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I am a quantum probabilist, with most of my work focusing on quantum stochastic differential equations (QSDEs), for example giving existence and uniqueness results, as well as studying properties of the solutions. I make use of many areas of functional analysis (semigroups, operator algebras and operator spaces) in order to discuss processes on C*-algebras and operator spaces, and QSDEs with unbounded coefficients. Such equations are used in quantum optics, and quantum control and statistics.

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Research Interests

Quantum probability Quantum stochastic calculus Operator algebras Operator spaces

My main interest is in the field of quantum probability. This is a noncommutative generalisation of traditional probability theory in which the commutative algebra of (bounded) random variables is replaced by a more general operator algebra. The application of traditional probability to quantum mechanics is problematic in the sense that it cannot adequately deal with phenomenon such as Heisenberg's Uncertainty Principle; the quantised version of probability seeks to rectify this.

The passage of any mathematical theory from its traditional form to a quantised version throws up a number of questions. Ones that are relevant for quantum probability include

• Does the law of large numbers still work?
• Are there conditional expectations?
• What is independence?

The answers for the above are yes, not always, and you have a choice, respectively. Once you make a choice for independence it is then possible to discuss central limit theorems and deduce what should be considered to be a normal random variable. With this then comes the chance to talk about quantised versions of Brownian motion (and other stochastic processes), and related quantum stochastic calculi.

Research Grants

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Peer Reviewed Journals

	Year	Publication
	(2014)	'An algebraic construction of quantum flows with unbounded generators' Belton, Alexander C.R.; Wills, Stephen J. (2014) 'An algebraic construction of quantum flows with unbounded generators'. Annales De L'Institut Henri Poincaré (B) Probabilités Et Statistiques, [Details]
	(2014)	'Quantum stochastic cocycles and completely bounded semigroups on operator spaces' Lindsay, J. Martin; Wills, Stephen J. (2014) 'Quantum stochastic cocycles and completely bounded semigroups on operator spaces'. International Mathematics Research Notices, 2014 (11):3096-3139 [DOI] [Details]
	(2010)	'E-semigroups subordinate to CCR flows' Wills, Stephen J.; (2010) 'E-semigroups subordinate to CCR flows'. Communications on Stochastic Analysis, 4 (4):579-592 [Details]
	(2007)	'Quantum stochastic operator cocycles via associated semigroups' Lindsay, J. Martin; Wills, Stephen J. ; (2007) 'Quantum stochastic operator cocycles via associated semigroups'. Mathematical Proceedings of the Cambridge Philosophical Society, 142 (3):535-556 [Details]
	(2007)	'On the generators of quantum stochastic operator cocycles' Wills, Stephen J. ; (2007) 'On the generators of quantum stochastic operator cocycles'. Markov Processes and Related Fields, 13 (1):191-211 [Details]
	(2006)	'Construction of some quantum stochastic operator cocycles by the semigroup method' Lindsay, J. Martin; Wills, Stephen J. ; (2006) 'Construction of some quantum stochastic operator cocycles by the semigroup method'. Proceedings of the Indian Academy of Sciences: Mathematical Sciences, 116 (4):519-529 [Details]
	(2003)	'Solving quantum stochastic differential equations with unbounded coefficients' Fagnola, Franco; Wills, Stephen J. ; (2003) 'Solving quantum stochastic differential equations with unbounded coefficients'. Journal of Functional Analysis, 198 (2):279-310 [Details]
	(2003)	'Dilation of Markovian cocycles on a von Neumann algebra' Goswami, Debashish; Lindsay, J. Martin; Sinha, Kalyan B.; Wills, Stephen J. ; (2003) 'Dilation of Markovian cocycles on a von Neumann algebra'. Pacific Journal of Mathematics, 211 (2):221-247 [Details]
	(2003)	'Homomorphic Feller cocycles on a C*-algebra' Lindsay, J. Martin; Wills, Stephen J. ; (2003) 'Homomorphic Feller cocycles on a C*-algebra'. Journal of the London Mathematical Society, 68 (2) (1):255-272 [Details]
	(2001)	'A stochastic Stinespring theorem' Goswami, Debashish; Lindsay, J. Martin; Wills, Stephen J. ; (2001) 'A stochastic Stinespring theorem'. Mathematische Annalen, 319 (4):647-673 [Details]
	(2001)	'Existence of Feller cocycles on a C*-algebra' Lindsay, J. Martin; Wills, Stephen J. ; (2001) 'Existence of Feller cocycles on a C*-algebra'. Bulletin of the London Mathematical Society, 33 (5):613-621 [Details]
	(2000)	'Mild solutions of quantum stochastic differential equations' Fagnola, Franco ; Wills, Stephen J. ; (2000) 'Mild solutions of quantum stochastic differential equations'. Electronic Communications in Probability, 5 :158-171 [Details]
	(2000)	'Markovian cocycles on operator algebras adapted to a Fock filtration' Lindsay, J. Martin; Wills, Stephen J.; (2000) 'Markovian cocycles on operator algebras adapted to a Fock filtration'. Journal of Functional Analysis, 178 (2):269-305 [Details]
	(2000)	'Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise' Lindsay, J. Martin; Wills, Stephen J. ; (2000) 'Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise'. Probability Theory and Related Fields, 116 (4):505-543 [Details]

Conference Publications

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Teaching Interests

2005-06: MA4403, MS2001, MS3001
2006-07: MA1008, MA2055, MA4403, MS2001
2007-08: MA1008, MA4052, MA4403, MS2005
2008-09: MA1008, MA4052, MA4403, MS2005
2009-10: MA1008, MA2051, MA4052, MS2005/MS5005
2010-11: MA1008, MA2051, MA4052, MS2005/MS5005
2011-12: MA1008, MA2051, MA4052, MS2005/MS5005
2012-13: MA1008, MA2051, MA4052
2013-14: MA1008, MA3051, MA4052, MS2002
2014-15: MA1008, MA3051, MS2002