Time-dependent Time Fractional Equations and Probabilistic Representation

Abstract

We study the nonlocal initial-value problem of the form \begin{align*} \sL u(x, t) &= h(x, t) \quad \text{for}\ (x,t)\in \R^d\times (0,\infty), \\ u(x,t) &= f(x) \quad \text{on}\ \R^d\times (-\infty, 0] \end{align*} where $\sL$ is an integro-differential operator given by \begin{eqnarray*} && \hskip -0.2truein \mathcal L \varphi(x,t) \nonumber \\ & =& \frac{1}{2}\sum_{i,j=1}^{d} a_{ij}(x,t)\:\partial_{x_ix_j}^{2} \varphi(x,t) + \sum_{i=1}^{d} b_i(x,t)\:\partial_{x_i} \varphi (x,t) % + \gamma (x, t) \partial_t u(x, t) \\ & &+ \int_{\R^d\times \R\setminus\{(0,0)\}} \bigg[ \varphi (x+y,t-s)-\varphi (x,t) -\nabla_x \varphi(x,t) \cdot y \: \mathbf 1_{\{|y|\le1\}} \bigg] J(x,t;dy,ds). \nonumber \end{eqnarray*} For the case where the jump measure takes form $J(x,t; dy, ds) = j(x,t)\delta_0(dy)\nu(ds)$ for some L\'evy measure $\nu$ on $\R$, if $f\in C_{b}^{2,\alpha}(\R^d)$ and $h\in \text{Lip}(\R^d\times \R)$ satisfies $h(\cdot, t) \in C^{2,\alpha}(\R^d)$ for each $t\in \R$ for some $\alpha \in (0,1)$, then the above parabolic equation has a unique classical solution. See Theorem \ref{TimeFracEqn} for precise statement. When the joint process $(X,Z)$ generated by $\sL$ is a L\'{e}vy process, i.e. $a_{ij}(x,t)=a_{ij}$, $b_i(x,t) = b_i$ are constants and $J(x,t; dy, ds) = J(dy, ds)$ is a L\'{e}vy measure on $\R^{d} \times \R$, and if $f\in C_{b}^{2}(\R^d)$ and $h(x,t) = \int_{\R^d \times [t,\infty)} (f(x+y)-f(x))J(dy, ds)$, then the above parabolic equation also has a unique classical solution. In this case, the solution $u$ is a bounded and continuous function on $\R^d\times \R$ and $u(\cdot, t)\in C_{b}^{2}(\R^d)$ for each $t\in \R$. See Theorem \ref{MainTheorem4LevyCase} for precise statement. Our method is probabilistic and direct. Probabilistic representation of solutions to the time-fractional equations is given.