Stochastic regularization effects of semi-martingales on random functions (Q335875)

The authors address the question of writing the time-average of a function \(f\) along some stochastic process \(X\) in terms of a stochastic integral of the solution \(F\) to the Fokker-Planck equation associated with \(X\). More precisely, if the process \(X\) solves the stochastic equation NEWLINE\[NEWLINEX_t=x+\int^t_0b(s, X_s)ds+W_t\quad (\text{for a Brownian motion }W\text{ in }\mathbb R^d)NEWLINE\]NEWLINE and if (for some fixed positive \(T\)) \(F\) solves the backward Fokker-Planck equation NEWLINE\[NEWLINEF(t,x)=(\frac{1}{2}\Delta+b(s,x)\cdot\nabla)F(s,x)ds-\int^T_tf(s,x)ds,NEWLINE\]NEWLINE then \textit{F. Flandoli} et al. [Bull. Sci. Math. 134, No. 4, 405--422 (2010; Zbl 1198.60023)] established that almost surely NEWLINE\[NEWLINE\int^T_0f(s,X_s)ds=-F(0,x)-\int^t_0\nabla F(s,X_s)\cdot dW_s.NEWLINE\]NEWLINE This holds for deterministic functions \(f\), and can be used to show a stronger regularity of the time-average \(x\mapsto\int^T_0f(s,X_s)ds\) than that of \(f\) alone.NEWLINENEWLINEThe purpose of the authors is here to extend this type of representation to random functions \(f\) and \(b\). Namely, under a series of conditions on the random functions \(f=f (t,x,\omega)\) and \(b=b(t,x,\omega)\) (non-trivial examples are given), the authors provide their solution in the following two steps.NEWLINENEWLINEFirst, they consider the process \(X\) solving NEWLINE\[NEWLINEX_t=x+\int^t_0b(s,X_s,W_{\min\{\cdot,s\}})ds+W_t\quad(\text{for a Brownian motion }W\text{ in }\mathbb R^d)NEWLINE\]NEWLINE together with its semigroup \((P_{t,s})_{t\leq s\leq T}\), and establish the existence of a unique Malliavin-differentiable, strong adapted process \((F,Z)\) solving NEWLINE\[NEWLINEF(t,x)=-\int^T_tP_{t,s}f(s,x)ds-\int^T_tP_{t,s}Z(s,x)\cdot dW_s,NEWLINE\]NEWLINE twhich is given by NEWLINE\[NEWLINEF(t,x)=\mathbb E\biggl[-\int^T_tP_{t,s}f(s,x)ds\biggl|\mathcal F_t\biggr],\, Z(t,x)=\mathbb E\biggl[-\int^T_tD_tP_{t,s}f(s,x)ds\biggl|\mathcal F_t\biggr].NEWLINE\]NEWLINE Second, they establish their main result: almost surely, NEWLINE\[NEWLINE\in^T_0f(s,X_s)ds=-F(0,x)-\int^T_0(\nabla F(s,X_s)+Z(s,X_s))\cdot dW_s-\int^T_0\operatorname{div}Z(s,X_s)ds.NEWLINE\]