scholarly journals On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients

2020 ◽  
Vol 72 (9) ◽  
pp. 1254-1285
Author(s):  
A. Pilipenko ◽  
A. Kulik
Keyword(s):  
Selection Problem ◽  
Extremal Solutions ◽  
Averaging Principle ◽  
Small Noise ◽  
Lipschitz Continuous ◽  
Limit Process ◽  
Locally Lipschitz ◽  
And Diffusion ◽  
Locally Lipschitz Continuous

UDC 519.21 In this paper we solve a selection problem for multidimensional SDE where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane It is assumed that the drift has a Hoelder asymptotics as approaches and the limit ODE does not have a unique solution.We show that if the drift pushes the solution away from then the limit process with certain probabilities selects some extremal solutions to the limit ODE. If the drift attracts the solution to then the limit process satisfies an ODE with some averaged coefficients. To prove the last result we formulate an averaging principle, which is quite general and new.

LIPSCHITZ CONTINUITY FOR H-CONVEX FUNCTIONS IN CARNOT GROUPS

2006 ◽  
Vol 08 (01) ◽  
pp. 1-8 ◽  
Author(s):  
MINGBAO SUN ◽  
XIAOPING YANG
Keyword(s):  
Convex Functions ◽  
Lipschitz Continuity ◽  
Carnot Group ◽  
Carnot Groups ◽  
Lipschitz Continuous ◽  
Locally Lipschitz ◽  
Locally Lipschitz Continuous

For a Carnot group G of step two, we prove that H-convex functions are locally bounded from above. Therefore, H-convex functions on a Carnot group G of step two are locally Lipschitz continuous by using recent results by Magnani.

Über die C2-Kompaktheit der Bahn von Lösungen semflinearer parabolischer Systeme

1982 ◽  
Vol 93 (1-2) ◽  
pp. 99-103 ◽  
Author(s):  
Reinhard Redlinger
Keyword(s):  
Boundary Conditions ◽  
Regular Solution ◽  
Parabolic System ◽  
Lipschitz Continuous ◽  
Semilinear Parabolic System ◽  
Locally Lipschitz ◽  
Semilinear Parabolic ◽  
Locally Lipschitz Continuous

SynopsisThe semilinear parabolic systemut+A(x, D)u=g(u) in (0, ∞) × Ω, Ω⊂ℝnbounded,u∈ ℝN, with homogeneous boundary conditionsB(x, D)u=0 on (0, ∞)×∂Ω is considered. The non-linearitygis assumed to be locally Lipschitz-continuous. It is shown that the orbit of a bounded regular solutionuis relatively compact in.

Limit measures and ergodicity of fractional stochastic reaction–diffusion equations on unbounded domains

2021 ◽  
pp. 2140012
Author(s):  
Zhang Chen ◽  
Bixiang Wang
Keyword(s):  
Invariant Measures ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Lipschitz Continuous ◽  
Bounded Interval ◽  
Locally Lipschitz ◽  
Stochastic Reaction ◽  
And Diffusion

This paper deals with invariant measures of fractional stochastic reaction–diffusion equations on unbounded domains with locally Lipschitz continuous drift and diffusion terms. We first prove the existence and regularity of invariant measures, and then show the tightness of the set of all invariant measures of the equation when the noise intensity varies in a bounded interval. We also prove that every limit of invariant measures of the perturbed systems is an invariant measure of the corresponding limiting system. Under further conditions, we establish the ergodicity and the exponentially mixing property of invariant measures.

scholarly journals RADIAL SYMMETRY OF POSITIVE SOLUTIONS TO EQUATIONS INVOLVING THE FRACTIONAL LAPLACIAN

2014 ◽  
Vol 16 (01) ◽  
pp. 1350023 ◽  
Author(s):  
PATRICIO FELMER ◽  
YING WANG
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
Radial Symmetry ◽  
Open Unit ◽  
Open Unit Ball ◽  
Lipschitz Continuous ◽  
Local Character ◽  
Locally Lipschitz ◽  
Moving Planes ◽  
Locally Lipschitz Continuous

The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem [Formula: see text] where (-Δ)αdenotes the fractional Laplacian, α ∈ (0, 1), and B1denotes the open unit ball centered at the origin in ℝNwith N ≥ 2. The function f : [0, ∞) → ℝ is assumed to be locally Lipschitz continuous and g : B1→ ℝ is radially symmetric and decreasing in |x|. In the second place we consider radial symmetry of positive solutions for the equation [Formula: see text] with u decaying at infinity and f satisfying some extra hypothesis, but possibly being non-increasing.Our third goal is to consider radial symmetry of positive solutions for system of the form [Formula: see text] where α1, α2∈(0, 1), the functions f1and f2are locally Lipschitz continuous and increasing in [0, ∞), and the functions g1and g2are radially symmetric and decreasing. We prove our results through the method of moving planes, using the recently proved ABP estimates for the fractional Laplacian. We use a truncation technique to overcome the difficulty introduced by the non-local character of the differential operator in the application of the moving planes.

Stochastic approximation with non-additive measurement noise

1998 ◽  
Vol 35 (02) ◽  
pp. 407-417 ◽  
Author(s):  
Han-Fu Chen
Keyword(s):  
Growth Rate ◽  
Stochastic Approximation ◽  
Strong Consistency ◽  
Additive Noise ◽  
Measurement Noise ◽  
Mixing Process ◽  
Lipschitz Continuous ◽  
Mixing Coefficients ◽  
Locally Lipschitz ◽  
Locally Lipschitz Continuous

The Robbins–Monro algorithm with randomly varying truncations for measurements with non-additive noise is considered. Assuming that the function under observation is locally Lipschitz-continuous in its first argument and that the noise is a φ-mixing process, strong consistency of the estimate is shown. Neither growth rate restriction on the function, nor the decreasing rate of the mixing coefficients are required.

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