johanna.mcbryde@icms.org.uk      0131 650 9816

SMSTC Semester 2 - Mondays 9:00-11:00

This is taught as part of the MIGSAA CDT programme and offered across SMSTC.

Aims

  • to show that SPDEs naturally arise in real world problems
  • to provide the students with basic concepts, techniques and results of the theory of SPDEs
  • to show a range of applications

Content

  1. Basic stochastic analysis: Ito's formula, white noise, spacetime white noise, SDEs. Examples of stochastic PDEs (SPDEs).
  2. Classic problems of nonlinear filtering theory, SPDEs arising in nonlinear filtering
  3. L_2 theory of linear parabolic SPDEs, solvability and regularity in Sobolev spaces
  4. Basic results of the L_p theory of SPDEs
  5. Classical solutions. Method of random characteristics
  6. Some classes of nonlinear SPDEs
  7. Applications in signal processing (filtering) and in population models

After a brief presentation of the probabilistic framework of the course, we motivate our interest in stochastic PDEs. In particular, we show that to solve basic problems in nonlinear filtering theory we have to investigate certain classes of linear and also nonlinear stochastic PDEs. With this and other applications in mind, we develop first a theory, called L_2 theory, on existence, uniqueness and regularity of the solutions to linear parabolic SPDEs, and then more advanced results from the L_p theory will be presented. Hence we get existence and uniqueness of classical solutions to linear (and semilinear) parabolic SPDEs. Moreover we show that these solutions can also be obtained by the method of random characteristics, which generalises the well known method of characteristics used in solving first order (deterministic) PDEs. Next we generalise the L_2 theory to solve nonlinear stochastic evolution equations. Finally we apply the theory of SPDEs presented in the course to solve the main problems of filtering theory we started with. Moreover, we show some other applications in physics and population genetics.

Outcomes

  • familiarity with filtering (data processing) problems and their connections with SPDEs
  • basic knowledge of concepts, results and techniques in solving SPDEs
  • familiarity with a range of applications of SPDEs

Prerequisites

  • Basic notions and results of probability theory and stochastic analysis
  • Elements of functional analysis concerning linear operators on Hilbert and Banach spaces
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Mail : johanna.mcbryde@icms.org.uk