Wednesday, November 15, 2023 - 15:15 in V3-201 + Zoom
Weighted $L^1$-semigroup approach for nonlinear Fokker—Planck equations and generalized Ornstein—Uhlenbeck processes
A talk in the Bielefeld Stochastic Afternoon series by
Marco Rehmeier
| Abstract: |
For the nonlinear Fokker--Planck equation
$$ \partial_tu = \Delta\beta(u)-\nabla \Phi \cdot \nabla \beta(u) -
\text{div}_{\varrho}\big(D(x)b(u)u\big),\quad (t,x) \in
(0,\infty)\times \mathbb{R}^d,$$
where $\varrho = \exp(-\Phi)$ is the density of a finite Borel measure
and $\nabla \Phi$ is unbounded, we construct mild solutions with
bounded initial data via the Crandall--Liggett semigroup approach in
the weighted space $L^1(\mathbb{R}^d,\mathbb{R};\varrho dx)$. By the
superposition principle, we lift these solutions to weak solutions to
the corresponding McKean--Vlasov SDE, which can be considered a model
for generalized nonlinear perturbed Ornstein--Uhlenbeck processes.
Finally, for these solutions we prove the nonlinear Markov property in
the sense of McKean.
Within the CRC this talk is associated to the project(s): A5 |
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