Conference Agenda

Let $E$ be a locally compact separable metric space, $D$ be an open subset of $E$ and $m$ be a Radon measure on $E$ with full support. Let $(L,\mathfrak D(L))$ be a self-adjoint operator that generates a Markov semigroup $(T_t)_{t>0}$ on $L^2(E;m)$ and regular Dirichlet form $(\mathcal E,\mathfrak D(\mathcal E))$ (i.e. $L$ is a Dirichlet operator). The goal is to study, within this general framework, the Dirichlet problem for semilinear equations \[ -Lu=f(\cdot,u)+\mu\quad\text{in }D. \] Here $f:E\times\mathbb R\rightarrow\mathbb R$ is a given function and $\mu\ll\mbox{Cap}$, where $\mbox{Cap}:2^E\to [0,\infty]$ is a Choquet capacity naturally associated with $L$ (note that $m\ll\mbox{Cap}$).

It is by now recognized that well-posed Dirichlet problem for the above equation must consist of two conditions: an exterior condition on $D^c:=\mathbb R^d\setminus D$ and a description of the asymptotic behavior of a solution at the boundary $\partial D$.

We give a probabilistic and analytical definition of a solution to the problem and show their equivalence. As a result, we prove Feynman-Kac formula for solutions of the studied problem. Next, based on the theory of Backward Stochastic Differential Equations, we prove an existence results in case $f$ is monotone with respect to $u$. We also provide some regularity results.

The talk is based on the paper [1].

Bibliography

[1] Klimsiak, T., Rozkosz A.: Dirichlet problem for semilinear partial integro-differential equations: the method of orthogonal projection. arXiv:2304.00393

We show existence of a stochastic parabolic obstacle problem with obstacle $\psi =0$ under homogeneous Dirichlet boundary conditions. In the penalized equation, the penalization term converges to a random Radon measure $\eta$ only. Since the solution $u$ of the obstacle problem is not continuous in space-time in general, this causes problems to give a proper definition of the minimalization condition of $\eta$. We show that $\eta$ does not charge sets of zero capacity and the solution is nonnegative almost everywhere with respect to the Lebesgue measure and $\eta(u)=0$ for some Borel-measurable representative. Uniqueness may be obtained for quasi continuous solutions.

We consider Brownian motion on manifolds with sticky reflection from the boundary and with or without diffusion along the boundary. For the invariant measure consisting of a convex combination of the volume measure in the interior and the Hausdorff measure on the boundary we present Poincaré and logarithmic Sobolev inequalities under general curvature assumptions on the manifold and its boundary. Additionally we also present a Cheeger-type inequality for the spectral gap. This is based on joint work with Max von Renesse and Feng-Yu Wang.

We consider the quenched Edwards--Wilkinson equation with a Gaussian disorder, which is white in the spacial component and colored in the height component. We comment on showing the existence of the solution for this singular SPDE and possibly discuss the phenomenon of pinning vs depinning.