scholarly journals Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven stochastic differential equations with smooth drift coefficient functions with at most polynomially growing derivatives

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Arnulf Jentzen ◽  
Benno Kuckuck ◽  
Thomas Müller-Gronbach ◽  
Larisa Yaroslavtseva
Keyword(s):  
Additive Noise ◽  
Strong Sense ◽  
Coefficient Function ◽  
Initial Value ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
First Order ◽  
Initial Values ◽  
Locally Lipschitz ◽  
Drift Coefficient

<p style='text-indent:20px;'>In the recent article [A. Jentzen, B. Kuckuck, T. Müller-Gronbach, and L. Yaroslavtseva, <i>J. Math. Anal. Appl. 502</i>, 2 (2021)] it has been proved that the solutions to every additive noise driven stochastic differential equation (SDE) which has a drift coefficient function with at most polynomially growing first order partial derivatives and which admits a Lyapunov-type condition (ensuring the existence of a unique solution to the SDE) depend in the strong sense in a logarithmically Hölder continuous way on their initial values. One might then wonder whether this result can be sharpened and whether in fact, SDEs from this class necessarily have solutions which depend in the strong sense locally Lipschitz continuously on their initial value. The key contribution of this article is to establish that this is not the case. More precisely, we supply a family of examples of additive noise driven SDEs, which have smooth drift coefficient functions with at most polynomially growing derivatives and whose solutions do not depend in the strong sense on their initial value in a locally Lipschitz continuous, nor even in a locally Hölder continuous way.</p>

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.

Stochastic approximation with non-additive measurement noise

1998 ◽  
Vol 35 (2) ◽  
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.

Mathematical Models of Papermaking

2001 ◽  
Vol 6 (1) ◽  
pp. 9-19 ◽  
Author(s):  
A. Buikis ◽  
J. Cepitis ◽  
H. Kalis ◽  
A. Reinfelds ◽  
A. Ancitis ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Initial Value Problems ◽  
Moisture Transfer ◽  
Bound Water ◽  
Drying Process ◽  
Initial Value ◽  
First Order ◽  
Heat And Moisture ◽  
The Mathematical Model ◽  
The Web

The mathematical model of wood drying based on detailed transport phenomena considering both heat and moisture transfer have been offered in article. The adjustment of this model to the drying process of papermaking is carried out for the range of moisture content corresponding to the period of drying in which vapour movement and bound water diffusion in the web are possible. By averaging as the desired models are obtained sequence of the initial value problems for systems of two nonlinear first order ordinary differential equations. 

On nonhomeomorphic mappings with the inverse Poletsky inequality

2020 ◽  
Vol 17 (3) ◽  
pp. 414-436
Author(s):  
Evgeny Sevost'yanov ◽  
Serhii Skvortsov ◽  
Oleksandr Dovhopiatyi
Keyword(s):  
Boundary Behavior ◽  
Hölder Continuity ◽  
Continuous Extension ◽  
Quasiconformal Mappings ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Finite Distortion ◽  
Mappings Of Finite Distortion ◽  
Lipschitz Continuous ◽  
Inverse Inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

scholarly journals No Infimum Gap and Normality in Optimal Impulsive Control Under State Constraints

2021 ◽  
Author(s):  
Giovanni Fusco ◽  
Monica Motta
Keyword(s):  
Original Problem ◽  
State Constraints ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Continuous Data ◽  
Lipschitz Continuous ◽  
Extended Sense ◽  
Unbounded Controls ◽  
Locally Lipschitz ◽  
Optimal Impulsive Control

AbstractIn this paper we consider an impulsive extension of an optimal control problem with unbounded controls, subject to endpoint and state constraints. We show that the existence of an extended-sense minimizer that is a normal extremal for a constrained Maximum Principle ensures that there is no gap between the infima of the original problem and of its extension. Furthermore, we translate such relation into verifiable sufficient conditions for normality in the form of constraint and endpoint qualifications. Links between existence of an infimum gap and normality in impulsive control have previously been explored for problems without state constraints. This paper establishes such links in the presence of state constraints and of an additional ordinary control, for locally Lipschitz continuous data.

scholarly journals A graphical solution of a differential equation with application to hill’s treatment of nerve excitation

1937 ◽  
Vol 123 (832) ◽  
pp. 382-395 ◽  
Keyword(s):  
Viscous Medium ◽  
Graphical Solution ◽  
Initial Value ◽  
First Order ◽  
Rate Dependent ◽  
Monomolecular Reaction ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Physical And Chemical ◽  
Nerve Excitation

Linear differential equations with constant coefficients are very common in physical and chemical science, and of these, the simplest and most frequently met is the first-order equation a dy / dt + y = f(t) , (1) where a is a constant, and f(t) a single-valued function of t . The equation signifies that the quantity y is removed at a rate proportional to the amount present at each instant, and is simultaneously restored at a rate dependent only upon the instant in question. Familiar examples of this equation are the charging of a condenser, the course of a monomolecular reaction, the movement of a light body in a viscous medium, etc. The solution of this equation is easily shown to be y = e - t / a { y 0 = 1 / a ∫ t 0 e t /a f(t) dt , (2) where y 0 is the initial value of y . In the case where f(t) = 0, this reduces to the well-known exponential decay of y .

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