#### Date of Submission

9-22-2010

#### Date of Award

2-22-2011

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Goswami, Debashish (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

Motivated by the major role played by probabilistic models in many areas of science, several quantum (i.e. non-commutative) generalizations of classical probability have been attempted by a number of mathematicians. The pioneering works of K.R. Parthasarathy, L. Accardi, R.L. Hudson, P.A. Meyer and others led to the development of one such non-commutative model called â€˜quantum probabilityâ€™ which has a very rich theory of quantum stochastic calculus a la Hudson and Parthasarathy. Within the framework of quantum stochastic calculus, the â€˜grand designâ€™ that engages us is the canonical construction and study of âˆ—-homomorphic flows (jt)tâ‰¥0 on a given C âˆ— or von-Neumann algebra of observables, say A, where jt : A â†’ A" âŠ— B(Î“(L 2 (R+, k0))) satisfies a differential equation of the formdjt(Â·) = jt(Î¸ Âµ Î½ (Â·))Î›Î½ Âµ (dt),Î› Î½ Âµ (dt) being the well-known quantum stochastic integrators in the Fock-space Î“(L 2 (R+, k0)) (see for example [52]), with k0 being a Hilbert space called the noise space. The vacuum expectation semigroup of the flow, generated by Î¸ 0 0 (Â·), is a contractive semigroup of completely positive maps on the said algebra. In the realm of classical probability, such semigroups are typically the expectation semigroups associated with Markov processes. Of particular importance are the so called heat semigroups on Riemannian manifolds, which are the expectation semigroups associated with manifold-valued Brownian motions. The quantum analogue of â€˜dilation problemâ€™, i.e. to construct a Markov process from its expectation semigroup, is very interesting and important in quantum probability too.There is an interesting confluence of Riemannian geometry and classical stochastic process, under the framework of â€˜stochastic geometryâ€™. In particular there are interesting connections between the geometry of a Riemannian manifold and the probabilistic information obtained from a Brownian motion taking value in the manifold. For example, the geometric invariants of the manifold such as mean curvaturevolume etc. can be obtained from the asymptotic analysis of exit time of the Brownian motion from balls of small volume.It is therefore natural to explore the possibility of extending this philosophy to the quantum set-up, i.e. the possibility of connecting quantum stochastic calculus with non-commutative geometry (a la Connes), leading to a development of â€˜quantum stochastic geometryâ€™. As Brownian motions on manifolds are Markov processes with unbounded generators, it is important for pursuing this programme to have a reasonable theory of quantum stochastic calculus with unbounded coefficients. In this thesis, we shall begin by studying different aspects of quantum stochastic calculus with unbounded coefficients, and in the end, we shall try to connect the theory with non-commutative geometry.In chapter 1, we discuss the basic defintions and results e.g. C* and vonNeumann algebras, quantum isometry group, compact quantum group, quantum stochastic calculus, quantum dynamical semigroup, quantum stop-time etc, that we will be using in this thesis.The first difficulty in dealing with quantum stochastic calculus with unbounded coefficients is the absence of a convenient method for proving the homomorphic property of the quantum stochastic flow.

#### Control Number

ISILib-TH302

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Das, Biswarup Dr., "Quantum Stochastic Flows: Trotter Product Formula, Dilations and Quantum Brownian Motion." (2011). *Doctoral Theses*. 61.

https://digitalcommons.isical.ac.in/doctoral-theses/61

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842837