A Rigorous Mathematical Construction of Feynman Path Integrals for the Schrödinger Equation with Magnetic Field

Abstract

A Feynman path integral formula for the Schrödinger equation with magnetic field is rigorously mathematically realized in terms of infinite dimensional oscillatory integrals. We show (by the example of a linear vector potential) that the requirement of the independence of the integral on the approximation procedure forces the introduction of a counterterm to be added to the classical action functional. This provides a natural explanation for the appearance of a Stratonovich integral in the path integral formula for both the Schrödinger and heat equation with magnetic field.

Appendix A: Proof of Lemma 2

Let us consider the sequence of random variables $\{g_n\}$ defined in (48), namely:

$$\begin{aligned} g_n (\omega )= \sum _{j=0}^{n-1} a(\sqrt{i\hbar }\omega (s_j))\left( \omega (s_{j+1})-\omega (s_j) \right) , \qquad \omega \in C_t \end{aligned}$$

and the stochastic integral

$$\begin{aligned} G (\omega )= \int _0^t a(\sqrt{i\hbar }\omega (s)) d\omega (s), \end{aligned}$$

where $s_j=\frac{jt}{n}$ and a is the Fourier transform of a complex bounded measure on ${\mathbb {R}}$ with compact support contained in the ball $B_R$ with radius $R \in {\mathbb {R}}^+$:

$$\begin{aligned} a(\sqrt{i\hbar }\omega (s))=\int _{{\mathbb {R}}}e^{i\sqrt{i\hbar }\xi \omega (s)}d\mu (\xi ). \end{aligned}$$

Without loss of generality, we can restrict ourselves to prove the convergence of $g_n$ to G in $L^p(C_t, {\mathbb {P}})$ for p even.

By the BDG inequalities (see, e.g., [42]) we have

$$\begin{aligned} {\mathbb {E}} \left[ | G-g_n |^{2p} \right] \le C_{2p} \cdot {\mathbb {E}} \left[ \left( \sum _{j=0}^{n-1} {\int _{s_j}^{s_{j+1}}} \left| a(\sqrt{i\hbar }\omega (s))-a(\sqrt{i\hbar }\omega (s_j))\right| ^2 ds\right) ^p \right] , \end{aligned}$$

(77)

with $C_{2p}$ a positive constant. Moreover we have:

$$\begin{aligned}&\left| a(\sqrt{i\hbar }\omega (s))-a(\sqrt{i\hbar }\omega (s_j))\right| ^2 \nonumber \\&\quad = ( \omega (s)-\omega (s_j))^2 \left| \int _0^1 \sqrt{i\hbar }a'\left( \sqrt{i\hbar }\left( \omega (s_j)+u(\omega (s)-\omega (s_j))\right) \right) du \right| ^2 \nonumber \\&\quad = ( \omega (s)-\omega (s_j))^2 \left| i\sqrt{i\hbar } \int _0^1 \int _{{\mathbb {R}}} \xi e^{i\sqrt{i\hbar }\xi (\omega (s_j)+u(\omega (s)-\omega (s_j)))} d\mu (\xi ) du \right| ^2 \\&\quad \le \hbar ({\omega (s)}-{\omega (s_j)})^2 \cdot {\mathcal {G}}(\omega (s),{\omega (s_j)}), \nonumber \end{aligned}$$

(78)

where

$$\begin{aligned}&{\mathcal {G}}(\omega (s),{\omega (s_j)}) \nonumber \\&\quad =\int _0^1 \int _0^1 \int _{{\mathbb {R}}}\int _{{\mathbb {R}}} |\xi _1||\xi _2|e^{-\frac{\sqrt{2}}{2}\xi _1({\omega (s_j)}+u_1({\omega (s)}-{\omega (s_j)}))} e^{-\frac{\sqrt{2}}{2}\xi _1({\omega (s_j)}+u_2({\omega (s)}-{\omega (s_j)}))}\nonumber \\&\qquad d|\mu |(\xi _1)d|\mu |(\xi _2)du_1du_2. \end{aligned}$$

(79)

Using (78), we can write the expectation (77) as follows

(80)

where in the latter inequality we used Schwarz inequality, with

$$\begin{aligned} I_n^1=&\sqrt{{\mathbb {E}}\left[ {\sum _{j_1,\dots , j_p=0}^{n-1}} \left( {\int _{s_{j_1}}^{s_{j_1+1}}}\cdots {\int _{s_{j_p}}^{s_{j_p+1}}} {(\omega (s_1)-\omega (s_{j_1}))}^4 \cdots {(\omega (s_p)-\omega (s_{j_p}))}^4 ds_1 \cdots ds_p \right) \right] }, \nonumber \\\end{aligned}$$

(81)

$$\begin{aligned} I_n^2=&\sqrt{{\mathbb {E}}\left[ {\sum _{j_1,\dots , j_p=0}^{n-1}} \left( {\int _{s_{j_1}}^{s_{j_1+1}}}\cdots {\int _{s_{j_p}}^{s_{j_p+1}}} {{\mathcal {G}}(\omega (s_1),\omega (s_{j_1}))}^2 \cdots {{\mathcal {G}}(\omega (s_p),\omega (s_{j_p}))}^2 ds_1 \cdots ds_p \right) \right] }.\nonumber \\ \end{aligned}$$

(82)

We will show that $I_n^1 \rightarrow 0$ for $n \rightarrow \infty$ and that $I_n^2$ is uniformly bounded in n.

Let us consider the integral $I_n^1$ given by (81).

All the expectations ${\mathbb {E}}\left[ {(\omega (s_1)-\omega (s_{j_1}))}^4 \cdots {(\omega (s_p)-\omega (s_{j_p}))}^4 \right]$ can be computed taking into account the coincidences of the indices $j_r$, with $r=1,\dots , p$ in the following way:

$$\begin{aligned}&{\int _{s_{j_1}}^{s_{j_1+1}}}\cdots {\int _{s_{j_p}}^{s_{j_p+1}}} {\mathbb {E}}\left[ {(\omega (s_1)-\omega (s_{j_1}))}^4 \cdots {(\omega (s_p)-\omega (s_{j_p}))}^4 \right] ds_1 \cdots ds_p \nonumber \\&\quad =\int _0^{\frac{t}{n}} \cdots \int _0^{\frac{t}{n}} {\mathbb {E}}\left[ \omega (s_1)^4 \cdots \omega (s_{p_1})^4 \right] ds_1 \cdots ds_{p_1} \int _0^{\frac{t}{n}} \nonumber \\&\qquad \cdots \int _0^{\frac{t}{n}} {\mathbb {E}}\left[ \omega (s_{p_1+1})^4\cdots \omega (s_{p_1+p_2})^4\right] ds_{{p_1}+1} \cdots ds_{p_1+p_2} \nonumber \\&\qquad \dots \qquad \int _0^{\frac{t}{n}} \cdots \int _0^{\frac{t}{n}} {\mathbb {E}}\left[ \omega (s_{p_{i-1}+1})^4 \cdots \omega (s_{p_{i-1}+p_i})^4 \right] ds_{p_{i-1}+1} \cdots ds_{p_{i-1}+p_i}, \end{aligned}$$

(83)

with $p_1+p_2+ \cdots + p_i=p$ and we have used that in distribution ${\omega (s)}-{\omega (s_j)} \sim \omega (s-s_j)$. Further, the generic term containing $\tilde{p}$ factors, for any $\tilde{p}=1, \dots , p$, can be computed as

$$\begin{aligned}&\int _0^{\frac{t}{n}} \cdots \int _0^{\frac{t}{n}} {\mathbb {E}}\left[ \omega (s_1)^4 \cdots \omega (s_{\tilde{p}})^4 \right] ds_1 \cdots ds_{\tilde{p}}\\&\quad = \tilde{p}! \idotsint \limits _{0<s_1<\cdots<s_{\tilde{p}}<t/n} {\mathbb {E}}\left[ \omega (s_1)^4 \cdots \omega (s_{\tilde{p}})^4\right] ds_1 \cdots ds_{\tilde{p}}. \end{aligned}$$

By a straightforward calculation² we can represent ${\mathbb {E}}\left[ \omega (s_1)^4 \cdots \omega (s_{\tilde{p}})^4 \right]$ as a homogeneous polynomial $P(s_1,s_2-s_1, \dots , s_{\tilde{p}}-s_{\tilde{p}-1})$ with $\deg (P)=2\tilde{p}$. We can rewrite it as $Q(s_1,s_2,\dots , s_{\tilde{p}})$, with $\deg (Q)=2\tilde{p}$ (its coefficients depending only on $\tilde{p}$). Thanks to the change of variables $t_i=\frac{s_i}{t/n}$, we have

$$\begin{aligned}&\idotsint \limits _{0<s_1<\cdots<s_{\tilde{p}}<t/n}Q(s_1,s_2,\dots , s_{\tilde{p}})ds_1\cdots ds_{\tilde{p}}\\&\quad =\idotsint \limits _{0<t_1<\cdots<t_{\tilde{p}}<1} \left( \frac{t}{n} \right) ^{3\tilde{p}} Q(t_1,t_2,\dots , t_{\tilde{p}}) dt_1 \cdots dt_{\tilde{p}}= C \cdot \left( \frac{t}{n} \right) ^{3\tilde{p}}, \end{aligned}$$

with

$$\begin{aligned} C=\int _{0<t_1< \cdots< t_{\tilde{p}}<1} Q(t_1,t_2,\dots , t_{\tilde{p}}) dt_1 \cdots dt_{\tilde{p}}. \end{aligned}$$

Applying the same argument for all terms in (83) we get

$$\begin{aligned}&{\int _{s_{j_1}}^{s_{j_1+1}}}\cdots {\int _{s_{j_p}}^{s_{j_p+1}}} {\mathbb {E}}\left[ {(\omega (s_1)-\omega (s_{j_1}))}^4 \cdots {(\omega (s_p)-\omega (s_{j_p}))}^4 \right] ds_1 \cdots ds_p\\&\quad =C_1 \cdots C_i \cdot \left( \frac{t}{n}\right) ^{3(p_1+\cdots +p_i)}={\widetilde{C}}\cdot \left( \frac{t}{n}\right) ^{3p}, \end{aligned}$$

with $\tilde{C}=C_1 \cdots C_i$. Thus all the contributions can be estimated by ${\widetilde{K}}_p \cdot \left( \frac{t}{n}\right) ^{3p}$, where ${\widetilde{K}}_p$ is the maximum of the constants computed as ${\widetilde{C}}$. Eventually, using ${\sum _{j_1,\dots , j_p=0}^{n-1}} 1 =n^p$ we get

$$\begin{aligned} I_n^1 \le \sqrt{{\sum _{j_1,\dots , j_p=0}^{n-1}} {\widetilde{K}}_p\cdot \left( \frac{t}{n}\right) ^{3p}}=\sqrt{\left( \frac{t}{n}\right) ^{3p}\cdot {\widetilde{K}}_p \cdot n^p}=\widetilde{{\mathcal {K}}}_p \cdot \frac{t^{\frac{3p}{2}}}{n^p} \xrightarrow {n \rightarrow \infty } 0. \end{aligned}$$

Concerning $I_n^2$, recalling the definition (79) of ${\mathcal {G}}$, we have to study

$$\begin{aligned} I_n^2=\sqrt{{\sum _{j_1,\dots , j_p=0}^{n-1}} \left( {\int _{s_{j_1}}^{s_{j_1+1}}}\cdots {\int _{s_{j_p}}^{s_{j_p+1}}} {\mathbb {E}}\left[ {{\mathcal {G}}(\omega (s_1),\omega (s_{j_1}))}^2 \cdots {{\mathcal {G}}(\omega (s_p),\omega (s_{j_p}))}^2 \right] ds_1 \cdots ds_p \right) }. \end{aligned}$$

By writing explicitly the functions ${\mathcal {G}}(\cdot ,\cdot )$, we get the following bound:

$$\begin{aligned}&\left( I_n^2 \right) ^2\le {\sum _{j_1,\dots , j_p=0}^{n-1}} \left( {\int _{s_{j_1}}^{s_{j_1+1}}}\cdots {\int _{s_{j_p}}^{s_{j_p+1}}} \int _0^1 \cdots \int _0^1 \int _{{\mathbb {R}}} \cdots \int _{{\mathbb {R}}} {\mathbb {E}}\left[ \prod _{i=1}^p |\xi _i||\tilde{\xi }_i||\zeta _i||\tilde{\zeta }_i| \right. \right. \\&\quad e^{-\frac{\sqrt{2}}{2}\xi _i(\omega (s_{j_i}))+u_i(\omega (s_i)-\omega (s_{j_i}))} \cdot e^{-\frac{\sqrt{2}}{2}\tilde{\xi }_i(\omega (s_{j_i}))+\tilde{u}_i(\omega (s_i)-\omega (s_{j_i}))} \cdot e^{-\frac{\sqrt{2}}{2}\zeta _i(\omega (s_{j_i}))+v_i(\omega (s_i)-\omega (s_{j_i}))} \\&\quad e^{-\frac{\sqrt{2}}{2}\tilde{\zeta }_i(\omega (s_{j_i}))+\tilde{v}_i(\omega (s_i)-\omega (s_{j_i}))} \Bigg ] d|\mu |(\xi _i)d|\mu |(\tilde{\xi }_i)d|\mu |(\zeta _i)d|\mu |(\tilde{\zeta }_i)du_id\tilde{u}_idv_id\tilde{v}_i \Bigg ). \end{aligned}$$

Since by assumption the support of the measure $\mu$ is contained in a ball $B_R$ of radius R, we can bound $|\xi _i||\tilde{\xi }_i||\zeta _i||\tilde{\zeta }_i|\le R^{4}$ on the support of $\mu$ obtaining :

$$\begin{aligned}&\left( I_n^2 \right) ^2 \le R^{4p} {\sum _{j_1,\dots , j_p=0}^{n-1}} \left( {\int _{s_{j_1}}^{s_{j_1+1}}}\cdots {\int _{s_{j_p}}^{s_{j_p+1}}} \int _0^1 \cdots \int _0^1 \int _{{\mathbb {R}}} \right. \\&\quad \cdots \int _{{\mathbb {R}}} {\mathbb {E}}\left[ \prod _{i=1}^p e^{-\frac{\sqrt{2}}{2}\omega (s_i)(\xi _i+\tilde{\xi }_i+\zeta _i+\tilde{\zeta }_i)}\right. \\&\quad e^{-\frac{\sqrt{2}}{2}(\omega (s_i)-\omega (s_{j_i}))(\xi _i u_i+\tilde{\xi }_i \tilde{u}_i + \zeta _i v_i + \tilde{\zeta }_i \tilde{v}_i)} \Bigg ] d|\mu |(\zeta _i)d|\mu |(\tilde{\zeta }_i)du_id\tilde{u}_i dv_i d\tilde{v}_i \Bigg ). \end{aligned}$$

We notice that the term under the expectation can be computed as

$$\begin{aligned} \exp \left[ P(s_1, s_{j_1},\dots ,s_p,s_{j_p},\xi _1,\tilde{\xi }_1 , \dots , v_p, \tilde{v}_{p})\right] , \end{aligned}$$

where P is a polynomial function, which maximum $M_P$ for $s_i,s_{k_i} \in [0,t]$, $u_i,\tilde{u}_i,v_i,\tilde{v}_i \in [0,1]$, and $\xi _i, \tilde{\xi }_i, \zeta _i, \tilde{\zeta }_i \in {{\,\mathrm{supp}\,}}(\mu )$, for all $i = 1 \dots p$. Finally, by integrating and summing with respects to all variables, we get a finite term of the order $t^p\cdot |\mu |^{4p}\cdot M$, proving a uniform bound for $I_n^2$. Hence

$$\begin{aligned} g_n(\omega ) \xrightarrow {L^{2p}(C_t,{\mathbb {P}})} \int _0^t a(\sqrt{i\hbar }\omega (s))d\omega (s), \qquad \omega \in C_t. \end{aligned}$$

Let us consider now the sequence of random variables $\{h_n\}$ defined by (49) namely:

$$\begin{aligned} h_n(\omega )=\sum _{j=0}^{n-1} \frac{1}{2}\cdot a'( \sqrt{ i\hbar } {\omega (s_j)})({\omega (s_{j+1})}-{\omega (s_j)})^2, \qquad \omega \in C_t, \end{aligned}$$

and set $a'(\sqrt{i\hbar }\omega (s))\equiv \phi (\omega (s))$, for any $s \in [0,t]$. Let H be the random variable defined by

$$\begin{aligned} H(\omega ) =\frac{1}{2} \int _0^t \phi (\omega (s)) ds, \qquad \omega \in C_t. \end{aligned}$$

We have:

$$\begin{aligned} H(\omega )-h_n(\omega )&= \frac{1}{2}\sum _{j=0}^{n-1}\left( \int _{s_j}^{s_{j+1}} \phi (\omega (s)) ds - \phi (\omega (s_j))\left( \omega (s_{j+1})-\omega (s_j)\right) ^2\right) \\&=\frac{1}{2}\sum _{j=0}^{n-1}\Bigg ( \int _{s_j}^{s_{j+1}} \phi (\omega (s_j))ds + \int _{s_j}^{s_{j+1}}\phi '(\omega (s_j))(\omega (s)-\omega (s_j))ds\\&\quad +\int _{s_j}^{s_{j+1}} \int _0^1\left( \omega (s)-\omega (s_j))^2 \phi ''(\omega (s_j)+u(\omega (s)\right. \\ {}&\quad \left. -\omega (s_j)))(1-u)\right) duds - \phi (\omega (s_j)) \left( \omega (s_{j+1})-\omega (s_j) \right) ^2 \Bigg ). \end{aligned}$$

Hence

$$\begin{aligned} \Vert H-h_n\Vert _{L^{p}(C_t, {\mathbb {P}})}\le \frac{1}{2}\left( \Vert J_n^1\Vert _{L^{p}(C_t, {\mathbb {P}})}+\Vert J_n^2\Vert _{L^{2p}(C_t, {\mathbb {P}})}+\Vert J_n^3\Vert _{L^{p}(C_t, {\mathbb {P}})}\right) , \end{aligned}$$

where:

$$\begin{aligned} J^1_n(\omega )= & {} \sum _{j=0}^{n-1} \phi (\omega (s_j)) \left( (s_{j+1}-s_j)-(\omega (s_{j+1})-\omega (s_j))^2 \right) ; \\ J_n^2 (\omega )= & {} \sum _{j=0}^{n-1} \phi '(\omega (s_j)) \int _{s_j}^{s_{j+1}} (\omega (s)-\omega (s_j))ds; \\ J^3_n(\omega )= & {} \sum _{j=0}^{n-1}\int _{s_j}^{s_{j+1}} \int _0^1\left( \omega (s)-\omega (s_j))^2 \phi ''(\omega (s_j)+u(\omega (s)-\omega (s_j)))(1-u)\right) duds. \end{aligned}$$

Without loss of generality we can consider the case where the function $\phi :{\mathbb {R}}\rightarrow {\mathbb {C}}$ is real valued, since the general case follows easily by the inequality $\Vert J_n^1\Vert _{L^{p}}\le \Vert Re(J_n^1)\Vert _{L^{p}}+\Vert Im(J_n^1)\Vert _{L^{p}}$.

The $L^{p}$ norm of the function $J_n^1$ can be estimated as:

$$\begin{aligned}&{\mathbb {E}}[|J^1_n|^{2p}]\\&\quad =\sum _{j_1,...,j_{2p}=0}^{n-1}{\mathbb {E}}\Bigg [ \phi (\omega (s_{j_1}))\cdots \phi (\omega (s_{j_{2p}}))\left( (s_{{j_1}+1}-s_{j_1})-(\omega (s_{j_1+1})-\omega (s_{j_1}))^2 \right) \\&\qquad \cdots \left( (s_{{j_{2p}}+1}-s_{j_{2p}})-(\omega (s_{j_{2p}+1})-\omega (s_{j_{2p}}))^2 \right) \Bigg ]\\&\quad \le (2p)!\sum _{0\le j_1\le ...\le j_{2p}\le n-1}{\mathbb {E}}\Bigg [ \phi (\omega (s_{j_1}))\cdots \phi (\omega (s_{j_{2p}}))\left( (s_{{j_1}+1}-s_{j_1})-(\omega (s_{j_1+1})\right. \\&\qquad - \left. \omega (s_{j_1}))^2 \right) \\&\qquad \cdots \left( (s_{{j_{2p}}+1}-s_{j_{2p}})-(\omega (s_{j_{2p+1}})-\omega (s_{j_{2p}}))^2 \right) \Bigg ]\ . \end{aligned}$$

Since ${\mathbb {E}}[((s_{j+1}-s_{j})-(\omega (s_{j}+1)-\omega (s_j))^2 )]=0$, the sum above contains only the $n^{2p-1}$terms where $j_1\le \dots \le j_{2p-1}=j_{2p}$. Indeed, if $j_1\le \dots \le j_{2p-1}<j_{2p}$:

$$\begin{aligned}&{\mathbb {E}}\Bigg [\prod _{i=1}^{2p} \phi (\omega (s_{j_i}))\left( (s_{{j_1}+1}-s_{j_1})-(\omega (s_{j_1+1})-\omega (s_j))^2 \right) \Bigg ]\\&\quad ={\mathbb {E}}\Bigg [\prod _{i=1}^{2p-1} \phi (\omega (s_{j_i}))\left( (s_{{j_1}+1}-s_{j_1})-(\omega (s_{j_1+1})-\omega (s_j))^2 \right) \phi (\omega (s_{j_{p}}))\Bigg ] \cdot \\&\qquad {\mathbb {E}}\Bigg [\left( (s_{{j_{2p}}+1}-s_{j_{2p}})-(\omega (s_{j_{2p+1}})-\omega (s_{j_{2p}}))^2 \right) \Bigg ] = 0. \end{aligned}$$

Direct computation shows that all the terms in this sum are of order $O((s_{{j}+1}-s_{j})^{2p})=O(1/n^{p})$ or less. Indeed, taking into account the possible coincidences of indexes, all the terms are of the form

$$\begin{aligned}&{\mathbb {E}}\Bigg [ \phi (\omega (s_{k_1}))^{p_1}\left( (s_{{k_1}+1}-s_{k_1})-(\omega (s_{k_1+1})-\omega (s_{k_1}))^2 \right) ^{p_1}\cdots \nonumber \\&\quad \cdots \phi (\omega (s_{k_r}))^{p_r}\left( (s_{{k_r}+1}-s_{k_r})-(\omega (s_{k_r+1})-\omega (s_{k_r}))^2 \right) ^{p_r}\Bigg ], \end{aligned}$$

(84)

where $p_1+\dots +p_r=2p$ and $k_1<k_2< \cdots <k_r$. By writing $\phi (x)=\int e^{i\sqrt{i}\xi x} d\nu (\xi )$, $x \in R$ with $\nu$ complex Borel measure on ${\mathbb {R}}$ supported in the ball $B_R$, the integral (84) can be estimates as:

$$\begin{aligned}&\int _{{\mathbb {R}}^{2p}}{\mathbb {E}}\left[ \left( (s_{{k_r}+1}-s_{k_r})-(\omega (s_{k_r+1})-\omega (s_{k_r}))^2 \right) ^{p_r}\right] \nonumber \\&\quad \prod _{\alpha =0}^{r-2}\Bigg ({\mathbb {E}}\left[ e^{i\sqrt{i}(\omega (s_{k_{r-\alpha }})-\omega (s_{k_{r-\alpha -1}+1}))\sum _{l=1}^{\sum _{\beta =0}^\alpha p_{r-\beta }}\xi _l}\right] \Bigg ) \nonumber \\&\Bigg (\prod _{\alpha =1}^{r-1}{\mathbb {E}}\left[ e^{i\sqrt{i}(\omega (s_{k_{r-\alpha }+1})-\omega (s_{k_{r-\alpha }}))\sum _{l=1}^{\sum _{\beta =0}^\alpha p_{r-\beta }}\xi _l}\left( (s_{{k_{r-\alpha }}+1}-s_{k_{r-\alpha }})-(\omega (s_{k_{r-\alpha }+1})\right. \right. \nonumber \\&\quad \left. \left. -\omega (s_{k_{r-\alpha }}))^2 \right) ^{p_{r-\alpha }}\right] \Bigg ) \nonumber \\&{\mathbb {E}}\left[ e^{i\sqrt{i}\omega (s_{k_1})\sum _{l=1}^{2p}\xi _l}\right] d\nu (\xi _1)\dots d\mu (\xi _{2p}). \end{aligned}$$

(85)

Now, since $\omega (t_1)-\omega (t_2)$ has the same law as $(t_1-t_2)^{\frac{1}{2}}X$, with X a standard normal random variable and for all $\zeta \in {\mathbb {R}}$, $0\le t_1\le t_2$, $k\in {\mathbb {N}}$, we have:

$$\begin{aligned}&{\mathbb {E}}[e^{i\sqrt{i} \zeta (\omega (t_1)-\omega (t_2)) }]=e^{-\frac{i}{2}(t-s)\xi ^2},\\&{\mathbb {E}}[e^{i\sqrt{i} \zeta X }X^{2k}]=H_{2k}(\sqrt{i} \zeta )e^{-\frac{i}{2}\zeta ^2}, \end{aligned}$$

with $H_n$ denoting the $n^{th}$ Hermite polynomial. Hence:

$$\begin{aligned}&\Bigg | {\mathbb {E}}\left[ e^{i\sqrt{i}(\omega (s_{k_{r-\alpha }})-\omega (s_{k_{r-\alpha -1}+1})) \sum _{l=1}^{\sum _{\beta =0}^\alpha p_{r-\beta }}\xi _l}\right] \Bigg |=1\\&\Bigg |{\mathbb {E}}\left[ e^{i\sqrt{i}(\omega (s_{k_{r-\alpha }+1})-\omega (s_{k_{r-\alpha }}))\sum _{l=1}^{\sum _{\beta =0}^\alpha p_{r-\beta }}\xi _l}\left( (s_{{k_{r-\alpha }}+1}-s_{k_{r-\alpha }})\right. \right. \\&\qquad \left. \left. -(\omega (s_{k_{r-\alpha }+1})-\omega (s_{k_{r-\alpha }}))^2 \right) ^{p_{r-\alpha }}\right] \Bigg | \\&\quad \le (s_{{k_{r-\alpha }}+1}-s_{k_{r-\alpha }})^{p_{r-\alpha }}P_{\alpha ,p_{r-\alpha }}(\xi _1, \dots ,\xi _{2p}), \end{aligned}$$

with $P_{\alpha ,p_{r-\alpha }}:{\mathbb {R}}^{2p}\rightarrow {\mathbb {R}}$ suitable polynomial functions. By setting

$$\begin{aligned} M:=\max _{r, p_1, \dots p_r}\prod _{\alpha =1}^{r-1}\max _{\xi _1,\dots \xi _p\in B_R}|P_{\alpha ,p_{r-\alpha }}(\xi _1, \dots ,\xi _p)|, \end{aligned}$$

we get ${\mathbb {E}}[|J^1_n|^{2p}]\le M\frac{t^{2p}}{n}|\nu (B_R)|^{2p}$, obtaining the required convergence result:

$$\begin{aligned} \lim _{n\rightarrow \infty }{\mathbb {E}}[|J^1_n|^{2p}]\rightarrow 0. \end{aligned}$$

The same argument produces an analogous estimate for ${\mathbb {E}}[|J^2_n|^{2p}]$. Indeed, always assuming without loss of generality that the function $\phi$ is real valued, we get:

$$\begin{aligned}&{\mathbb {E}}[|J^2_n|^{2p}]= \sum _{j_1,\dots ,j_{2p} =0}^{n-1}{\mathbb {E}}\Bigg [ \phi '(\omega (s_{j_1}))\cdots \phi '(\omega (s_{j_{2p}})) \int _{s_{j_1}}^{s_{{j_1}+1}} (\omega (u_1)-\omega (s_{j_1}))du_1\\&\qquad \cdots \int _{s_{j_{2p}}}^{s_{{j_{2p}}+1}} (\omega (u_{2p})-\omega (s_{j_{2p}}))du_{2p} \Bigg ] \\&\quad \le (2p)!\sum _{0\le j_1\le \dots \le j_{2p}\le n-1}^{n-1}{\mathbb {E}}\Bigg [ \phi '(\omega (s_{j_1}))\cdots \phi '(\omega (s_{j_{2p}})) \int _{s_{j_1}}^{s_{{j_1}+1}} (\omega (u_1)\\&\qquad -\omega (s_{j_1}))du_1\\&\qquad \cdots \int _{s_{j_{2p}}}^{s_{{j_{2p}}+1}} (\omega (u_{2p})-\omega (s_{j_{2p}}))du_{2p}\Bigg ] . \end{aligned}$$

Again, since ${\mathbb {E}}[\int _{s_j}^{s_j+1}(\omega (u)-\omega (s_j))du]=0$, we can consider only the $n^{2p-1}$ terms with $j_1\le \cdots \le j_{2p-1}= j_{2p}$. All terms have the same structure as the integrals appearing in (80) and by using the same arguments applied for the estimates of integrals (81) and (82), we obtain $\lim _{n\rightarrow \infty }{\mathbb {E}}[|J^2_n|^{2p}]=0$. Furthermore, the same argument applies also to the term $J_n^3$, yielding $\lim _{n\rightarrow \infty }{\mathbb {E}}[|J^3_n|^{2p}]=0$.

Thus

$$\begin{aligned} h_n \xrightarrow {L^{p}(\Omega ,{\mathbb {P}})} \int _{0}^t \phi (\omega (s))ds. \end{aligned}$$

We estimate the last term $r_n$ (defined in (49)) by means of the Cauchy–Schwarz inequality as follows:

(86)

where

$$\begin{aligned} {\mathcal {F}}({\omega (s_j)},{\omega (s_j)}j)=&\int _0^1\int _0^1 \int _{{\mathbb {R}}}\int _{{\mathbb {R}}}|\kappa _1||\kappa _2| e^{-\frac{\sqrt{2}}{2} \kappa _1 ({\omega (s_j)}+(\omega (s_{j+1})-\omega (s_j))u_1)} \\&e^{-\frac{\sqrt{2}}{2} \kappa _2 ({\omega (s_j)}+(\omega (s_{j+1})-\omega (s_j))u_2)} (1-u_1)^2(1-u_2)^2 du_1du_2d|\mu |(\kappa _1)d|\mu |(\kappa _2). \end{aligned}$$

Both factors appearing in the last line of (86) can be estimated by the same techniques applied in the study of the terms (81) and (82), obtaining $r_n \xrightarrow {L^p(\Omega ,{\mathbb {P}})}0$.

Eventually, we conclude that the sequence of random variables $f_n$ defined as

$$\begin{aligned} f_n(\omega )=\int _0^t {\mathbf{a}}\left( \sqrt{ i\hbar } \omega _n(s) \right) \cdot \dot{\omega }_n(s) ds, \end{aligned}$$

converges, as $n \rightarrow \infty$, in $L^p(\Omega ,{\mathbb {P}})$ to the random variable f defined as the Stratonovich stochastic integral

$$\begin{aligned} f(\omega )=\int _0^t {\mathbf{a}}( \sqrt{ i\hbar } \omega (s))\circ d\omega (s). \end{aligned}$$

$\square$