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Brownian motion and its applications to PDEs
PD Dr. Yana Kinderknecht
News.
Tutorials: each week on Wednesday, 12:00-12:45, SR 1 (U.37). First meeting on 30.10.2019.
Lecturer.
PD Dr. Yana A. Kinderknecht , Geb. E2 4, Room. 209, Tel.: 302-4743Content.
- Einstein's explanation of Brownian motion provided the cornerstone which underlies deep connections between stochastic processes and evolution equations. In this lecture course, we introduce a mathematical model of a physical Brownian motion (using a random walk approach), we discuss different realisations and study basic properties of a (mathematical) Brownian motion. We overview some stochastical and analytical tools from Martingale Theory, Ito Calculus and Theory of Operator Semigroups. Using these tools, we obtain stochastic representations (or Feynman-Kac formulae) for solutions of some parabolic and Schrödinger-type equations. These stochastic representations are nothing else but expectation of some suitable functionals of Brownian motion. Such expressions can be understood as path integrals with respect to a probability measure (generated by Brownian motion) and can be used to model the corresponding evolution via numerical methods (Methods of Monte Carlo). Further, we go beyond Brownian motion and discuss other stochastic processes appearing as suitable scaling limits of (continuous time) random walks. Such processes are related to the so-called anomalous diffusion and include both some Markovian and some non-Markovian processes. The corresponding evolution equations contain fractional time- and/or space-derivatives.
The lecture course is designed in a self-contained manner and can be taken independently from / parallely to "Stochastik II".
- Einstein's explanation of Brownian motion provided the cornerstone which underlies deep connections between stochastic processes and evolution equations. In this lecture course, we introduce a mathematical model of a physical Brownian motion (using a random walk approach), we discuss different realisations and study basic properties of a (mathematical) Brownian motion. We overview some stochastical and analytical tools from Martingale Theory, Ito Calculus and Theory of Operator Semigroups. Using these tools, we obtain stochastic representations (or Feynman-Kac formulae) for solutions of some parabolic and Schrödinger-type equations. These stochastic representations are nothing else but expectation of some suitable functionals of Brownian motion. Such expressions can be understood as path integrals with respect to a probability measure (generated by Brownian motion) and can be used to model the corresponding evolution via numerical methods (Methods of Monte Carlo). Further, we go beyond Brownian motion and discuss other stochastic processes appearing as suitable scaling limits of (continuous time) random walks. Such processes are related to the so-called anomalous diffusion and include both some Markovian and some non-Markovian processes. The corresponding evolution equations contain fractional time- and/or space-derivatives.
Conditions.
Time and place: Thursdays, 12:00 -14:00, SR 9, Geb. E2 4.
Preliminary knowledge: Necessary: Analysis I-II, Probability. Not necessary but helpful: Functional Analysis, Stochastic Processes (Stochastik II).
Language: English or German (by agreement).
Certificate.
Credit points: 4,5 CP.
Criteria for obtaining the certificate: 1. Active participation in lectures and tutorials. 2. Passing of oral exam.
Literature.
Main literature:
[1] Schilling R.L., Partzsch L. Brownian motion. An introduction to stochastic processes. 2014.
[2] Lörinczi J., Hiroshima F., Betz V. Feynman-Kac-Type Theorems and Gibbs Measures on Path Space. 2019.Additional literature:
[1] Simon B. Functional Integration and Quantum Physics. 1996.
[2] Reed M., Simon B. Methods of Modern Mathematical Physics. Vol. 2. 1975.
[3] Karatzas I., Shreve S.E. Brownian Motion and Stochastic Calculus. 1988.
[4] Freidlin M. Functional Integration and Partial Differential Equations. 1985.
[5] Oksendal B. Stochastic Differential Equations.1998.
[6] Einstein A. Investigations on the Theory of Brownian Movement. 1956 (Dover, New York)
[7] Doss H. Sur une Resolution Stochastique de l'Equation de Schrödinger a Coefficients Analytiques. Comm. Math. Phys. 73 (1980), 247-264.
[8] Metzler R., Klafter J. The Random Walk's Guide to Anomalous Diffusion: a Fractional Dynamics Approach. Physics Reports 339 (2000), 1-77.