Abstract
For all $\kappa > 0$, we show that the support of SLE$_\kappa$ curves is the closure in the sup-norm of the set of Loewner curves driven by nice (e.g. smooth) functions. It follows that the support is the closure of the set of simple curves starting at $0$.
Highlights
1.1 Overview The support of a random variable X in a Polish space is the set of points x such that for any open neighborhood V of x, we have P(X ∈ V ) > 0
The random variable X will be a random process, namely the SLEκ trace, and our goal is to describe its support
Characterising the support of random processes such as Brownian motion and diffusions is an important research problem for stochastic differential equations, where it was initiated by Stroock and Varadhan [25] when they studied a strong maximum principle of a PDE operator
Summary
1.1 Overview The support of a random variable X in a Polish space is the set of points x such that for any open neighborhood V of x, we have P(X ∈ V ) > 0. The main difficulty in proving such statements is that the Schramm-Loewner map (or, in the SDE case, solution map) is not continuous, and even only almost surely defined. If it were, the support theorem for SLE would immediately follow from the well-known support theorem for Brownian motion. Note that the SLEκ curve is not a diffusion process, even though the Loewner equation with Brownian motion as an input can be seen as an SDE. By rounding off the edges and reparametrising by half-plane capacity (a precise argument is conducted in the proof of Proposition 6.4), we find a smooth Loewner curve γ (which in particular has a smooth driving function) with γ8 − γ ∞,[0,1] < ε. Proving Proposition 1.4 is the main part of this paper
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