# A Probabilistic Approach to Large Time Behaviour of Viscosity Solutions of Parabolic Equations with Neumann Boundary Conditions

A Probabilistic Approach to Large Time Behaviour of Viscosity Solutions of Parabolic Equations... This paper is devoted to the study of the large time behaviour of viscosity solutions of parabolic equations with Neumann boundary conditions. This work is the sequel of Hu et al. (SIAM J Control Optim 53:378–398, 2015) in which a probabilistic method was developed to show that the solution of a parabolic semilinear PDE behaves like a linear term $$\lambda T$$ λ T shifted with a function v, where $$(v,\lambda )$$ ( v , λ ) is the solution of the ergodic PDE associated to the parabolic PDE. We adapt this method in finite dimension by a penalization method in order to be able to apply an important basic coupling estimate result and with the help of a regularization procedure in order to avoid the lack of regularity of the coefficients in finite dimension. The advantage of our method is that it gives an explicit rate of convergence. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# A Probabilistic Approach to Large Time Behaviour of Viscosity Solutions of Parabolic Equations with Neumann Boundary Conditions

, Volume 74 (2) – Oct 23, 2015
30 pages

/lp/springer_journal/a-probabilistic-approach-to-large-time-behaviour-of-viscosity-su2MjBiKwk
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-015-9318-0
Publisher site
See Article on Publisher Site

### Abstract

This paper is devoted to the study of the large time behaviour of viscosity solutions of parabolic equations with Neumann boundary conditions. This work is the sequel of Hu et al. (SIAM J Control Optim 53:378–398, 2015) in which a probabilistic method was developed to show that the solution of a parabolic semilinear PDE behaves like a linear term $$\lambda T$$ λ T shifted with a function v, where $$(v,\lambda )$$ ( v , λ ) is the solution of the ergodic PDE associated to the parabolic PDE. We adapt this method in finite dimension by a penalization method in order to be able to apply an important basic coupling estimate result and with the help of a regularization procedure in order to avoid the lack of regularity of the coefficients in finite dimension. The advantage of our method is that it gives an explicit rate of convergence.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 23, 2015

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