)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We present the concepts of set-valued stochastic integrals in a plane and prove the existence of a solution to stochastic integral inclusions of the form $z_{s, t} \in φ_{s, t} + \int_{0}^{s} \int_{0}^{t} F_{u, v} (z_{u, v}) d u d v + \int_{0}^{s} \int_{0}^{t} G_{u, v} (z_{u, v}) d w_{u, v}$

Systems of Bellman Equations to Stochastic Differential Games with Discount Control

Alain Bensoussan, Jens Frehse (2008)

Bollettino dell'Unione Matematica Italiana

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We consider two dimensional diagonal elliptic systems $\Delta u+au=H(x,u,\nabla u)$ which arise from stochastic differential games with discount control. The Hamiltonians $H$ have quadratic growth in $\nabla u$ and a special structure which has notyet been covered by regularity theory. Without smallness condition on $H$ , the existence of a regular solution is established.

Initial measures for the stochastic heat equation

Daniel Conus, Mathew Joseph, Davar Khoshnevisan, Shang-Yuan Shiu (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a family of nonlinear stochastic heat equations of the form $\partial_{t} u = ℒ u + σ (u) \dot{W}$ , where $\dot{W}$ denotes space–time white noise, $ℒ$ the generator of a symmetric Lévy process on $𝐑$ , and $σ$ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_{0}$ . Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $ℒ f = c f^{''}$ for some $c g t; 0$ , we prove that if $u_{0}$ is a finite measure of compact support, then the...

Doubly stochastic matrices and the Bruhat order

Richard A. Brualdi, Geir Dahl, Eliseu Fritscher (2016)

Czechoslovak Mathematical Journal

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The Bruhat order is defined in terms of an interchange operation on the set of permutation matrices of order $n$ which corresponds to the transposition of a pair of elements in a permutation. We introduce an extension of this partial order, which we call the stochastic Bruhat order, for the larger class $Ω_{n}$ of doubly stochastic matrices (convex hull of $n \times n$ permutation matrices). An alternative description of this partial order is given. We define a class of special faces of $Ω_{n}$ induced by permutation...

The right tail exponent of the Tracy–Widom $β$ distribution

Laure Dumaz, Bálint Virág (2013)

Annales de l'I.H.P. Probabilités et statistiques

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The Tracy–Widom $β$ distribution is the large dimensional limit of the top eigenvalue of $β$ random matrix ensembles. We use the stochastic Airy operator representation to show that as $a \to \infty$ the tail of the Tracy–Widom distribution satisfies $P ({𝑇𝑊}_{β} g t; a) = a^{- (3 / 4) β + o (1)} exp (- \frac{2}{3} β a^{3 / 2}) .$

Transition semigroups for stochastic semilinear equations on Hilbert spaces

Anna Chojnowska-Michalik

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A large class of stochastic semilinear equations with measurable nonlinear term on a Hilbert space H is considered. Assuming the corresponding nonsymmetric Ornstein-Uhlenbeck process has an invariant measure μ, we prove in the $L^{p} (H, μ)$ spaces the existence of a transition semigroup $(P_{t})$ for the equations. Sufficient conditions are provided for hyperboundedness of $P_{t}$ and for the Log Sobolev Inequality to hold; and in the case of a bounded nonlinear term, sufficient and necessary conditions are obtained....

Invariance principle for the random conductance model with dynamic bounded conductances

Sebastian Andres (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We study a continuous time random walk $X$ in an environment of dynamic random conductances in $ℤ^{d}$ . We assume that the conductances are stationary ergodic, uniformly bounded and bounded away from zero and polynomially mixing in space and time. We prove a quenched invariance principle for $X$ , and obtain Green’s functions bounds and a local limit theorem. We also discuss a connection to stochastic interface models.

Stochastic approximation properties in Banach spaces

V. P. Fonf, W. B. Johnson, G. Pisier, D. Preiss (2003)

Studia Mathematica

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We show that a Banach space X has the stochastic approximation property iff it has the stochasic basis property, and these properties are equivalent to the approximation property if X has nontrivial type. If for every Radon probability on X, there is an operator from an $L_{p}$ space into X whose range has probability one, then X is a quotient of an $L_{p}$ space. This extends a theorem of Sato’s which dealt with the case p = 2. In any infinite-dimensional Banach space X there is a compact set K...

Double sine series with nonnegative coefficients and Lipschitz classes

Vanda Fülöp (2006)

Colloquium Mathematicae

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Denote by $f_{s s} (x, y)$ the sum of a double sine series with nonnegative coefficients. We present necessary and sufficient coefficient conditions in order that $f_{s s}$ belongs to the two-dimensional multiplicative Lipschitz class Lip(α,β) for some 0 < α ≤ 1 and 0 < β ≤ 1. Our theorems are extensions of the corresponding theorems by Boas for single sine series.

On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes

Nicolas Fournier (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order $α$ with drift and diffusion coefficients $b$ , $σ$ . When $α \in (1, 2)$ , we investigate pathwise uniqueness for this equation. When $α \in (0, 1)$ , we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $α \in (0, 1)$ or $α \in (1, 2)$ and on whether the driving stable process is symmetric or not. Our...

Sungularity as $L_{2}$ -regularity and vice versa

T. Pogany (1990)

Matematički Vesnik

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Lipschitz and uniform embeddings into $ℓ_{\infty}$

N. J. Kalton (2011)

Fundamenta Mathematicae

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We show that there is no uniformly continuous selection of the quotient map $Q : ℓ_{\infty} \to ℓ_{\infty} / c ₀$ relative to the unit ball. We use this to construct an answer to a problem of Benyamini and Lindenstrauss; there is a Banach space X such that there is a no Lipschitz retraction of X** onto X; in fact there is no uniformly continuous retraction from $B_{X * *}$ onto $B_{X}$ .

Discrete Approximations of Strong Solutions of Reflecting SDEs with Discontinuous Coefficients

Alina Semrau (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study $L^{p}$ convergence for the Euler scheme for stochastic differential equations reflecting on the boundary of a general convex domain D ⊆ ℝd. We assume that the equation has the pathwise uniqueness property and its coefficients are measurable and continuous almost everywhere with respect to the Lebesgue measure. In the case D=[0,∞) new sufficient conditions ensuring pathwise uniqueness for equations with possibly discontinuous coefficients are given.