scholarly journals On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
V. P. Kurenok
Keyword(s):  
Stochastic Equation ◽  
Existence Of Solutions ◽  
Stable Process ◽  
Sufficient Conditions ◽  
Stochastic Equations ◽  
One Dimensional ◽  
Symmetric Stable Process ◽  
Measurable Coefficients ◽  
Existence Proofs ◽  
Dimensional Equation

We consider a one-dimensional stochastic equation , , with respect to a symmetric stable process of index . It is shown that solving this equation is equivalent to solving of a 2-dimensional stochastic equation with respect to the semimartingale and corresponding matrix . In the case of we provide new sufficient conditions for the existence of solutions of both equations with measurable coefficients. The existence proofs are established using the method of Krylov's estimates for processes satisfying the 2-dimensional equation. On another hand, the Krylov's estimates are based on some analytical facts of independent interest that are also proved in the paper.

The Time Change Method and SDEs with Nonnegative Drift

2010 ◽  
Vol 53 (3) ◽  
pp. 503-515
Author(s):  
V. P. Kurenok
Keyword(s):  
Stochastic Equation ◽  
Existence Of Solutions ◽  
Stochastic Equations ◽  
Time Change ◽  
Stable Processes ◽  
Existence Proof ◽  
Symmetric Stable Processes ◽  
Measurable Coefficients

AbstractUsing the time change method we show how to construct a solution to the stochastic equation dXt = b(Xt–)dZt + a(Xt )dt with a nonnegative drift a provided there exists a solution to the auxililary equation dLt = [a–1/αb](Lt–) + dt where Z, ¯ are two symmetric stable processes of the same index α ∈ (0, 2]. This approach allows us to prove the existence of solutions for both stochastic equations for the values 0 < α < 1 and only measurable coefficients a and b satisfying some conditions of boundedness. The existence proof for the auxililary equation uses the method of integral estimates in the sense of Krylov.

scholarly journals EXISTENCE OF SIGN-CHANGING SOLUTIONS FOR THE NONLINEAR p-LAPLACIAN BOUNDARY VALUE PROBLEM

2013 ◽  
Vol 11 (02) ◽  
pp. 1350004 ◽  
Author(s):  
WEI-CHENG LIAN ◽  
WEI-CHUAN WANG ◽  
YAN-HSIOU CHENG
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
One Dimensional ◽  
Laplacian Equation ◽  
Separated Boundary Conditions ◽  
Sign Changing Solutions ◽  
Nodal Properties

We study the nonlinear one-dimensional p-Laplacian equation [Formula: see text] where p > 1, y(p-1) = |y|p-1 sgn y = |y|p-2y, with linear separated boundary conditions. We give sufficient conditions for the existence of solutions with prescribed nodal properties concerning the behavior of f(s)/s(p-1) when s is at infinity and zero, respectively. These results are more general and complementary than previous known ones for the case when p = 2 and q is nonnegative.

On Multidimensional SDEs Without Drift and with A Time-Dependent Diffusion Matrix

2000 ◽  
Vol 7 (4) ◽  
pp. 643-664 ◽  
Author(s):  
H. J. Engelbert ◽  
V. P. Kurenok
Keyword(s):  
Stochastic Process ◽  
Weak Solutions ◽  
Sufficient Conditions ◽  
Stochastic Equations ◽  
Diffusion Matrix ◽  
Homogeneous Case ◽  
Initial State ◽  
Existence Of Weak Solutions ◽  
One Dimensional ◽  
Arbitrary Initial State

Abstract We study multidimensional stochastic equations where x o is an arbitrary initial state, W is a d-dimensional Wiener process and is a measurable diffusion coefficient. We give sufficient conditions for the existence of weak solutions. Our main result generalizes some results obtained by A. Rozkosz and L. Słomiński [Stochastics Stochasties Rep. 42: 199–208, 1993] and T. Senf [Stochastics Stochastics Rep. 43: 199–220, 1993] for the existence of weak solutions of one-dimensional stochastic equations and also some results by A. Rozkosz and L. Słomiński [Stochastic Process. Appl. 37: 187–197, 1991], [Stochastic Process. Appl. 68: 285–302, 1997] for multidimensional equations. Finally, we also discuss the homogeneous case.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

On Approximate Large Deviations for 1D Diffusion

2003 ◽  
Vol 10 (2) ◽  
pp. 381-399
Author(s):  
A. Yu. Veretennikov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Large Deviations ◽  
Stochastic Differential Equation ◽  
Approximation Scheme ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Euler Approximation ◽  
One Dimensional

Abstract We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of a one-dimensional stochastic differential equation on the torus is close to that for an exact solution of this equation.

scholarly journals Limit of 𝑝-Laplacian obstacle problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Raffaela Capitanelli ◽  
Maria Agostina Vivaldi
Keyword(s):  
Asymptotic Behavior ◽  
Uniform Convergence ◽  
Explicit Expression ◽  
Positive Measure ◽  
Sufficient Conditions ◽  
Obstacle Problems ◽  
Dimensional Case ◽  
Asymptotic Behavior Of Solutions ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as {p\to\infty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for data f (that do not change sign in Ω) possibly vanishing in a set of positive measure.

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

2021 ◽  
pp. 108128652110238
Author(s):  
Barış Erbaş ◽  
Julius Kaplunov ◽  
Isaac Elishakoff
Keyword(s):  
Ad Hoc ◽  
Moving Load ◽  
Low Frequency ◽  
Elastic Beam ◽  
Winkler Foundation ◽  
One Dimensional ◽  
Beam Equations ◽  
Dimensional Equation ◽  
Phase And Group Velocities ◽  
Group Velocities

A two-dimensional mixed problem for a thin elastic strip resting on a Winkler foundation is considered within the framework of plane stress setup. The relative stiffness of the foundation is supposed to be small to ensure low-frequency vibrations. Asymptotic analysis at a higher order results in a one-dimensional equation of bending motion refining numerous ad hoc developments starting from Timoshenko-type beam equations. Two-term expansions through the foundation stiffness are presented for phase and group velocities, as well as for the critical velocity of a moving load. In addition, the formula for the longitudinal displacements of the beam due to its transverse compression is derived.

scholarly journals Existence of Solutions of Nonlinear Stochastic Volterra Fredholm Integral Equations of Mixed Type

2010 ◽  
Vol 2010 ◽  
pp. 1-16 ◽  
Author(s):  
K. Balachandran ◽  
J.-H. Kim
Keyword(s):  
Integral Equations ◽  
Existence And Uniqueness ◽  
Mixed Type ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Stochastic Integral ◽  
Sufficient Conditions ◽  
Fredholm Integral Equations ◽  
Equations Of Mixed Type ◽  
Stochastic Integral Equations

We establish sufficient conditions for the existence and uniqueness of random solutions of nonlinear Volterra-Fredholm stochastic integral equations of mixed type by using admissibility theory and fixed point theorems. The results obtained in this paper generalize the results of several papers.

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