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Convergence and regularity of probability laws by using an interpolation method. (English) Zbl 1377.60066
The authors discuss the problem of the existence of a (regular) density of probability measures. After having established a general criterion, they apply this to several settings: (i) path dependent SDEs driven by a Brownian motion and with Dini (log-Hölder)-continuous coefficients; (ii) the stochastic heat equation driven by white noise and with Dini-continuous coefficients; (iii) piecewise deterministic Markov processes coming from a jump-type SDE having smooth coefficients satisfying certain growth and non-degeneracy conditions.
The main result of the paper concerns the existence of regular densities. Roughly speaking, if a probability measure \(\mu\) can be approximated by measures \(\mu_n\) having densities \(\phi_n(x)\,dx\), criteria are given such that \(\mu\) inherits a (regular) density: for example, if \(\phi_n \in C^{q+2m}(\mathbb R^d)\) such that \[ d_k(\mu,\mu_n) \|\phi_n\|^\alpha_{q+2m,2m,p}\leq C,\quad \alpha > (q+2k+d/p_*)/2m \] then \(\mu\) has a density \(\phi(x)\,dx\) with \(\phi\in W^{q,p}\). Here, \(p\in (1,\infty)\), \(p_*\) is the conjugate index of \(p\), \(m,q,k\in \mathbb N\), \(W^{q,p}\) is the standard Sobolev space over \(\mathbb R^d\), \(\|\phi\|_{k,m,p}\) is a classical \(L^p\)-Sobolev norm of order \(k\in\mathbb N\) with a polynomial weight function \((1+|x|)^m\) and, finally, \(d_k\) is a distance between measures defined by \[ d_k(\mu,\mu_n) = \sup\left\{\left|\int f\,d\mu - \int f\,d\nu\right| : \sum_{0\leq |\alpha|\leq k}\|\partial^\alpha f\|_\infty\leq 1\right\} \] (so, \(d_0\) is the TV-distance and \(d_1\) is the Fortet-Mourier distance).
The proof of the above result relies on a more general result cast using Orlicz and Sobolev-Orlicz spaces. Let \(\mathsf{e}\) be a suitably regular Young function (defining the Orlicz space \(L^{\mathsf{e}}\)) and denote by \(\beta(t) = \beta_{\mathsf{e}}(t) = t/\mathsf{e}^{-1}(t)\). Then \[ \pi_{q,k,m,\mathsf{e}}(\mu,(\mu_n)_n) = \sum_{n=0}^\infty 2^{n(q+k)}\beta(2^{nd})d_k(\mu,\mu_n) + \sum_{n=0}^\infty 2^{-2nm}\|\phi_n\|_{2m+q,2m,\mathsf{e}} \] (the latter is the Sobolev-Orlicz norm) and \[ \rho_{q,k,m,\mathsf{e}}(\mu) := \inf\pi_{q,k,m,\mathsf{e}}(\mu,(\mu_n)_n) \] with the inf extending over all sequences of absolutely continuous (w.r.t. Lebesgue measure) measures. \[ \mathcal S_{q,k,m,\mathsf{e}} := \left\{\mu : \rho_{q,k,m,\mathsf{e}}(\mu)<\infty\right\} \] Then, the abstract main result says that \(\mathcal S_{0,k,m,\mathsf{e}}\subset L^{\mathsf{e}}\) and the density \(\phi\) of \(\mu\) satisfies the inequality \(\|\phi\|_{L^{\mathsf{e}}}\leq C\rho_{0,k,m,\mathsf{e}}(\mu)\) with a universal constant \(C\). There are also higher-order analogues of this result.
The key estimate, essentially an interpolation result, is the remark that one has \[ \|f\|_{q,\mathsf{e}}\leq C\rho_{q,k,m,\mathsf{e}}(\mu) \] for all measures of the form \(\mu(dx) = f(x)\,dx\) and \(f\in C^{2m+q}(\mathbb R^d)\) and with a universal constant \(C\).

MSC:
60H07 Stochastic calculus of variations and the Malliavin calculus
46B70 Interpolation between normed linear spaces
28A15 Abstract differentiation theory, differentiation of set functions
46E10 Topological linear spaces of continuous, differentiable or analytic functions
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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