The Malliavin regularity

Malliavin regularity refers to properties related to the smoothness of random variables within the context of the Malliavin calculus, a branch of stochastic analysis. This calculus extends the classical differential calculus to functionals of stochastic processes, providing tools to study the regularity of these functionals, which is crucial in the analysis of stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs).

1. Malliavin Calculus Overview

Objective: It is used to investigate the differentiability of random variables that are functionals of stochastic processes. This involves defining derivatives with respect to random variables in a probability space.
Applications: Malliavin calculus is particularly useful in establishing results like the smoothness of probability density functions, integration by parts formulas, and providing criteria for the existence of densities.

2. Malliavin Derivative

Definition: For a given probability space $(\Omega, \mathcal{F}, \mathbb{P})$, the Malliavin derivative $D$ maps a random variable $F$ (function of a Wiener process $W(t)$ ) to a stochastic process $D_t F$ that represents its derivative with respect to $W(t)$ . If $F$ is a functional defined on a time interval $[0, T]$, then $D_t F$ is defined for $t \in [0, T]$ and describes the sensitivity of $F$ to changes in $W(t)$ .
Mathematical Formulation: If $F$ is of the form:

$F = f\left(W(h_1), \ldots, W(h_n)\right),$

where $f$ is a smooth function and $h_i$ are deterministic functions, then the Malliavin derivative is given by:

$D_t F = \sum_{i=1}^n \frac{\partial f}{\partial x_i} \mathbf{1}_{[0, T]}(t) h_i(t).$

3. Malliavin Regularity

Concept: Malliavin regularity refers to the smoothness properties of a random variable $F$ in the Sobolev space $\mathbb{D}^{k, p}$ . Here, $k$ indicates the number of derivatives considered, and $p$ represents the $L^p$ norm used for integrability.
Sobolev Spaces $\mathbb{D}^{k, p}$ : A random variable $F$ belongs to the space $\mathbb{D}^{k, p}$ if:

$\|F\|{k, p} = \left( \mathbb{E}\left[ |F|^p + \sum{j=1}^{k} \int_{0}^{T} |D^j_t F|^p \, dt \right] \right)^{1/p} < \infty.$

4. Significance in SPDEs and Stochastic Analysis

Existence of Densities: A major application of Malliavin regularity is proving that a random variable has a smooth probability density function. If $F$ is sufficiently regular (e.g., in $\mathbb{D}^{1,2}$ ), it often satisfies Hörmander’s condition, which guarantees the existence of a density.
Regularization Effects: In SPDEs, the Malliavin calculus can show how noise influences the regularity of solutions. For instance, even if the deterministic version of a PDE has limited regularity, the introduction of a stochastic term may improve the solution's regularity under certain conditions.

5. Technical Properties

Chain Rule: If $F = g(G)$ where $g$ is a smooth function and $G$ is Malliavin differentiable, then:

$D_t F = g'(G) D_t G.$

Integration by Parts: This is a crucial result that allows the derivation of estimates for the distribution of random variables. For a sufficiently regular random variable $F$ and a test function $\varphi$ :

$\mathbb{E}[F \varphi(F)] = \mathbb{E}[\varphi'(F) \langle DF, -DL^{-1}F \rangle],$

where $L$ is the Ornstein–Uhlenbeck operator.

6. Malliavin Matrix

Definition: For a vector of random variables $\mathbf{F} = (F_1, \ldots, F_n)$ , the Malliavin matrix $\gamma_{\mathbf{F}}(t)$ is defined as:

$\gamma_{\mathbf{F}}(t) = \left( \langle D_t F_i, D_t F_j \rangle \right)_{1 \leq i, j \leq n}.$