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In this paper, we establish small time large deviation principles for the quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone.

We construct vector-valued Sobolev spaces $$F_\mathbb{B}^{r,p} $$ , related to a general sub-Markovian semi-group and we give conditions for c r,p-quasi-everywhere convergence.

We consider the formal SDE $$\textrm{d} X_t = b(t,X_t)\textrm{d} t + \textrm{d} Z_t, \qquad X_0 = x \in \mathbb {R}^d, (\text {E})$$ where $$b\in L^r ([0,T],\mathbb {B}_{p,q}^\beta (\mathbb {R}^d,\mathbb {R}^d))$$ is a time-inhomogeneous Besov drift and $$Z_t$$ is a symmetric d-dimensional $$\alpha $$ -stable process, $$\alpha \in (1,2)$$ , whose spectral measure is absolutely continuous w.r.t. the Lebesgue measure on the sphere. Above, $$L^r$$ and $$\mathbb {B}_{p,q}^\beta $$ respectively denote Lebesgue and Besov spaces. We show that, when $$\beta > \frac{1-\alpha + \frac{\alpha }{r} + \frac{d}{p}}{2}$$ , the martingale solution associated with the formal generator of (E) admits a density which enjoys two-sided heat kernel bounds as well as gradient estimates w.r.t. the backward variable. Our proof relies on a suitable mollification of the singular drift aimed at using a Duhamel-type expansion. We then use a normalization method combined with Besov space properties (thermic characterization, duality and product rules) to derive estimates.

Let X be a Borel right Markov process, let m be an excessive measure for X, and let $\widehat{X}$ be the moderate Markov dual process associated with X and m. The potential theory of co-excessive measures (i.e., measures that are excessive for $\widehat{X}$ ) is developed with special emphasis on the Riesz decomposition. This is then applied to obtain the Riesz decomposition of excessive functions (of X) by exploiting the correspondence between such functions and co-excessive measures. The potential theory of co-excessive measures also enables us to discuss Walsh’s interior réduite under minimal conditions. Many of the tools of the theory of Markov processes are employed in this development. For example, Kuznetsov measures, Ray compactifications, h-transforms, and duality theory for Borel right processes.

This first result of this paper is about the Laplace transform of u(X T ) where u is harmonic on some bounded domain Ω, X t is Brownian motion and T is the exit time from Ω. The following results focus on exit times from balls and Faber–Krahn and reverse Faber–Krahn type inequalities for balls. We also study the behaviour of the first Dirichlet eigenvalue for complex balls under complex interpolation. The method of proof heavily relays on the log-concavity of gaussian measures.

In this paper, we obtain two-sided estimates on the heat kernels for elliptic operators with singular coefficients equipped with mixed boundary conditions.

This paper investigates the existence, nonexistence, and qualitative properties of p-harmonic functions in the upper half-space $$\mathbb {R}^N_+$$ $$(N\ge 3)$$ satisfying nonlinear boundary conditions for $$1