Related papers: Diffusion Processes on $p$-Wasserstein Space over …

We construct a recurrent diffusion process with values in the space of probability measures over an arbitrary closed Riemannian manifold of dimension $d\ge 2$. The process is associated with the Dirichlet form defined by integration of the…

By using the spectrum of the underlying symmetric diffusion operator, the convergence in $L^p$-Wasserstein distance $\mathbb W_p (p\ge 1)$ is characterized for the empirical measure $\mu_t$ of non-symmetric subordinated diffusion processes…

Given any closed Riemannian manifold $M$, we construct a reversible diffusion process on the space ${\mathcal P}(M)$ of probability measures on $M$ that is (i) reversible w.r.t.~the entropic measure ${\mathbb P}^\beta$ on ${\mathcal P}(M)$,…

The paper studies a class of Ornstein-Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron-Martin space. It…

We provide a general construction scheme for $\mathcal L^p$-strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated…

Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\partial M$, let $V\in C^2(M)$ such that $\mu({\rm d} x):={\rm e}^{V(x)}{\rm d} x$ is a probability measure, and let $X_t$ be the diffusion process generated…

The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron--Martin space. In…

Let $\mathcal P_2$ be the space of probability measures on $\R^d$ having finite second moment, and consider the Riemannian structure on $\mathcal P_2$ induced by the intrinsic derivative on the $L^2$-tangent space. By using stochastic…

For a class of stochastic differential equations with reflection for which a certain ${\mathbb{L}}^p$ continuity condition holds with $p>1$, it is shown that any weak solution that is a strong Markov process can be decomposed into the sum…

Let $\mathbb P$ be any Borel probability measure on the $L^2$-Wasserstein space $(\mathscr{P}_2(M),W_2)$ over a closed Riemannian manifold $M$. We consider the Dirichlet form $\mathcal E$ induced by $\mathbb P$ and by the Wasserstein…

For a class of (non-symmetric) diffusion processes on a length space, which in particular include the (reflecting) diffusion processes on a connected compact Riemannian manifold, the exact convergence rate is derived for $({\mathbb E}…

We investigate long-time behaviors of empirical measures associated with subordinated Dirichlet diffusion processes on a compact Riemannian manifold $M$ with boundary $\partial M$ to some reference measure, under the quadratic Wasserstein…

Von Renesse and the author (Ann. Prob. '09) developed a second order calculus on the Wasserstein space P([0,1]) of probability measures on the unit interval. The basic objects of interest had been Dirichlet form, semigroup and continuous…

We construct a new random probability measure on the sphere and on the unit interval which in both cases has a Gibbs structure with the relative entropy functional as Hamiltonian. It satisfies a quasi-invariance formula with respect to the…

Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\partial M$, let $V\in C^2(M)$ such that $\mu(dx):=e^{V(x)} d x$ is a probability measure, and let $X_t$ be the diffusion process generated by…

We study the particle method to approximate the gradient flow on the $L^p$-Wasserstein space. This method relies on the discretization of the energy introduced by [3] via nonoverlapping balls centered at the particles and preserves the…

Consider a set of continuous maps from the interval $[0,1]$ to a domain in ${\mathbb R}^d$. Although the topological boundary of this set in the path space is not smooth in general, by using the theory of functions of bounded variation (BV…

For a Markov process associated with a diffusion type Dirichlet form an upper bound is shown for the law of the finite dimensional distributions of the process. Under some more assumptions on the underlaying space this is also shown for the…

The Dirichlet form associated with the intrinsic gradient on Poisson space is known to be quasi-regular on the complete metric space $\ddot\Gamma=$ $\{Z_+$-valued Radon measures on $\IR^d\}$. We show that under mild conditions, the set…

By adopting the coupling by reflection and choosing an auxiliary function which is convex near infinity, we establish the exponential convergence of diffusion semigroups $(P_t)_{t\ge0}$ with respect to the standard $L^p$-Wasserstein…