** **### PUBLICATIONS OF RUTH J. WILLIAMS

** ARTICLES **

For items marked as preprint or accepted, click on the
highlighted word
to see an abstract of the paper and possibly a version in
postscript.

** REFLECTING BROWNIAN MOTIONS: EXISTENCE, UNIQUENESS AND ANALYSIS
**

S. R. S. Varadhan and R. J. Williams, Brownian motion in a wedge with oblique reflection,
Communications in Pure and Applied
Mathematics, 38 (1985), 405-443.
R. J. Williams, Recurrence classification and invariant measure for reflected
Brownian motion in a wedge,
Annals of Probability, 13 (1985), 758-778.
R. J. Williams, Reflected Brownian motion in a wedge: semimartingale property,
Z. Wahr. verw.
Geb. 69 (1985), 161-176.
R. J. Williams, Local time and excursions of reflected Brownian motion in a
wedge, Publ. RIMS Kyoto Univ. 23 (1987), 297-319.
L. N. Trefethen and R. J. Williams,
Conformal mapping solution of Laplace's equation on a polygon
with oblique derivative boundary conditions,
J. Computational and Applied
Mathematics, 14 (1986), 227-249.
Y. Kwon and R. J. Williams, Reflected Brownian motion in a cone with radially homogeneous reflection field, Trans. Amer. Math. Soc. 327 (1991),
739-780.
J. M. Harrison and R. J. Williams, Multidimensional reflected Brownian motions having exponential
stationary distributions, Annals
of Probability, 15 (1987), 115-137.
R. J. Williams, Reflected Brownian motion with skew symmetric data in a
polyhedral domain, Probability Theory and Related Fields,
75 (1987), 459-485.
M. I. Reiman and R. J. Williams, A boundary property of semimartingale reflecting Brownian
motions, Probability Theory and Related
Fields 77 (1988), 87-97, 80 (1989), 633.
R. J. Williams, On time-reversal of reflected Brownian motions, in
Seminar on Stochastic Processes, 1987, eds. E. Cinlar, K. L. Chung, R. K. Getoor, Birkhauser, Boston, 1988, pp. 265-276.
R. J. Williams,
Asymptotic variance parameters for the boundary local times of reflected
Brownian motion
on a compact interval, Journal of Applied Probability, 29 (1992),
996-1002.
L. M. Taylor and R. J. Williams,
Existence and uniqueness of semimartingale reflecting Brownian motions
in an orthant,
Probability Theory and Related Fields, 96 (1993), 283-317.
J. G. Dai and R. J. Williams, Existence and uniqueness of semimartingale
reflecting Brownian motions in convex polyhedrons,
Theory of Probability
and Its Applications, 40 (1995), 1-40. Correctional note, 50 (2006), 346-347.
Paul Dupuis and Ruth J. Williams, Lyapunov functions for semimartingale
reflecting Brownian motions,
Annals of Probability, 22 (1994), 680-702.
M. Menshikov and R. J. Williams,
Passage-time moments for continuous non-negative stochastic
processes and applications,
Advances in
Applied
Probability, 28 (1996), 747-762.
R. J. Williams, Semimartingale reflecting Brownian motions in the orthant,
Stochastic Networks, IMA Volumes
in Mathematics and Its Applications, Volume 71,
eds. F. P. Kelly and R. J. Williams, Springer-Verlag,
New York, 1995, pp. 125-137.
R. J. Williams, An invariance principle for semimartingale
reflecting Brownian motions in an orthant,
Queueing Systems: Theory and Applications, 30 (1998), 5-25.
W. Kang and R. J. Williams, An invariance principle for semimartingale
reflecting Brownain motions in domains with piecewise smooth boundaries,
Annals of Applied
Probability, 17 (2007), 741--779.

** FLUID AND DIFFUSION MODELS FOR QUEUEING NETWORKS **

J. M. Harrison and R. J. Williams,
Brownian models of open queueing networks with homogeneous customer
populations, Stochastics, 22 (1987), 77-115.
J. M. Harrison, R. J. Williams and H. Chen,
Brownian models of closed queueing networks with homogeneous
customer populations, Stochastics and
Stochastics Reports,
29 (1990), 37-74.
R. J. Williams, Brownian models of multiclass queueing networks,
Proceedings 29th IEEE Conference on Decision
and Control, December 1990, 573-574.
J. M. Harrison and R. J. Williams,
On the quasireversibility of a multiclass Brownian service station,
Annals of Probability, 18 (1990), 1249-1268.
J. M. Harrison and R. J. Williams,
Brownian models of feedforward queueing networks: quasireversibility
and product form solutions,
Annals of Applied Probability,
2 (1992), 263-293.
J. M. Harrison and R. J. Williams,
A multiclass closed queueing network with unconventional
heavy traffic behavior,
Annals of Applied Probability, 6
(1996), 1-47.
R. J. Williams, On the approximation of queueing networks
in heavy traffic, in
Stochastic Networks: Theory and Applications, Proceedings of
Royal Society Workshop,
Edinburgh, 1995, F. P. Kelly, S. Zachary, I. Ziedins (eds),
Oxford University Press, Oxford, 1996;
pp. 35-56.
R. J. Williams, Some recent developments for queueing networks,
in Probability
Towards 2000, L. Accardi and C. C. Heyde (eds), Springer, 1998; 340-356.
R. J. Williams, Diffusion approximations for open multiclass queueing networks:
sufficient conditions involving state space collapse,
Queueing Systems: Theory and Applications, 30 (1998),
27-88.
R. J. Williams, Reflecting diffusions and queueing
networks, Invited paper, Proceedings of the
International Congress of Mathematicians, Berlin, 1998.
H. C. Gromoll, A. L. Puha and R. J. Williams,
The fluid limit of a heavily loaded
processor sharing queue,
Annals of Applied Probability, 12 (2002), 797-859.
A. L. Puha and R. J. Williams,
Invariant states and rates of convergence for a critical fluid model of a processor sharing
queue,
Annals of Applied Probability, 14 (2004), 517-554.

* Related work: * H. C. Gromoll, Diffusion approximation for a processor sharing queue in heavy traffic,
Annals of Applied Probability, 14 (2004), 555-611.

A. L. Puha, A. L. Stolyar and R. J. Williams,
The fluid limit of an overloaded processor sharing queue,
Mathematics of Operations Research,
31 (2006), 316-350.

A. L. Puha and R. J. Williams,
Asymptotic behavior of a critical fluid model for a processor sharing queue via relative entropy, Stochastic Systems,
6 (2016), 251-300.

** ANALYSIS AND CONTROL OF STOCHASTIC PROCESSING NETWORKS **
R. J. Williams,
On dynamic scheduling of a parallel server system with
complete resource pooling,
in "Analysis of Communication Networks: Call Centres, Traffic and Performance", D. R. McDonald and S. R. E. Turner
(eds.), Fields Institute Communications Volume 28, American Mathematical Society, 2000, pp. 49-71.
S. L. Bell and R. J. Williams,
Dynamic scheduling of a system with two parallel servers:
asymptotic optimality of a continuous review threshold policy
in heavy traffic, Proceedings of the
38th IEEE Conference on Decision and Control, Phoenix, December
1999, 2255-2260.
S. L. Bell and R. J. Williams,
Dynamic scheduling of a system with two parallel servers in
heavy traffic with resource pooling:
asymptotic optimality of a threshold policy,
Annals of Applied Probability, 11 (2001), 608-649.
S. L. Bell and R. J. Williams,
Dynamic scheduling of a parallel server system in heavy traffic with
complete resource pooling: asymptotic optimality of a
threshold policy,
Electronic J. of Probability, 10 (2005), 1044-1115.
M. Bramson and R. J. Williams,
On dynamic scheduling of stochastic networks in heavy traffic and some new results for the workload process,
Proceedings of the
39th IEEE Conference on Decision and Control, December 2000, 516-521.
M. Bramson and R. J. Williams,
Two workload properties for Brownian networks,
Queueing Systems, 45 (2003), 191-221.
F. P. Kelly and R. J. Williams,
Fluid model for a network operating under a fair bandwidth-sharing policy,
Annals of Applied Probability,
14 (2004), 1055-1083.
J. M. Harrison and R. J. Williams,
Workload reduction of a generalized Brownian network,
Annals
of Applied Probability, 15 (2005), 2255-2295.
W. Kang, F. P. Kelly, N. H. Lee and R. J. Williams,
Fluid and Brownian approximations for an Internet congestion control model,
Proceedings of the 43rd IEEE Conference on Decision and
Control, December 2004, 3938-3943.
J. M. Harrison and R. J. Williams,
Workload interpretation for Brownian models of stochastic
processing networks,
Mathematics of Operations Research, 32 (2007), 808--820.
S. Bhardwaj, R. J. Williams and A. S. Acampora,
On the performance of a two-user downlink system in heavy traffic,
IEEE Transactions on Information Theory,
Vol. 53 (2007), 1851-1859.
H. C. Gromoll and R. J. Williams,
Fluid limits for networks with bandwidth
sharing and general document size distributions,
Annals of Applied Probability,
19 (2009), 243-280.
H. C. Gromoll and R. J. Williams,
Fluid model for a data network with alpha fair bandwidth
sharing and general document size distributions: two examples
of stability, IMS Collections, Markov
Processes and Related Topics: A Festschrift for Thomas G. Kurtz, Vol. 4
(2008), 253--265.
R. Atar, A. Budhiraja and R. J. Williams,
HJB equations for certain singularly controlled diffusions,
Annals of Applied Probability, 17 (2007), 1745-1776.
W. N. Kang, F. P. Kelly, N. H. Lee and R. J. Williams,
State space collapse and diffusion approximation for a network
operating under a fair bandwidth sharing policy.
Annals of Applied Probability, 19 (2009), 1719-1780.
S. Bhardwaj and R. J. Williams,
Diffusion approximation for a heavily loaded multi-user wireless
communication system with cooperation,
Queueing Systems, 62 (2009), 345--382.
F. P. Kelly and R. J. Williams,
Heavy traffic on a controlled motorway,
Festschrift volume in honor
of J. F. C. Kingman,
N. H. Bingham and C. M. Goldie (eds.), London
Mathematical Society, Lecture Notes Series, 2009.
W. N. Kang and R. J. Williams,
Diffusion approximation for an input-queued switch operating under a maximum weight matching policy, Stochastic
Systems, 2 (2012), 277-321.
V. Pesic and R. J. Williams,
Dynamic Scheduling for Parallel Server Systems in Heavy Traffic:
Graphical Structure, Decoupled Workload Matrix and Some Sufficient Conditions for Solvability of the Brownian Control Problem,
Stochastic Systems, 6 (2016), 26-89.
R. J. Williams, Stochastic Processing Networks, to appear in Annual Review of Statistics and Its Application.

** SYMMETRIC REFLECTED DIFFUSIONS **

R. J. Williams and W. A. Zheng, On reflecting Brownian motion -- a weak convergence approach,
W. A. Zheng, Annales de l'Institut Henri Poincare, 26 (1990),
461-488.
R. J. Williams,
Reflected Brownian motion: Hunt process and semimartingale representation
, in Barcelona Seminar on Stochastic Analysis, September 1991, Progress in
Probability Series, Birkhauser,
Boston, 1993, pp.
216-221.
Z. Q. Chen, P. J. Fitzsimmons and R. J. Williams,
Reflecting Brownian motions: quasimartingales and strong
Caccioppoli sets,
Potential Analysis, 2 (1993), 219-243.

E. Pardoux and R. J. Williams, Symmetric reflected diffusions,
Annales de l'Institut Henri Poincare, 30 (1994), 13-62.

** DIFFUSIONS AND PARTIAL DIFFERENTIAL EQUATIONS **

Z. Q. Chen, R. J. Williams and Z. Zhao,
On the existence of positive solutions
of semilinear elliptic equations with Dirichlet
boundary conditions,
Mathematische Annalen, 298 (1994), 543-556.

Z. Q. Chen, R. J. Williams and Z. Zhao,
A Sobolev inequality and Neumann heat kernel estimate
for unbounded domains,
Mathematical Research Letters, 1
(1994), 177-184.

Z. Q. Chen, R. J. Williams and Z. Zhao,
Non-negative solutions for semilinear
elliptic equations with boundary conditions --
a probabilistic approach,
Proceedings
of Symposia in Pure Mathematics: Stochastic Analysis, Vol. 57,
eds. M. Cranston and M. Pinsky,
American Mathematical Society, Providence, RI, 1995, pp. 65-82.

Z. Q. Chen, R. J. Williams and Z. Zhao,
On the existence of positive solutions for semilinear
elliptic equations with Neumann boundary conditions,
Probability Theory
and Related Fields, 101 (1995), 251-276.

Z. Q. Chen, R. J. Williams and Z. Zhao,
On the existence of positive solutions for semilinear
elliptic equations with singular lower
order coefficients and Dirichlet boundary conditions,
Math. Annalen., 315 (1999), 735-769.

S. N. Evans and R. J. Williams, Transition operators of diffusions
reduce zero-crossing,
Transactions of the American Mathematical Society, 351 (1999), 1377-1389.
.

** FEYNMAN-KAC GAUGE AND THE SCHRODINGER EQUATION **

R. J. Williams, A Feynman-Kac gauge for solvability of the Schrodinger
equation, Advances in Applied Mathematics, 6 (1985), 1-3.
K. L. Chung, P. Li and R. J. Williams,
Comparison of probability and classical methods for the
Schrodinger equation,
Expositiones Mathematicae, 4 (1986), 271-278.
R. J. Williams, On K. L. Chung's research on stopped Feynman-Kac functionals
and the Schrodinger equation. Preprint.

** BROWNIAN MOTION ANALYSIS **

R. J. Williams, Brownian motion with polar drift, Transactions
of the American Mathematical Society,
292 (1985), 225-246.
K. Burdzy and R. J. Williams,
On Brownian excursions in Lipschitz domains, Part I: Local path
properties, Transactions
of the American Mathematical Society, 298 (1986),
289-306.
K. Burdzy, E. H. Toby and R. J. Williams,
On Brownian excursions in Lipschitz domains, Part II: Local
asymptotic distributions, in
Seminar on Stochastic Processes, 1988, eds. E. Cinlar, K. L. Chung,
R. K. Getoor, Birkhauser, Boston, 1989, pp. 55-85.

** STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATIONS **

R. J. Williams, Some connections between Brownian motion and analysis, via stochastic
calculus, Topics in
Contemporary Probability and Its Applications,
CRC Press, Boca Raton, ed.
J. L. Snell, 1995, pp. 75-88.

P. J. Brockwell and R. J. Williams,
On the existence and application of continuous-time
threshold autoregressions of order two,
Advances in Applied Probability,
29 (1997), 205-227.

J. R. Movellan, P. Mineiro and R. J. Williams,
A
Monte Carlo EM approach for partially observable diffusion processes:
theory and applications to neural networks,
Neural Computation,
Volume 14, Number 7, July 1, 2002.
H. Deng, M. Krstic and R. J. Williams, Stabilization of stochastic
nonlinear systems driven by noise of unknown covariance,
IEEE Transactions on Automatic Control, 46 (2001), 1237-1253.

M. S. Kinnally and R. J. Williams, On existence and uniqueness of stationary distributions for stochastic delay differential equations with positivity constraints,
Electronic J. of Probability, Vol. 15 (2010), 409-451.

D. Lipshutz and R. J. Williams, Existence, uniqueness and stability of slowly oscillating
periodic solutions for delay differential equations with
non-negativity constraints, SIAM J. on Mathematical Analysis, 47 (2015), 4467-4535.
** BIOCHEMICAL REACTION NETWORKS AND SYNTHETIC BIOLOGY **

G. Craciun, J. W. Helton, and R. J. Williams,
Homotopy methods for counting reaction network equilibria,
Mathematical Biosciences, Volume 216, Issue 2, December 2008, pages 140-149.
W. H. Mather, N. A. Cookson, J. Hasty, L. S. Tsimring
and R. J. Williams,
Correlation resonance generated by coupled enzymatic processing,
Biophysical Journal, 99 (2010), 3172-3181.
W. H. Mather, J. Hasty, L. S. Tsimring and R. J. Williams,
Factorized time-dependent distributions for certain multiclass queueing networks and an application to enzymatic processing networks,
Queueing
Systems, 69 (2011), 313-328.
N. A. Cookson, W. H. Mather, T. Danino, O. Mondragon-Palomino, R. J. Williams, L. S. Tsimring and J. Hasty,
Queueing up for enzymatic processing: correlated signaling through coupled degradation,
Molecular Systems Biology 7 (2011), Article number: 561, doi:10.1038/msb.2011.94.
W. H. Mather, J. Hasty, L. Tsimring and R. J. Williams,
Translational cross talk in gene networks,
Biophysical Journal, 104 (2013), 2564--2572, published June 4, 2013.
DOI: 10.1016/j.bpj.2013.04.049.
P. J. Steiner, R. J. Williams, Jeff Hasty, and L. S. Tsimring,
Criticality and adaptivity in enzymatic networks,
Biophysical Journal, Vol. 111, Issue 5, 2016, 1078-1087.
DOI: http://dx.doi.org/10.1016/j.bpj.2016.07.036.
** GAME THEORY **

R. J. Williams, Sufficient conditions for Nash equilibria in N-person games
over reflexive Banach spaces, J. Optimization Theory and
Applications, 30 (1980), 383-394.

R. J. Williams, Mixed strategy solutions for N-person quadratic games, J. Optimization
Theory and Applications, 30 (1980), 569-582.

** BOOKS **

An Introduction to Stochastic Integration, K. L. Chung and R. J.
Williams,
Birkhauser, Boston, 2nd edition, 1990. For a list of some
errata, click here.

Proceedings, Seminar on Stochastic Processes, 1989, Editors: E. Cinlar,
K. L. Chung,
R. K. Getoor, Managing Editors: P. J. Fitzsimmons and R. J. Williams,
Birkhauser, Boston, 1990.

Proceedings, Seminar on Stochastic Processes, 1990, Editor: E. Cinlar,
Managing Editors: P. J. Fitzsimmons and R. J. Williams,
Birkhauser, Boston, 1991.

Stochastic Networks, IMA Volumes
in Mathematics and Its Applications, Volume 71,
eds. F. P. Kelly and R. J. Williams, Springer-Verlag,
New York, 1995.
Introduction to the Mathematics of Finance, R. J. Williams,
American
Mathematical Society, 2006. For some additional exercises and
a list of errata for this book,
click here.
Selected Works of Kai Lai Chung, F. AitSahlia, E. P. Hsu, R. J. Williams (eds.), World Scientific, Singapore, 2008.