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We prove a comparison theorem for the spatial mass of the solutions of two exterior parabolic problems, one of them having symmetrized geometry, using approximation of the Schwarz symmetrization by polarizations, as it was introduced in Brock and Solynin (Trans Am Math Soc 352(4):1759–1796, 2000). This comparison provides an alternative proof, based on PDEs, of the isoperimetric inequality for the...
We discuss numerical aspects related to a new class of NonLinear Stochastic Differential Equation (NLSDE) in the sense of McKean, which are supposed to represent non conservative nonlinear Partial Differential Equations (PDEs). We propose an original interacting particle system for which we discuss the propagation of chaos. We consider a time-discretized approximation of this particle system to which...
We consider the variant of a stochastic parabolic Ginzburg-Landau equation that allows for the formation of point defects of the solution. The noise in the equation is multiplicative of the gradient type. We show that the family of the Jacobians associated to the solution is tight on a suitable space of measures. Our main result is the characterization of the limit points of this family. They are...
Convergence of a full discretization of a second order stochastic evolution equation with nonlinear damping is shown and thus existence of a solution is established. The discretization scheme combines an implicit time stepping scheme with an internal approximation. Uniqueness is proved as well.
We consider the Navier–Stokes equations in $${\mathbb {R}}^d$$ R d ( $$d=2,3$$ d = 2 , 3 ) with a stochastic forcing term which is white noise in time and coloured in space; the spatial covariance of the noise is not too regular, so Itô calculus cannot be applied in the space of finite energy vector fields. We prove existence of weak solutions for $$d=2,3$$ d = 2 , 3 ...
We establish sublinear growth of correctors in the context of stochastic homogenization of linear elliptic PDEs. In case of weak decorrelation and “essentially Gaussian” coefficient fields, we obtain optimal (stretched exponential) stochastic moments for the minimal radius above which the corrector is sublinear. Our estimates also capture the quantitative sublinearity of the corrector (caused by the...
In this article, we are concerned with a multidimensional degenerate parabolic–hyperbolic equation driven by Lévy processes. Using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that Lévy noise depends only on the solution. This result is used to show the...
While small ball, or lower tail, asymptotic for Gaussian measures generated by solutions of stochastic ordinary differential equations is relatively well understood, a lot less is known in the case of stochastic partial differential equations. The paper presents exact logarithmic asymptotics of the small ball probabilities in a scale of Sobolev spaces when the Gaussian measure is generated by the...
In this article we present a way of treating stochastic partial differential equations with multiplicative noise by rewriting them as stochastically perturbed evolutionary equations in the sense of Picard and McGhee (Partial differential equations: a unified Hilbert space approach, DeGruyter, Berlin, 2011), where a general solution theory for deterministic evolutionary equations has been developed...
We study the existence and uniqueness of the stochastic viscosity solutions of fully nonlinear, possibly degenerate, second order stochastic pde with quadratic Hamiltonians associated to a Riemannian geometry. The results are new and extend the class of equations studied so far by the last two authors.
In this article, we address the generalized BBM equation with white noise dispersion which reads $$\begin{aligned} du-du_{xx}+u_x \circ dW+ u^pu_xdt=0, \end{aligned}$$ d u - d u x x + u x ∘ d W + u p u x d t = 0 , in the Stratonovich formulation, where W(t) is a standard real valued Brownian motion. We first investigate the well-posedness of the initial value problem for...
In this paper, we introduce and analyze a new low-rank multilevel strategy for the solution of random diffusion problems. Using a standard stochastic collocation scheme, we first approximate the infinite dimensional random problem by a deterministic parameter-dependent problem on a high-dimensional parameter domain. Given a hierarchy of finite element discretizations for the spatial approximation,...
Given a separable and real Hilbert space $${\mathbb {H}}$$ H and a trace-class, symmetric and non-negative operator $$\mathscr {G}\,{:}\,{\mathbb {H}}\rightarrow {\mathbb {H}}$$ G : H → H , we examine the equation $$\begin{aligned} dX_t = -X_t dt + b(X_t) dt + \sqrt{2} dW_t, \quad X_0=x\in {\mathbb {H}}, \end{aligned}$$ d X t = - X t d t + b ( X t ) d t + 2...
In this paper, we address the long time behavior of solutions of the stochastic Schrödinger equation $$du+(\lambda u+i\Delta u +i\alpha |u|^{2\sigma }u)dt=\Phi dW_t$$ d u + ( λ u + i Δ u + i α | u | 2 σ u ) d t = Φ d W t in $${{\mathbb {R}}}^{d}$$ R d . We prove the existence of an invariant measure in $$H^{1}$$ H 1 for $$\sigma <2/(2-d)$$ σ <...
Motivated by applications to SPDEs we extend the Itô formula for the square of the norm of a semimartingale y(t) from Gyöngy and Krylov (Stochastics 6(3):153–173, 1982) to the case $$\begin{aligned} \sum _{i=1}^m \int _{(0,t]} v_i^{*}(s)\,dA(s) + h(t)=:y(t)\in V \quad dA\times {\mathbb {P}}\text {-a.e.}, \end{aligned}$$ ∑ i = 1 m ∫ ( 0 , t ] v i ∗ ( s ) d A ( s ) + h ...
We prove an invariance principle for the two-dimensional lattice parabolic Anderson model with small potential. As applications we deduce a Donsker type convergence result for a discrete random polymer measure, as well as a universality result for the spectrum of discrete random Schrödinger operators on large boxes with small potentials. Our proof is based on paracontrolled distributions and some...
The goal of this paper is to develop provably efficient importance sampling Monte Carlo methods for the estimation of rare events within the class of linear stochastic partial differential equations. We find that if a spectral gap of appropriate size exists, then one can identify a lower dimensional manifold where the rare event takes place. This allows one to build importance sampling changes of...
We consider the linear stochastic heat equation on $$\mathbb {R}^\ell $$ R ℓ , driven by a Gaussian noise which is colored in time and space. The spatial covariance satisfies general assumptions and includes examples such as the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter $$H\in (\frac{1}{4}, \frac{1}{2}]$$ H ∈ ( 1 4 ...
It is discussed the convergence of a Douglas–Rachford type splitting algorithm for the infinite dimensional stochastic differential equation $$\begin{aligned} dX+A(t)(X)dt=X\,dW \text{ in } (0,T);\ X(0)=x, \end{aligned}$$ d X + A ( t ) ( X ) d t = X d W in ( 0 , T ) ; X ( 0 ) = x , where $$A(t):V\rightarrow V'$$ A ( t ) : V → V ′ is a nonlinear, monotone, coercive...
Even though the heat equation with random potential is a well-studied object, the particular case of time-independent Gaussian white noise in one space dimension has yet to receive the attention it deserves. The paper investigates the stochastic heat equation with space-only Gaussian white noise on a bounded interval. The main result is that the space-time regularity of the solution is the same for...
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