1 Introduction
The FeynmanKac equation describes the distribution of the functionals of the trajectories of the particles, where the functional is defined as with being a trajectory of a particle and a prescribed function depending on specific applications Kac1949
. There are two kinds of FeynmanKac equations: one is for the forward FeynmanKac equation, governing the joint probability density of the functional and position; and another one is for the backward equation, just focusing on the distribution of the functionals. If the particles are with powerlaw waiting time and/or jump length distribution(s), the governing equations for the distribution of the functionals are socalled fractional FeynmanKac equations
Agmon1984 ; Carmi2010 ; Wang2018 , since the fractional substantial derivative is involved in the equations. More generalizations of the FeynmanKac equations include the models governing the distribution of the functionals of the particles undergoing the reaction and diffusion processes and of the particles with multiple internal states Hou2018 ; Xu2018 ; Xu20182 .Here we solve the following backward fractional FeynmanKac equation, presented in Carmi2010 , describing the functional distribution of the particles with powerlaw waiting time, i.e.,
(1.1) 
where and
is the joint probability density function of finding the particle on
at time with the initial position of the particle at ; is the Fourier pair of ; ; stands for Laplace operator; is a bounded domain and is assumed to be bounded in in this paper; is a fixed final time; denotes the RiemannLiouville fractional substantial derivative, whose definition Li2015 is(1.2)  
where denotes the RiemannLiouville fractional derivative with the definition Podlubny1999
So far there have been many works for fractional partial differential equations, including the finite difference method, finite element method, spectral method, and so on
Acosta2018 ; Bazhlekova2015 ; chen2009 ; Cheng2015 ; Deng2008 ; Deng2013 ; Ervin2006 ; li2012 ; sun2006 , but there are relatively less researches on solving fractional FeynmanKac equation numerically Chen2018 ; Deng2015 ; Deng:17 ; Deng2017 ; Nie2019 . The main reasons are that fractional substantial derivative is a timespace coupled nonlocal operator and the equation covers the complex parameters which bring about many challenges on regularity and numerical analyses. To our best knowledge, numerical approximation on fractional substantial derivative is given in Chen2015 ; Ref. Deng2015 numerically solves the forward and backward fractional FeynmanKac equations with the assumptions that the solution is regular, is a positive constant, and ( means the real part of ); Ref. Deng2017 presents the error estimate for the backward fractional FeynmanKac equation with and ; Ref. Deng:17 provides an efficient timestepping method to solve the forward fractional FeynmanKac equation and makes error analysis in the measure norm. In this paper, we use the finite element method in space and convolution quadrature introduced in Lubich1988 ; Lubich19882 in time to solve the backward fractional FeynmanKac equation (1.1). The main contributions are as follows.
We first provide Sobolev regularity for the solution of Eq. (1.1), i.e., Theorem 2.1 gives that the solution when is bounded in and . Compared with the previous works Chen2018 ; Deng2015 ; Deng2017 , we construct numerical scheme without any assumption on the regularity of solution in temporal and spatial directions.

Then we modify the approximation of the RiemannLiouville fractional derivative got by convolution quadrature to approximate the RiemannLiouville fractional substantial derivative, which skillfully overcome the trouble brought by the noncommutativity of the RiemannLiouville fractional derivative and in error estimate for fully discrete scheme, i.e., in Eq. (3.1).

Next, a suitable modify based on the Laplace transform representation of solution is presented to guarantee the accuracy of secondorder backward difference scheme (3.9) (see Sec. 3).

Besides, motivated by the error estimate in space in Bazhlekova2015 ; Jin2016 , a general idea is to get the estimate of the difference between and (for their detailed definitions, see Sec. 3 and Sec. 4). Generally, the sufficient regularity on is required to ensure the accuracy of the approximation. Here, we use for the fully discrete scheme (4.1) in Sec. 4 instead of , which weakens the requirement of regularity on to keep the accuracy of the numerical scheme.

Finally, we provide a complete error analysis for the proposed numerical scheme and obtain the optimal convergence rates in  and norm.
The rest of the paper is organized as follows. We first provide some preliminaries and a regularity estimate for the solution of Eq. (1.1) in Sec. 2. Section 3 presents the approximation of the RiemannLiouville fractional substantial derivative by backward Euler and secondorder backward difference convolution quadratures and gives the error estimates of the time semidiscrete schemes. In Sec. 4, we use the finite element method to discretize the Laplace operator and provide the error estimate for the fully discrete scheme with the nonsmooth initial data. In Sec. 5, we verify the effectiveness of the algorithm by numerical experiments. We conclude the paper with some discussions in the last section.
2 Preliminaries
First, we introduce with a zero Dirichlet boundary condition. For any , denote the space with the norm Thomee2006
where
are the eigenvalues ordered nondecreasingly and the corresponding eigenfunctions (normalized in the
norm) of operator . Thus , , and . For and , we define sectors and in the complex plane asand the contour is defined by
oriented with an increasing imaginary part, where denotes the imaginary unit and . Then we denote as the operator norm from to and define and as and respectively in the following. Throughout this paper, denotes a generic positive constant, whose value may differ at each occurrence; and let arbitrary small.
Similar to the skill used in Chen2018 ; Deng2015 ; Deng2017 , Eq. (1.1) can also be converted into
(2.1) 
where denotes Caputo fractional substantial derivative defined by Li2015
and means the Caputo fractional derivative with its definition Podlubny1999
Then we recall the Laplace transform for the fractional substantial derivative.
Lemma 2.1 (Li2015 )
The Laplace transform of the RiemannLiouville fractional substantial derivative with is given by
and the Laplace transform of the Caputo fractional substantial derivative with is given by
where and ‘’ stands for taking the Laplace transform. And in the following we denote as .
According to Lemma 2.1, the solution of Eq. can be written as
(2.2) 
Remark 2.1
By the definition of , it is easy to see that
Before we provide the regularity estimate for the solution of Eq. (2.1), the following lemma about is also needed.
Lemma 2.2 (Deng:17 )
Let be bounded in . By choosing sufficiently close to and sufficiently large (depending on the value ), we have the following results:

For all and , we have , and
(2.3) where and denote two positive constants. So and are both analytic function of .

The operator is welldefined, bounded, and analytic with respect to , satisfying
(2.4) (2.5)
Theorem 2.1
Assume is bounded in . If and is the solution of Eq. (2.1), then we have the estimate
3 Temporal discretization and error analysis
In this section, we first use the backward Euler and secondorder backward difference convolution quadratures introduced in Lubich1988 ; Lubich19882 to discretize the RiemannLiouville fractional substantial derivative and get the first and secondorder schemes in time. Then we provide the complete error analysis.
Let the time step size with , , , and . Firstly, we use the relationship between Caputo fractional derivative and RiemannLiouville fractional derivative
to reformulate Eq. (2.1) with RiemannLiouville fractional substantial derivative, i.e.,
(3.1) 
3.1 Backward Euler scheme and error estimate
We use backward Euler convolution quadrature to discretize the time fractional substantial derivative and get the firstorder accuracy in time. Introduce as the numerical approximation of solution . Then we can obtain the temporal semidiscrete scheme
(3.2) 
where
(3.3) 
and
Multiplying on both sides of the first formula of (3.2) and summing from to lead to
(3.4) 
Simple calculation implies
which is followed by . Furthermore, we have
Using Cauchy’s integral formula yields
(3.5)  
where , , , and the second equality follows by taking . Deforming the contour to , one has
(3.6) 
Next, we provide a lemma about defined in (3.5).
Lemma 3.1 (Deng:17 )
Let be bounded in . By choosing sufficiently close to and sufficiently large depending on , there exists a positive constant depending on and such that the following estimates hold when :

For , we have , and

The operator is welldefined, bounded, and analytic with respect to , satisfying
where . Here, means the imaginary part of and the real part of .

For the real number , the following estimate holds
Theorem 3.1
3.2 Secondorder backward difference scheme and error estimate
In this subsection, we use secondorder backward difference convolution quadrature to discretize the time fractional substantial derivative and obtain the secondorder accuracy in time. Similarly, introduce as the numerical approximation of the solution , and let
(3.7) 
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