Quasi-linear stochastic partial differential equations with irregular coefficients: Malliavin regularity of the solutions

Abstract

We study quasi-linear stochastic partial differential equations with discontinuous drift coefficients. Existence and uniqueness of a solution is already known under weaker conditions on the drift, but we are interested in the regularity of the solution in terms of Malliavin calculus. We prove that when the drift is bounded and measurable the solution is directional Malliavin differentiable.

Appendix

Consider the matrices $\Sigma$ and M from Sect. 4. The purpose of this section in to show that the function $f_m(s_1, \ldots s_m) := \mathrm{per}( M \Sigma M^T)$ is such that for $\beta \in (0,1)$ we have

$$\begin{aligned} \int _{0 < s_1 < \cdots s_m < t} |f_m(s_1, \ldots , s_m)|^{\beta } ds_1 \ldots ds_m \le C^m \end{aligned}$$

for some constant $C = C(t,\beta )$.

We start by noting that

$$\begin{aligned} M\Sigma M^T = \left( \begin{array}{rrrrrr} a_m &{} b_m &{} 0 &{} \cdots &{} \cdots &{} 0 \\ b_m &{} a_{m-1} &{} b_{m-1}&{} \cdots &{} \cdots &{} 0 \\ 0 &{} b_{m-1} &{} a_{m-2} &{} \cdots &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} &{} \ddots &{} \cdots &{} \vdots \\ 0 &{} \cdots &{} \cdots &{} 0 &{} a_{2} &{} b_{2} \\ 0 &{} \cdots &{} \cdots &{} 0 &{} b_{2} &{} a_1 \\ \end{array} \right) \end{aligned}$$

where

$$\begin{aligned} a_j = \left\{ \begin{array}{ll} (s_j - s_{j-1})^{-1/2} +(s_{j-1} - s_{j-2})^{-1/2} &{} \quad \text { for } j=m, \ldots ,3 \\ (s_{2} - s_1)^{-1/2} + s_1^{-1/2} &{} \quad \text { for } j= 2\\ s_1^{-1/2} &{}\quad \text { for } j= 1 \\ \end{array} \right. \end{aligned}$$

and $b_j = -(s_{j} - s_{j-1})^{-1/2}$ for $j = m, \ldots , 2$ and $b_{2} = -s_1^{-1/2}$.

Using the definition of the permanent of a matrix we see that we have the following recursive relation

$$\begin{aligned} f_m(s_1, \ldots , s_m) =&\left( (s_{m} - s_{m-1})^{-1/2} +(s_{m-1} - s_{m-2})^{-1/2} \right) f_{m-1}( s_1, \ldots ,s_{m-1}) \\&+ (s_{m-1}- s_{m-2})^{-1} f_{m-2}(s_1, \ldots ,s_{m-2}) \\ \end{aligned}$$

with

$$\begin{aligned} f_1(s_1) = s_1^{-1/2} \, \, \text { and } \, \, f_2(s_1, s_2) = (s_2 - s_1)^{-1/2}s_1^{-1/2} + 2s_1^{-1} . \end{aligned}$$

We write $f_m(s_1,\ldots ,s_m) = p_m(s_1^{-1/2}, (s_2 - s_{1})^{-1/2}, \ldots ,(s_m - s_{m-1})^{-1/2})$ where $p_m$ is the polynomial recursively defined by

$$\begin{aligned} p_m(x_1, \ldots , x_m) = ( x_m + x_{m-1} ) p_{m-1}( x_1, \ldots , x_{m-1}) + x_{m-1}^2 p_{m-2}(x_1 \ldots ,x_{m-2}) \end{aligned}$$

with

$$\begin{aligned} p_1(x_1) = x_1 \text { and } \, \, p_2(x_1, x_2) = x_2x_1 + 2x_1^2 . \end{aligned}$$

If we denote by $deg_{x_i}p_m$ the degree of the polynomial in the variable $x_i$, for $i=1, \ldots m$ we see from the recursive relation that

$$\begin{aligned} deg_{x_m}p_m = 1 \, \text { and } \, deg_{x_j} p_m \le 2, \text { for } j=1, \ldots , m-1 . \end{aligned}$$

Moreover, if we denote by $\gamma _m$ the number of terms in this polynomial, it is clear from the recursive relation that

$$\begin{aligned} \gamma _m = 2 \gamma _{m-1} + \gamma _{m-2} \end{aligned}$$

and

$$\begin{aligned} \gamma _1 = 1 \, \, \text { and } \, \, \gamma _2 = 2 . \end{aligned}$$

So that we have $\gamma _m \le C^m$ for C large enough.

It follows that we may write

$$\begin{aligned} p_m(x_1, \ldots , x_m) = \sum _{\alpha } c_{\alpha } x^{\alpha } \end{aligned}$$

where the sum is taken over all multiindices $\alpha \in \mathbb {N}^m$ with $\alpha _i \le 2$ and $\alpha _1 \le 1$. Here we have denoted $x^{\alpha } = x_1^{\alpha _1} \ldots x_m^{\alpha _m}$. Moreover, there are at most $C^m$ terms in this sum with C as above and one can show that $|c_{\alpha }| \le 3^m$ for all $\alpha$.

Consequently

$$\begin{aligned} |f(s_1, \ldots , s_m)|^{\beta } \le 3^m \sum _{\alpha } s_1^{-\beta \alpha /2}|s_2 - s_1|^{-\beta \alpha _1/2} \cdots |s_m - s_{m-1}|^{-\beta \alpha _{m-1}/2} . \end{aligned}$$

Since $\frac{\beta \alpha _i}{2} < 1$ for all $i = 1, \ldots , m$, each of the above terms are integrable over $0 < s_1 < \cdots < s_m < t$, and there are at most $C^m$ such terms. The result follows.