Content: "Stochastics IV: Interacting particles and (stochastic) PDEs"
Physical systems are often described by partial diff erential equations (PDEs). For example, the flow of heat through a medium is governed by the heat equation, and the dynamics of viscous fl uids are described by the Navier-Stokes equation. These equations are derived with physical arguments that operate on a macroscopic (observable) scale and do not involve the behavior of single molecules. On the other hand, if we would zoom into the physical systems, we would see only single molecules, moving seemingly chaotically. The question we try to approach in this course is whether we can derive the equations governing the observable behavior of certain physical systems by considering simple random dynamics for the single molecules and then zooming out to the observable scale. Concretely, we will consider interacting particle systems such as exclusion processes and zero range processes as the microscopic models.

• Continuous time Markov chains: characterization and construction;
• Martingale problems;
• Invariant measures;
• Simple exclusion and zero range processes;
• Tightness on Skorokhod space;
• Hydrodynamic limits;
• Fluctuation results;
• Local equilibrium, entropy methods

Literature:

• Kipnis, Landim: Scaling Limits of Interacting Particle Systems, Springer, 1999
• Further literature will be announced during the lecture, and there will also be lecture notes made available in Whiteboard.

Prerequisite: Stochastics I, II.
Recommended: Stochastics III and Functional Analysis.

Content: "Stochastics IV: Interacting particles and (stochastic) PDEs"
Physical systems are often described by partial diff erential equations (PDEs). For example, the flow of heat through a medium is governed by the heat equation, and the dynamics of viscous fl uids are described by the Navier-Stokes equation. These equations are derived with physical arguments that operate on a macroscopic (observable) scale and do not involve the behavior of single molecules. On the other hand, if we would zoom into the physical systems, we would see only single molecules, moving seemingly chaotically. The question we try to approach in this course is whether we can derive the equations governing the observable behavior of certain physical systems by considering simple random dynamics for the single molecules and then zooming out to the observable scale. Concretely, we will consider interacting particle systems such as exclusion processes and zero range processes as the microscopic models.

• Continuous time Markov chains: characterization and construction;
• Martingale problems;
• Invariant measures;
• Simple exclusion and zero range processes;
• Tightness on Skorokhod space;
• Hydrodynamic limits;
• Fluctuation results;
• Local equilibrium, entropy methods

Literature:

• Kipnis, Landim: Scaling Limits of Interacting Particle Systems, Springer, 1999
• Further literature will be announced during the lecture, and there will also be lecture notes made available in Whiteboard.

Prerequisite: Stochastics I, II.
Recommended: Stochastics III and Functional Analysis.

Content: "Stochastics IV: Interacting particles and (stochastic) PDEs"
Physical systems are often described by partial diff erential equations (PDEs). For example, the flow of heat through a medium is governed by the heat equation, and the dynamics of viscous fl uids are described by the Navier-Stokes equation. These equations are derived with physical arguments that operate on a macroscopic (observable) scale and do not involve the behavior of single molecules. On the other hand, if we would zoom into the physical systems, we would see only single molecules, moving seemingly chaotically. The question we try to approach in this course is whether we can derive the equations governing the observable behavior of certain physical systems by considering simple random dynamics for the single molecules and then zooming out to the observable scale. Concretely, we will consider interacting particle systems such as exclusion processes and zero range processes as the microscopic models.

• Continuous time Markov chains: characterization and construction;
• Martingale problems;
• Invariant measures;
• Simple exclusion and zero range processes;
• Tightness on Skorokhod space;
• Hydrodynamic limits;
• Fluctuation results;
• Local equilibrium, entropy methods

Literature:

• Kipnis, Landim: Scaling Limits of Interacting Particle Systems, Springer, 1999
• Further literature will be announced during the lecture, and there will also be lecture notes made available in Whiteboard.

Prerequisite: Stochastics I, II.
Recommended: Stochastics III and Functional Analysis.

Content: "Stochastics IV: Interacting particles and (stochastic) PDEs"
Physical systems are often described by partial diff erential equations (PDEs). For example, the flow of heat through a medium is governed by the heat equation, and the dynamics of viscous fl uids are described by the Navier-Stokes equation. These equations are derived with physical arguments that operate on a macroscopic (observable) scale and do not involve the behavior of single molecules. On the other hand, if we would zoom into the physical systems, we would see only single molecules, moving seemingly chaotically. The question we try to approach in this course is whether we can derive the equations governing the observable behavior of certain physical systems by considering simple random dynamics for the single molecules and then zooming out to the observable scale. Concretely, we will consider interacting particle systems such as exclusion processes and zero range processes as the microscopic models.

• Continuous time Markov chains: characterization and construction;
• Martingale problems;
• Invariant measures;
• Simple exclusion and zero range processes;
• Tightness on Skorokhod space;
• Hydrodynamic limits;
• Fluctuation results;
• Local equilibrium, entropy methods

Literature:

• Kipnis, Landim: Scaling Limits of Interacting Particle Systems, Springer, 1999
• Further literature will be announced during the lecture, and there will also be lecture notes made available in Whiteboard.

Prerequisite: Stochastics I, II.
Recommended: Stochastics III and Functional Analysis.

Content: "Stochastics IV: Interacting particles and (stochastic) PDEs"
Physical systems are often described by partial diff erential equations (PDEs). For example, the flow of heat through a medium is governed by the heat equation, and the dynamics of viscous fl uids are described by the Navier-Stokes equation. These equations are derived with physical arguments that operate on a macroscopic (observable) scale and do not involve the behavior of single molecules. On the other hand, if we would zoom into the physical systems, we would see only single molecules, moving seemingly chaotically. The question we try to approach in this course is whether we can derive the equations governing the observable behavior of certain physical systems by considering simple random dynamics for the single molecules and then zooming out to the observable scale. Concretely, we will consider interacting particle systems such as exclusion processes and zero range processes as the microscopic models.

• Continuous time Markov chains: characterization and construction;
• Martingale problems;
• Invariant measures;
• Simple exclusion and zero range processes;
• Tightness on Skorokhod space;
• Hydrodynamic limits;
• Fluctuation results;
• Local equilibrium, entropy methods

Literature:

• Kipnis, Landim: Scaling Limits of Interacting Particle Systems, Springer, 1999
• Further literature will be announced during the lecture, and there will also be lecture notes made available in Whiteboard.

Prerequisite: Stochastics I, II.
Recommended: Stochastics III and Functional Analysis.

Content: "Stochastics IV: Interacting particles and (stochastic) PDEs"
Physical systems are often described by partial diff erential equations (PDEs). For example, the flow of heat through a medium is governed by the heat equation, and the dynamics of viscous fl uids are described by the Navier-Stokes equation. These equations are derived with physical arguments that operate on a macroscopic (observable) scale and do not involve the behavior of single molecules. On the other hand, if we would zoom into the physical systems, we would see only single molecules, moving seemingly chaotically. The question we try to approach in this course is whether we can derive the equations governing the observable behavior of certain physical systems by considering simple random dynamics for the single molecules and then zooming out to the observable scale. Concretely, we will consider interacting particle systems such as exclusion processes and zero range processes as the microscopic models.

• Continuous time Markov chains: characterization and construction;
• Martingale problems;
• Invariant measures;
• Simple exclusion and zero range processes;
• Tightness on Skorokhod space;
• Hydrodynamic limits;
• Fluctuation results;
• Local equilibrium, entropy methods

Literature:

• Kipnis, Landim: Scaling Limits of Interacting Particle Systems, Springer, 1999
• Further literature will be announced during the lecture, and there will also be lecture notes made available in Whiteboard.

Prerequisite: Stochastics I, II.
Recommended: Stochastics III and Functional Analysis.

Refer to German description. Courses of Mathematis Education are part of the German teacher-training and held in German only.

Content: "Stochastics IV: Interacting particles and (stochastic) PDEs"
Physical systems are often described by partial diff erential equations (PDEs). For example, the flow of heat through a medium is governed by the heat equation, and the dynamics of viscous fl uids are described by the Navier-Stokes equation. These equations are derived with physical arguments that operate on a macroscopic (observable) scale and do not involve the behavior of single molecules. On the other hand, if we would zoom into the physical systems, we would see only single molecules, moving seemingly chaotically. The question we try to approach in this course is whether we can derive the equations governing the observable behavior of certain physical systems by considering simple random dynamics for the single molecules and then zooming out to the observable scale. Concretely, we will consider interacting particle systems such as exclusion processes and zero range processes as the microscopic models.

• Continuous time Markov chains: characterization and construction;
• Martingale problems;
• Invariant measures;
• Simple exclusion and zero range processes;
• Tightness on Skorokhod space;
• Hydrodynamic limits;
• Fluctuation results;
• Local equilibrium, entropy methods

Literature:

• Kipnis, Landim: Scaling Limits of Interacting Particle Systems, Springer, 1999
• Further literature will be announced during the lecture, and there will also be lecture notes made available in Whiteboard.

Prerequisite: Stochastics I, II.
Recommended: Stochastics III and Functional Analysis.