Differentiability of semigroups of stochastic differential equations with H\"older-continuous diffusion coefficients

TL;DR

This paper investigates the differentiability properties of semigroups generated by stochastic differential equations with H"older-continuous diffusion coefficients, providing bounds useful for high-dimensional applications.

Contribution

It establishes upper bounds for the derivatives of semigroups associated with SDEs having non-differentiable diffusion coefficients, often with dimension-independent constants.

Findings

Derived upper bounds for $C^m$-norms of semigroups

Constants often dimension-independent

Applicable to high-dimensional and infinite-dimensional SDEs

Abstract

Differentiability of semigroups is useful for many applications. Here we focus on stochastic differential equations whose diffusion coefficient is the square root of a differentiable function but not differentiable itself. For every $m\in\{0,1,2\}$ we establish an upper bound for a $C^m$-norm of the semigroup of such a diffusion in terms of the $C^m$-norms of the drift coefficient and of the squared diffusion coefficient. The constants in our upper bound are often dimension-independent. Our estimates are thus suitable for analyzing certain high-dimensional and infinite-dimensional degenerate stochastic differential equations.