Annales Academię Scientiarum Fennicę

Mathematica

Volumen 33, 2008, 413-427.

# EQUALITY CASES IN THE SYMMETRIZATION
INEQUALITIES FOR BROWNIAN TRANSITION
FUNCTIONS AND DIRICHLET HEAT KERNELS

## Dimitrios Betsakos

Aristotle University of Thessaloniki,
Department of Mathematics

54124 Thessaloniki, Greece; betsakos 'at' math.auth.gr

**Abstract.**
We prove equality statements
for the symmetrization inequalities
for Brownian transition functions and Dirichlet heat kernels. The
proofs involve the equality statements for the related
polarization inequalities which we also prove. These results lead
to symmetrization inequalities for Green functions, condenser
capacities, and exit times of Brownian motion.

**2000 Mathematics Subject Classification:**
Primary 35K05, 35B05, 31B15, 60J65.

**Key words:**
Heat kernel, polarization, symmetrization,
transition probability, Brownian motion, capacity, Green function.

**Reference to this article:** D. Betsakos:
Symmetrization inequalities for Brownian transition functions and
Dirichlet heat kernels.
Ann. Acad. Sci. Fenn. Math. 33 (2008), 413-427.

Full document as PDF file

Copyright © 2008 by Academia Scientiarum Fennica