Related papers: Malliavin Calculus for Degenerate Diffusions

Let $(W,H,\mu)$ be the classical Wiener space on $\R^d$. Assume that $X=(X_t)$ is a diffusion process satisfying the stochastic differential equation $dX_t=\sigma(t,X)dB_t+b(t,X)dt$, where $\sigma:[0,1]\times C([0,1],\R^n)\to \R^n\otimes…

We study score-based diffusion modelling in infinite-dimensional separable Hilbert spaces through Malliavin calculus, extending the analysis of generative models beyond the finite-dimensional setting. The forward diffusion process is…

Suppose $B$ is a Brownian motion and $B^n$ is an approximating sequence of rescaled random walks on the same probability space converging to $B$ pointwise in probability. We provide necessary and sufficient conditions for weak and strong…

In this paper, we use Malliavin calculus to show the existence and continuity of density functions of $d$-dimensional non-colliding particle systems such as hyperbolic particle systems and Dyson Brownian motion with smooth drift. For this…

We introduce a new framework based on Malliavin calculus to derive exact analytical expressions for the score function $\nabla \log p_t(x)$, i.e., the gradient of the log-density associated with the solution to stochastic differential…

By using the Malliavin calculus and solving a control problem, Bismut type derivative formulae are established for a class of degenerate diffusion semigroups with non-linear drifts. As applications, explicit gradient estimates and Harnack…

In the first paper of this series, I investigated whether a wavefunction model of a heavy particle and a collection of light particles might generate "Brownian-Motion-Like" trajectories of the heavy particle. I concluded that it was…

We use Malliavin operators in order to prove quantitative stable limit theorems on the Wiener space, where the target distribution is given by a possibly multidimensional mixture of Gaussian distributions. Our findings refine and generalize…

We study several important fine properties for the family of fractional Brownian motions with Hurst parameter $H$ under the $(p,r)$-capacity on classical Wiener space introduced by Malliavin. We regard fractional Brownian motions as Wiener…

Malliavin Calculus is about Sobolev-type regularity of functionals on Wiener space, the main example being the Ito map obtained by solving stochastic differential equations. Rough path analysis is about strong regularity of solution to…

Let $(X_t)$ be a reflected diffusion process in a bounded convex domain in $\mathbb R^d$, solving the stochastic differential equation $$dX_t = \nabla f(X_t) dt + \sqrt{2f (X_t)} dW_t, ~t \ge 0,$$ with $W_t$ a $d$-dimensional Brownian…

In this paper, we introduce some fundamental notions related to the so-called stochastic derivatives with respect to a given $\sigma$-field $\mathcal{Q}$. In our framework, we recall well-known results about Markov--Wiener diffusions. We…

In this work, by using the Malliavin calculus, under H\"ormander's condition, we prove the existence of distributional densities for the solutions of stochastic differential equations driven by degenerate subordinated Brownian motions.…

We consider a general one-dimensional overdamped diffusion model described by the It\^{o} stochastic differential equation (SDE) ${dX_t=\mu(X_t,t)dt+\sigma(X_t,t)dW_t}$, where $W_t$ is the standard Wiener process. We obtain a specific…

We study the dynamical properties of the Brownian diffusions having $\sigma {\rm Id}$ as diffusion coefficient matrix and $b=\nabla U$ as drift vector. We characterize this class through the equality $D^2_+=D^2_-$, where $D_{+}$ (resp.…

We construct surface measures in a Hilbert space endowed with a probability measure $\nu$. The theory fits for invariant measures of some stochastic partial differential equations such as Burgers and reaction--diffusion equations. Other…

The covariant form of the multivariable diffusion-drift process is described by the covariant Fokker--Planck equation using the standard toolbox of Riemann geometry. The covariant form of the equivalent Langevin stochastic differential…

In the L\'evy construction of Brownian motion, a Haar-derived basis of functions is used to form a finite-dimensional process $W^{N}$ and to define the Wiener process as the almost sure path-wise limit of $W^{N}$ when $N$ tends to infinity.…

We obtain estimates on the first-order Malliavin derivative of mild solutions, evaluated at fixed points in time and space, to a class of parabolic dissipative stochastic PDEs on bounded domain of $\mathbb{R}^d$. In particular, such…

We study one-dimensional stochastic differential equations of form $dX_t = \sigma(X_t)dY_t$, where $Y$ is a suitable H\"older continuous driver such as the fractional Brownian motion $B^H$ with $H>\frac12$. The innovative aspect of the…