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Equality cases in the symmetrization inequalities for Brownian transition functions and Dirichlet heat kernels. (English) Zbl 1160.35308

The author studies the equality cases in the symmetrization inequalities for Brownian transition functions and Dirichlet heat kernels. For instance one studies the equality cases for the inequality
\[ P_{t}(x,B)\leq P_{t}^{D^{\#}}(x^{\#},B^{\#}),\quad x\in D. \]
Here \(P_{t}(x,B)=P_{x}(X_{t}\in B;t<T_{D})\) is the transition function in the Borel subset \(B\subset\mathbb R^{n}\) of the Brownian motion in the domain \(D\subset\mathbb R^{n}\), starting from \(x=(x_{1},\dots,x_{n})\in D\) before the exit time \(T_{D}\) and \(x^{\#}=(x_{1},\ldots,x_{n-1},0)\) denotes the orthogonal projection of \(x\) on \(\Pi=\{x_{n}=0\}\).
The proofs are based on the approach to symmetrization via polarization. Results concerning symmetrization inequalities for Green functions, condenser capacities and exit times of Brownian motion are then deduced.

MSC:

35A08 Fundamental solutions to PDEs
35K05 Heat equation
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
60J65 Brownian motion
35R60 PDEs with randomness, stochastic partial differential equations
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