Lower bounds for densities of uniformly elliptic random variables on Wiener space

Abstract

In this article, we generalize the lower bound estimates for uniformly elliptic diffusion processes obtained by Kusuoka and Stroock. We define the concept of uniform elliptic random variable on Wiener space and show that with this definition one can prove a lower bound estimate of Gaussian type for its density. We apply our results to the case of the stochastic heat equation under the hypothesis of unifom ellipticity of the diffusion coefficient.

Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise

“…5, we establish a lower bound on p t,x , which proves Theorem 1.1(b). This upper (respectively lower) bound is a fairly direct extension to d ≥ 1 of a result of Bally and Pardoux [2] (respectively [7]) when d = 1. In Sect.…”

“…, u(t, x k )) is absolutely continuous with respect to Lebesgue measure, with a smooth and strictly positive density on {σ = 0} k , provided σ and b are infinitely differentiable functions which are bounded together with their derivatives of all orders. A Gaussian-type lower bound for this density is established by Kohatsu-Higa [7] under a uniform ellipticity condition. Morien [8] showed that the density function is also Hölder-continuous as a function of (t, x).…”

“…The rest of the proof of Theorem 1.1(b) follows along the same lines as in [7] for d = 1. We only sketch the remaining main points where the fact that d > 1 is important.…”

“…Using the terminology of [7], the first term is a process of order 1 and the next two terms are residues of order 1 (as in [7]). In the next step, we write the residues of order 1 as the sum of processes of order 2 and residues of order 2 and 3 as follows:…”

Density Formula and Concentration Inequalities with Malliavin Calculus

“…This was generalized by Kohatsu-Higa [9] in Wiener space via the concept of uniformly elliptic random variables; these random variables proved to be well-adapted to studying diusion equations. E. Nualart [13] showed that fractional exponential moments for a divergence-integral quantity known to be useful for bounding densities from above (see formula (1.1) below), can also be useful for deriving a scale of exponential lower bounds on densities; the scale includes Gaussian lower bounds.…”