Abstract
We establish two results concerning a class of geometric rough paths $\mathbf{X}$ which arise as Markov processes associated to uniformly subelliptic Dirichlet forms. The first is a support theorem for $\mathbf{X}$ in $\alpha$-H\"older rough path topology for all $\alpha \in (0,1/2)$, which answers in the positive a conjecture of Friz-Victoir (2010). The second is a H\"ormander-type theorem for the existence of a density of a rough differential equation driven by $\mathbf{X}$, the proof of which is based on analysis of (non-symmetric) Dirichlet forms on manifolds.
Highlights
Consider a symmetric Dirichlet form on L2(Rd, λ) d E(f, g) =ai,j(∂if )(∂jg)dλ, Rd i,j=1 (1.1)where λ is the Lebesgue measure and a is a measurable, uniformly elliptic function taking values in the space of symmetric d × d matrices
Where λ is the Lebesgue measure and a is a measurable, uniformly elliptic function taking values in the space of symmetric d × d matrices. It is well-known that there exists a symmetric Markov process X in Rd associated with E; see [14] for a general construction of X and [23] for fundamental analytic properties of E
We are interested in differential equations of the form dYt = V (Yt)dXt, Y0 = y0 ∈ Re, (1.2)
Summary
Comparing our situation to the case of Gaussian rough paths, where such support theorems are known with sharp Hölder exponents (see e.g., [13, Sec. 15.8], and [11] for recent improvements), the difficulty lies in the lack of a Gaussian structure, in particular the absence of a Cameron-Martin space Our solution to this problem relies almost entirely on elementary techniques. We first show that any stochastic process (taking values in a Polish space) admits explicit lower bounds on the probability of keeping a small α-Hölder norm, provided that it satisfies lower and upper bounds on certain transition probabilities comparable to Brownian motion This is made precise by conditions (1) and (2) and Theorem 2.5. We note that our current result gives no quantitative information about the density beyond its existence (not even for the couple (X, Y)), and we strongly suspect that the method can be improved to yield further information ( Lp bounds and regularity results in the spirit of the De Giorgi–Nash–Moser theorem)
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