# A Characterization of the Reflected Quasipotential

A Characterization of the Reflected Quasipotential Recent interest in the reflected quasipotential comes from the queueing theory literature, specifically the analysis of so-called $$(b,A,D)$$ ( b , A , D ) reflected Brownian motion where it is the large deviation rate function for the stationary distribution. Our purpose here is to characterize the reflected quasipotential in terms of a first-order Hamilton–Jacobi equation. Using conventional dynamic programming ideas, along with a complementarity problem formulation of the effect of the Skorokhod map on absolutely continuous paths, we will derive necessary conditions in the form of viscosity-sense boundary conditions. It turns out that even with these boundary conditions solutions are not unique. Thus a unique characterization needs to refer to some additional property of $$V(\cdot )$$ V ( · ) . We establish such a characterization in two dimensions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# A Characterization of the Reflected Quasipotential

, Volume 72 (3) – Dec 1, 2015
34 pages

/lp/springer_journal/a-characterization-of-the-reflected-quasipotential-06vSEh1pXK
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/s00245-014-9286-9
Publisher site
See Article on Publisher Site

### Abstract

Recent interest in the reflected quasipotential comes from the queueing theory literature, specifically the analysis of so-called $$(b,A,D)$$ ( b , A , D ) reflected Brownian motion where it is the large deviation rate function for the stationary distribution. Our purpose here is to characterize the reflected quasipotential in terms of a first-order Hamilton–Jacobi equation. Using conventional dynamic programming ideas, along with a complementarity problem formulation of the effect of the Skorokhod map on absolutely continuous paths, we will derive necessary conditions in the form of viscosity-sense boundary conditions. It turns out that even with these boundary conditions solutions are not unique. Thus a unique characterization needs to refer to some additional property of $$V(\cdot )$$ V ( · ) . We establish such a characterization in two dimensions.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Dec 1, 2015

### References

• Explicit solutions for variational problems in the quadrant
Avram, F; Dai, JG; Hasenbein, JJ

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Unlimited reading Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere. ### Stay up to date Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates. ### Organize your research It’s easy to organize your research with our built-in tools. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. ### DeepDyve Freelancer ### DeepDyve Pro Price FREE$49/month

\$360/year
Save searches from