Langevin Equations in the Small-Mass Limit: Higher-Order Approximations

Abstract

We study the small-mass (overdamped) limit of Langevin equations for a particle in a potential and/or magnetic field with matrix-valued and state-dependent drift and diffusion. We utilize a bootstrapping argument to derive a hierarchy of approximate equations for the position degrees of freedom that are able to achieve accuracy of order $m^{\ell /2}$ over compact time intervals for any $\ell \in \mathbb {Z}^+$. This generalizes prior derivations of the homogenized equation for the position degrees of freedom in the $m\rightarrow 0$ limit, which result in order $m^{1/2}$ approximations. Our results cover bounded forces, for which we prove convergence in $L^p$ norms and unbounded forces, in which case we prove convergence in probability.

Appendices

Appendix A: Assumptions Implying Homogenization as $m\rightarrow 0$

In this appendix, we give a list of properties that, as shown in [9], are sufficient to guarantee that the solutions to the SDE Eqs. (31)–(32) satisfy the properties Eqs. (8), (9), and (11) (note that what we call $\tilde{F}$ here was simply called F in [9]). Some of the assumptions below are strengthened, as compared to [9], in order to meet the needs of the current paper; we remark on this further below.

We assume that

• 1. $\gamma :[0,\infty )\times \mathbb {R}^n\rightarrow \mathbb {R}^{n\times n}$ is $C^3$.

  • (a) The values of $\gamma$ are symmetric matrices.

  • (b) The eigenvalues of $\gamma$ are uniformly bounded below by some $\lambda >0$.

  • (c) $\gamma$ is bounded.

  • (d) For all $T>0$ and all multi-indices $\alpha$ with $1\le |\alpha |\le 3$, $\partial _{q^\alpha }\gamma$ is bounded uniformly for $(t,q)\in [0,T]\times \mathbb {R}^{n}$.

  • (e) For all $T>0$ and all multi-indices $\alpha$ with $0\le |\alpha |\le 2$, $\partial _{q^\alpha }\partial _t\gamma$ is bounded uniformly for $(t,q)\in [0,T]\times \mathbb {R}^{n}$.

• 2. $\psi :[0,\infty )\times \mathbb {R}^n\rightarrow \mathbb {R}^n$ is $C^4$.

  • (a) For all $T>0$ and all multi-indices $\alpha$ with $1\le |\alpha |\le 4$, $\partial _{q^\alpha }\psi$ is bounded uniformly for $(t,q)\in [0,T]\times \mathbb {R}^{n}$.

  • (b) For all $T>0$ and all multi-indices $\alpha$ with $0\le |\alpha |\le 3$, $\partial _{q^\alpha }\partial _t\psi$ is bounded uniformly for $(t,q)\in [0,T]\times \mathbb {R}^{n}$.

• 3. $\tilde{F}:[0,\infty )\times \mathbb {R}^n\rightarrow \mathbb {R}^n$ is continuous.

  • (a) $\tilde{F}$ is bounded.

  • (b) $\tilde{F}$ is Lipschitz in q uniformly in t.

• 4. $\sigma :[0,\infty )\times \mathbb {R}^n\rightarrow \mathbb {R}^{n\times k}$ is continuous.

  • (a) $\sigma$ is bounded.

  • (b) $\sigma$ is Lipschitz in q uniformly in t.

• 5. $V:[0,\infty )\times \mathbb {R}^n\rightarrow \mathbb {R}^n$ is $C^2$.

  • (a) $\nabla _q V$ is Lipschitz in q uniformly in t.

  • (b) For all $T>0$, $\nabla _qV$ is bounded uniformly for $(t,q)\in [0,T]\times \mathbb {R}^{n}$.

  • (c) There exist $a,b\ge 0$ such that $\tilde{V}(t,q)\equiv a+b\Vert q\Vert ^2+V(t,q)$ is non-negative for all t, q.

  • (d) There exist $M,C> 0$ such that

    $$\begin{aligned} |\partial _tV(t,q)|\le M+C(\Vert q\Vert ^2+\tilde{V}(t,q)) \end{aligned}$$

    (132)

    and

    $$\begin{aligned} \Vert -\partial _t\psi (t,q)+\tilde{F}(t,q)\Vert ^2\le M+C(\Vert q\Vert ^2+\tilde{V}(t,q)) \end{aligned}$$

    (133)

    for all t, q.

• 5. There exists $C>0$ such that the (random) initial conditions satisfy $\Vert u^m_0\Vert ^2 \le Cm$ for all $m>0$ and all $\omega \in \Omega$ and $E[\Vert q^m_0\Vert ^p]<\infty$, $E[\Vert q_0\Vert ^p]<\infty$, and $E[\Vert q_0^m-q_0\Vert ^p]^{1/p}=O(m^{1/2})$ for all $p>0$.

The various global-in-time properties are used to prove non-explosion of solutions, while the properties over compact time intervals are used to prove convergence to the homogenized SDE in [9]. The reason we needed to strengthen certain regularity properties here, as compared to [9], is so we can prove the required Lipschitz properties of the remainder terms, Eq. (47), on compact time intervals; this is in contrast to [9], where one only had to show that the remainder terms converge to zero as $m\rightarrow 0$. For example, the third line of Eq. (94) includes a $\partial _{q^c}Q^{ikl}$ term, which in turn involves $\partial _{q^c}\partial _{q^b} \tilde{\gamma }^{-1}$. To ensure this is Lipschitz in q, we have assumed that $\tilde{\gamma }$ is $C^3$ with third derivative being bounded on compact time intervals; more precisely, we have assumed this of both $\gamma$ and $\partial _j\psi$, as these are used to construct $\tilde{\gamma }$. This is why we require conditions on the third derivative of $\gamma$ and the fourth derivative of $\psi$, as opposed to [9] where we only required conditions on derivatives up to order two and three, respectively. Similar remarks apply to the other objects.

Appendix B: Properties of the Fundamental Solution

Our derivations will require the use of several properties of the fundamental solution of a linear ordinary differential equation (ODE). Specifically, we need to consider the process obtained by pathwise solving the linear ODE

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Phi _t^m=-\frac{1}{m}\tilde{\gamma }(t,y_t)\Phi ^m_t,\quad \Phi ^m_0=I, \end{aligned}$$

(134)

where y is a continuous semimartingale. The process $\Phi ^m_t$ is adapted and pathwise $C^1$; we will call it the fundamental-solution process, as each of its paths is the fundamental solution to a linear ODE.

The symmetric part of $\tilde{\gamma }$, denoted by $\gamma$, is assumed to have eigenvalues bounded below by $\lambda >0$ (see Appendix A). This implies the following crucial bound

$$\begin{aligned} \Vert \Phi ^m_t(\Phi ^m_s)^{-1}\Vert \le e^{-\lambda (t-s)/m}\,\,\, \text { for all }t\ge s \end{aligned}$$

(135)

(see, for example, p.86 of [26]). Note that while the left hand side is random, the upper bound is not. As we have stated it, this bound requires the use of the the $\ell ^2$ operator norm. Otherwise, there is an additional constant multiplying the exponential.

We will also need the following bound on the difference between the fundamental solutions corresponding to two linear ODEs. See the Appendix to [13].

Lemma B1

Let $B_i:[0,T]\rightarrow \mathbb {R}^{n\times n}$; $i=1,2$, be continuous and suppose their symmetric parts have eigenvalues bounded above by $\mu$, uniformly in t. Consider the fundamental solutions, $\Phi _i(t)$, satisfying

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Phi _i(t)=B_i(t)\Phi _i(t),\,\, \Phi _i(0)=I. \end{aligned}$$

(136)

Then for any $0\le t\le T$ we have the bound

$$\begin{aligned} \Vert \Phi _1(t)-\Phi _2(t)\Vert \le e^{\mu t} \int _0^t \Vert B_1(s)-B_2(s)\Vert \mathrm{d}s . \end{aligned}$$

(137)

We will need the following lemma concerning stochastic convolutions, adapted from Lemma 5.1 in [8]:

Lemma B2

Let $B_s$ be a continuous adapted $\mathbb {R}^{n\times n}$-valued processes. Let $\Phi (t)$ be the fundamental-solution process, pathwise satisfying

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Phi (t)=B(t)\Phi (t),\,\, \Phi (0)=I. \end{aligned}$$

(138)

Let $V_s$ be a continuous adapted $\mathbb {R}^{n\times k}$-valued processes. Then we have the P-a.s. equality

$$\begin{aligned}&\Phi (t)\int _0^t\Phi ^{-1}(s) V_s \mathrm{d}W_s\nonumber \\&\quad =\Phi (t)\int _0^t V_s \mathrm{d}W_s-\Phi (t)\int _0^t \Phi ^{-1} (s)B(s) \left( \int _s^t V_r \mathrm{d}W_r\right) \mathrm{d}s\text { for all } t. \end{aligned}$$

(139)

The following lemma will assist us in bounding processes having the form of the last term in Eq. (139). The proof is very similar to that of Lemma 5.1 in [8], but we provide it for completeness.

Lemma B3

Let $V_s$ be a continuous adapted $\mathbb {R}^{n\times k}$-valued process and $\alpha >0$.

Then for every $j\in \mathbb {Z}_0$ there exists $C_j>0$ such that for all $T>0$, $\delta >0$ we have the P-a.s. bound

$$\begin{aligned}&\sup _{t\in [0,T]}\int _0^t (t-s)^je^{-\alpha (t-s)}\Vert \int _s^t V_r \mathrm{d}W_r\Vert \mathrm{d}s\nonumber \\&\quad \le \frac{C_j}{\alpha ^{j+1}}\bigg (\max _{\ell =1,...,N}\sup _{\tau \in [(\ell -1)\delta ,\min \{(\ell +1)\delta ,T\}]}\Vert \int _{(\ell -1)\delta }^{\tau }V_r\mathrm{d}W_r\Vert \nonumber \\&\qquad +e^{-\alpha \delta /2} \sup _{\tau \in [0,T]}\Vert \int _0^\tau V_r \mathrm{d}W_r\Vert \bigg ), \end{aligned}$$

(140)

where $N=\min \{\ell \in \mathbb {Z}:\ell \delta \ge T\}$. We emphasize that $C_j$ depends only on j.

Proof

Suppose $\delta <T$. First split

$$\begin{aligned}&\sup _{t\in [0,T]}\int _0^t (t-s)^j e^{-\alpha (t-s)}\Vert \int _s^t V_r \mathrm{d}W_r\Vert \mathrm{d}s\nonumber \\&\quad \le \sup _{t\in [0,\delta ]}\int _0^t(t-s)^j e^{-\alpha (t-s)}\Vert \int _s^t V_r \mathrm{d}W_r\Vert \mathrm{d}s\nonumber \\&\qquad +\sup _{t\in [\delta ,T]}\int _0^t(t-s)^j e^{-\alpha (t-s)}\Vert \int _s^t V_r \mathrm{d}W_r\Vert \mathrm{d}s. \end{aligned}$$

(141)

The first term can be bounded as follows.

$$\begin{aligned}&\sup _{t\in [0,\delta ]}\int _0^t (t-s)^je^{-\alpha (t-s)}\Vert \int _s^t V_r \mathrm{d}W_r\Vert \mathrm{d}s\nonumber \\&\quad =\sup _{t\in [0,\delta ]}\int _0^t (t-s)^je^{-\alpha (t-s)}\Vert \int _0^t V_r \mathrm{d}W_r-\int _0^s V_r \mathrm{d}W_r\Vert \mathrm{d}s\nonumber \\&\quad \le \frac{2}{\alpha ^{j+1}}\sup _{0\le \tau \le \delta }\Vert \int _0^\tau V_r \mathrm{d}W_r\Vert \int _0^{\alpha \delta } u^je^{-u}\mathrm{d}u. \end{aligned}$$

(142)

In the second term we split the integral to obtain

$$\begin{aligned}&\sup _{t\in [\delta ,T]}\int _0^t(t-s)^j e^{-\alpha (t-s)}\Vert \int _s^t V_r \mathrm{d}W_r\Vert \mathrm{d}s\nonumber \\&\quad \le \sup _{t\in [\delta ,T]}\int _0^{t-\delta }(t-s)^j e^{-\alpha (t-s)}\Vert \int _s^t V_r \mathrm{d}W_r\Vert \mathrm{d}s\nonumber \\&\qquad +\sup _{t\in [\delta ,T]}\int _{t-\delta }^t(t-s)^j e^{-\alpha (t-s)}\Vert \int _s^t V_r \mathrm{d}W_r\Vert \mathrm{d}s\nonumber \\&\quad \le \frac{2}{\alpha ^{j+1}}\left( \int _{0}^{\infty }u^{2j} e^{-u}du\right) ^{1/2}e^{-\alpha \delta /2} \sup _{\tau \in [0,T]}\Vert \int _0^\tau V_r \mathrm{d}W_r\Vert \nonumber \\&\qquad +\sup _{t\in [\delta ,T]}\int _{t-\delta }^t(t-s)^j e^{-\alpha (t-s)}\Vert \int _s^t V_r \mathrm{d}W_r\Vert \mathrm{d}s. \end{aligned}$$

(143)

Let $N=\min \{\ell \in \mathbb {Z}:\ell \delta \ge T\}$. Then P-a.s.

$$\begin{aligned}&\sup _{t\in [\delta ,T]}\int _{t-\delta }^t(t-s)^j e^{-\alpha (t-s)}\Vert \int _s^t V_r \mathrm{d}W_r\Vert \mathrm{d}s\nonumber \\&\quad \le \max _{\ell =1,...,N-1}\sup _{t\in [\ell \delta , \min \{(\ell +1)\delta ,T\}]}\int _{(\ell -1)\delta }^{t}(t-s)^je^{-\alpha (t-s)}\Vert \int _s^tV_r\mathrm{d}W_r\Vert \mathrm{d}s\nonumber \\&\quad \le \frac{2}{\alpha ^{j+1}}\int _{0}^{\infty }u^je^{-u}du\max _{\ell =1,...,N-1}\sup _{\tau \in [(\ell -1)\delta ,\min \{(\ell +1)\delta ,T\}]}\Vert \int _{(\ell -1)\delta }^{\tau }V_r\mathrm{d}W_r\Vert . \end{aligned}$$

(144)

Combining Eqs. (142),  (143), and  (144) gives the P-a.s. bound

$$\begin{aligned}&\sup _{t\in [0,T]}\int _0^t (t-s)^j e^{-\alpha (t-s)}\Vert \int _s^t V_r \mathrm{d}W_r\Vert \mathrm{d}s\nonumber \\&\quad \le \frac{C_j}{\alpha ^{j+1}}\bigg (\max _{\ell =1,...,N-1}\sup _{\tau \in [(\ell -1)\delta ,\min \{(\ell +1)\delta ,T\}]}\Vert \int _{(\ell -1)\delta }^{\tau }V_r\mathrm{d}W_r\Vert \nonumber \\&\qquad +e^{-\alpha \delta /2} \sup _{\tau \in [0,T]}\Vert \int _0^\tau V_r \mathrm{d}W_r\Vert \bigg ). \end{aligned}$$

(145)

The case $\delta \ge T$ is covered by bounding $\max _{\ell =1,...,N-1}$ by $\max _{\ell =1,...,N}$. $\square$

Appendix C: Frequently Used Inequalities

For the convenience of the reader, here we collect several inequalities that are repeatedly used in our proofs. In proofs, we will refer to them via the abbreviations given in parentheses.

Hölder’s Inequality (see, for example, Theorem 6.2 in [27]):

Lemma C1

(H). Let $(X,\mathcal {M},\mu )$ be a measure space, $1<p,q<\infty$ with $1/p+1/q=1$, and f, g be measurable functions on X. Then

$$\begin{aligned} \int |fg|d\mu \le \left( \int |f|^pd\mu \right) ^{1/p}\left( \int |g|^qd\mu \right) ^{1/q}. \end{aligned}$$

(146)

When applied to counting measure on $\{1,...,N\}$, with $g_i=1$, Hölder’s Inequality gives the following useful bound on finite sums (one can also obtain it from Jensen’s inequality):

Lemma C2

(HFS). Let $1\le p<\infty$ and $f_i\ge 0$, $i=1,...,N$. Then

$$\begin{aligned} \left( \sum _{i=1}^N f_i\right) ^p\le N^{p-1} \sum _{i=1}^N f_i^p. \end{aligned}$$

(147)

Minkowski’s Inequality for Integrals (see Theorem 6.19 in [27]):

Lemma C3

(MI). Let $(X,\mathcal {M},\mu )$ and $(Y,\mathcal {N},\nu )$ be sigma-finite measure spaces, $1\le p<\infty$, and f be a product-measurable function on $X\times Y$ that satisfies one of the following two conditions:

• 1. $f\ge 0$,

• 2. $f(\cdot ,y)\in L^p(\mu )$ for $\nu$-a.e. y and $y\rightarrow \Vert f(\cdot ,y)\Vert _{L^p(\mu )}$ is in $L^1(\nu )$.

Then

$$\begin{aligned} \left( \left| \int f(x,y)\nu (dy)\right| ^p\mu (dx)\right) ^{1/p}\le \int \left( \int |f(x,y)|^p\mu (dx)\right) ^{1/p} \nu (dy). \end{aligned}$$

(148)

$L^p$-Triangle Inequality (also known as Minkowski’s inequality, see Theorem 6.5 in [27]):

Lemma C4

(T). Let $(X,\mathcal {M},\mu )$ be a measure space, $1\le p<\infty$, and f, g be measurable functions on X. Then

$$\begin{aligned} \left( \int |f+g|^pd\mu \right) ^{1/p}\le \left( \int |f|^pd\mu \right) ^{1/p}+\left( \int |g|^pd\mu \right) ^{1/p}. \end{aligned}$$

(149)

Burkholder–Davis–Gundy Inequality (see Theorem 3.28 in [23]):

Lemma C5

(BDG). For every $p>0$ there exists constants $k_p,K_p\in (0,\infty )$ such that for all $\mathbb {R}$-valued continuous local martingales, M, and all stopping times, T, we have

$$\begin{aligned} E\left[ \sup _{0\le s\le T}|M_s|^{p}\right] \le K_pE[\langle M\rangle _T^{p/2}], \end{aligned}$$

(150)

where $\langle M\rangle$ denotes the quadratic variation of M.

Recall that the quadratic variation of an Itô integral of a $\mathbb {R}^k$-valued continuous, adapted process, $a_t$, with respect to an $\mathbb {R}^k$-valued Wiener process, $W_t$, is (using summation convention) given by

$$\begin{aligned} \left\langle \int _0^{(\cdot )} a_j(s) \mathrm{d}W^j_s\right\rangle _{T} =\int _0^T \Vert a(s)\Vert ^2\mathrm{d}s \end{aligned}$$

(151)

($\Vert \cdot \Vert$ denotes the $\ell ^2$ norm). If M is $\mathbb {R}^n$-valued, then one can still use Lemma C5 to bound $E\left[ \sup _{0\le s\le T}\Vert M_s\Vert ^{p}\right]$ by first using

$$\begin{aligned} \sup _{0\le s\le T}\Vert M_s\Vert ^{p}\le D_{p,n}\sum _{j=1}^n \sup _{0\le s\le T}|M_s^j|^p, \end{aligned}$$

(152)

where $D_{p,n}$ is a constant, depending only on p and n.

Appendix D: SDEs with Semimartingale Forcing

Let $W_t$ be an $\mathbb {R}^k$-valued Wiener process on $(\Omega ,\mathcal {F},P,\mathcal {F}_t)$, a filtered probability space satisfying the usual conditions [23]. In this section, we give some of the background theory of SDEs of the form

$$\begin{aligned} X_t=N_t+\int _{0}^tb(s,X_s)\mathrm{d}s+\int _0^t\sigma (s,X_s)\mathrm{d}W_s, \end{aligned}$$

(153)

i.e., SDEs where the initial condition is generalized to a time-dependent, continuous semimartingale forcing term, $N_t$. Much of the following can be found in [24], with the generalization to SDEs with explosions adapted from [28]. Both of these references discuss the generalization where $W_t$ is replaced by a more general driving semimartingale, but we do not need that extension here.

The main existence and uniqueness result for Eq. (153) mirrors that of the more standard SDE theory:

Theorem D1

Let $U\subset \mathbb {R}^n$ be open and $\sigma :[0,\infty )\times U\rightarrow \mathbb {R}^{n\times k}$, $b:[0,\infty )\times U\rightarrow \mathbb {R}^n$ satisfy the following:

• 1. $b,\sigma$ are measurable.

• 2. For every $T>0$ and compact $C\subset U$ there exists $K_{T,C}>0$ such that for all $t\in [0,T]$, $x,y\in C$ we have

  $$\begin{aligned} \sup _{t\in [0,T],x\in C}\Vert b(t,x)\Vert +\sup _{t\in [0,T],x\in C}\Vert \sigma (t,x)\Vert \le K_{T,C}. \end{aligned}$$

  (154)

• 3. For every $T>0$ and compact $C\subset U$ there exists $L_{T,C}>0$ such that for all $t\in [0,T]$, $x,y\in C$ we have

  $$\begin{aligned} \Vert b(t,x)-b(t,y)\Vert +\Vert \sigma (t,x)-\sigma (t,y)\Vert \le L_{T,C}\Vert x-y\Vert . \end{aligned}$$

  (155)

  i.e. b(t, x) and $\sigma (t,x)$ are locally Lipschitz in x, uniformly in t on compact intervals.

Then for any continuous semimartingale $N_t$ with $N_{0}$ valued in U, the SDE

$$\begin{aligned} X_t=N_{t}+\int _{0}^tb(s,X_s)\mathrm{d}s+\int _{0}^t\sigma (s,X_s)\mathrm{d}W_s \end{aligned}$$

(156)

has a unique (pathwise) maximal solution up to a stopping time, e, called the explosion time. For every $\omega \in \Omega$, e satisfies one of the following:

• 1. $e(\omega )=\infty$,

• 2. There exists a subsequence $t_n\nearrow e(\omega )$ with $\lim _{n\rightarrow \infty } X_{t_n}(\omega )=\infty$,

• 3. There exists a subsequence $t_n\nearrow e(\omega )$ with $\lim _{n\rightarrow \infty } \mathrm{d}(X_{t_n}(\omega ),\partial U)=0$.

As with standard SDEs, non-explosion of solutions follows when the drift and diffusion are linearly bounded:

Corollary D2

Let $\sigma :[0,\infty )\times \mathbb {R}^n\rightarrow \mathbb {R}^{n\times k}$, $b:[0,\infty )\times \mathbb {R}^n\rightarrow \mathbb {R}^n$ be continuous and satisfy the local Lipschitz property Eq. (155). Suppose we also have the following linear growth bound:

For each $T>0$ there exists $L_T>0$ such that

$$\begin{aligned} \sup _{t\in [0,T]}(\Vert b(t,x)\Vert +\Vert \sigma (t,x)\Vert )\le L_T(1+\Vert x\Vert ). \end{aligned}$$

(157)

Then for any continuous semimartingale, $N_t$, the SDE

$$\begin{aligned} X_t=N_{t}+\int _{0}^tb(s,X_s)\mathrm{d}s+\int _{0}^t\sigma (s,X_s)\mathrm{d}W_s \end{aligned}$$

(158)

has a unique maximal solution and it is defined for all $t\ge 0$, i.e., its explosion time is $e=\infty$ a.s.

We will also need a generalization of the theory of Lyapunov functions to the current setting; it is needed to prove non-explosion for the hierarchy of approximating equations when the assumption of bounded forcing is relaxed.

Theorem D3

Let $U\subset \mathbb {R}^n$ be open, $W_t$ be an $\mathbb {R}^k$-valued Wiener process. Suppose $b:[0,\infty )\times U\rightarrow \mathbb {R}^n$ and $\sigma :[0,\infty )\times U\rightarrow \mathbb {R}^{n\times k}$ are continuous and satisfy the local Lipschitz property Eq. (155).

Let $X_{0}$ be an $\mathcal {F}_{0}$-measurable random variable valued in U, $a:[0,\infty )\times \Omega \rightarrow \mathbb {R}^n$ and $c:[0,\infty )\times \Omega \rightarrow \mathbb {R}^{n\times k}$ be pathwise continuous, adapted processes, and let $N_t$ be the continuous semimartingale

$$\begin{aligned} N_t=X_{0}+\int _{0}^t a_s\mathrm{d}s+\int _{0}^tc_s \mathrm{d}W_s. \end{aligned}$$

(159)

Suppose we have a $C^{1,2}$ function $V:[0,\infty )\times U\rightarrow [0,\infty )$ and measurable functions $C,M:[0,\infty )\rightarrow [0,\infty )$ that satisfy:

• 1. M(t) and C(t) are integrable on compact subsets of $[0,\infty )$.

• 2. For any t and any $R>0$ there exists a compact $C\subset U$ and $\delta >0$ such that $V(s,x)\ge R$ for all $(s,x)\in [t-\delta ,t]\times C^c$.

• 3.
  $$\begin{aligned} L[V](t,x)\equiv&\partial _tV(t,x)+b^i(t,x)\partial _{x^i}V(t,x)+\frac{1}{2}\Sigma ^{ij}(t,x)\partial _{x_i}\partial _{x^j}V(t,x)\\ \le&M(t)+C(t)V(t,x), \end{aligned}$$

  where $\Sigma ^{ij}=\sum _\rho \sigma ^i_\rho \sigma ^j_\rho$,

• 4. $\Vert \nabla _x V(t,x)\Vert \le M(t)+C(t)V(t,x)$,

• 5. $\Vert D^2_x V(t,x)\Vert (1+\Vert \sigma (t,x)\Vert )\le M(t)+C(t)V(t,x)$.

Then the unique maximal solution to the SDE

$$\begin{aligned} X_t=N_{t}+\int _{0}^tb(s,X_s)\mathrm{d}s+\int _{0}^t\sigma (s,X_s)\mathrm{d}W_s \end{aligned}$$

(160)

has explosion time $e=\infty$ a.s. We call V a Lyapunov function for the SDE Eq. (160).

Proof

Existence of a solution, $X_t$, up to explosion time, e, follows from Theorem D1. Let $U_n$ be precompact open sets with $\overline{U_n}\subset U_{n+1}\subset U$ and $\cup _n U_n=U$. By looking at the equation on the events $\{X_{0}\in U_{n}\setminus U_{n-1}\}$ it suffices to suppose $X_{0}$ is contained in a compact subset of U (say, $U_1$).

Define $\eta _m=\inf \{t:\Vert a_t\Vert \ge m\}\wedge \inf \{t:c_t\ge m\}$. $a_t$ and $c_t$ are continuous and adapted, so $\eta _m$ are stopping times. Since $\eta _m$ increase to infinity, proving that there is no explosion with $N_t$ replaced by $N^m_t\equiv N_t^{\eta _m}$ for each m will imply that $e=\infty$.

Therefore we can fix m and consider X, the solution to

$$\begin{aligned} X_t=N^{m}_{t}+\int _{0}^tb(s,X_s)\mathrm{d}s+\int _{0}^t\sigma (s,X_s)\mathrm{d}W_s, \end{aligned}$$

(161)

with explosion time e.

Define the stopping times $\tau _n=\inf \{t:X_t\in U_n^c\}\wedge n$ and note that $\tau _n<e$ a.s and $\Vert X^{\tau _n}_t\Vert \le \sup _{x\in \overline{U_n}}\Vert x\Vert$. The continuous semimartingales $X^{\tau _n}$ are solutions to

$$\begin{aligned} X^{\tau _n}_t=\,&X_{0}+\int _{0}^{t\wedge \tau _n} 1_{s\le \eta _m}a_s\mathrm{d}s+\int _{0}^{t\wedge \tau _n}1_{s\le \eta _m}c_s \mathrm{d}W_s\nonumber \\&+\int _{0}^{t\wedge \tau _n} b(s,X^{\tau _n}_s)\mathrm{d}s+\int _{0}^{t\wedge \tau _n} \sigma (s,X^{\tau _n}_s)\mathrm{d}W_s, \end{aligned}$$

(162)

hence Itô’s Lemma implies

$$\begin{aligned}&V(t\wedge \tau _n,X_t^{\tau _n})-V(0,X_{0})\nonumber \\&\quad =\int _{0}^{t\wedge \tau _n}\partial _sV(s,X_s^{\tau _n})\mathrm{d}s+\int _{0}^{t\wedge \tau _n}\partial _{x^i}V(s,X^{\tau _n}_s)\mathrm{d}(X^{\tau _n})^i_s\nonumber \\&\qquad +\frac{1}{2}\int _{0}^{t\wedge \tau _n}\partial _{x^i}\partial _{x^j}V(s,X^{\tau _n}_s)d[(X^{\tau _n})^i,(X^{\tau _n})^j]_s\nonumber \\&\quad \le \int _{0}^{t\wedge \tau _n} M(s)\mathrm{d}s+\int _{0}^{t\wedge \tau _n}C(s)V(s,X_s^{\tau _n})\mathrm{d}s\nonumber \\&\qquad +\int _{0}^{t\wedge \tau _n} \partial _{x^i}V(s,X^{\tau _n}_s)(1_{s\le \eta _m}(c_s)^i_j+\sigma ^i_j(s,X^{\tau _n}_s))dB^j_s\nonumber \\&\qquad +\int _{0}^{t\wedge \tau _n} 1_{s\le \eta _m}a_s^i\partial _{x^i}V(s,X^{\tau _n}_s)\mathrm{d}s\nonumber \\&\qquad +\frac{1}{2}\int _{0}^{t\wedge \tau _n}1_{s\le \eta _m}\partial _{x^i}\partial _{x^j}V(s,X^{\tau _n}_s)((\sigma c)^{ij}(s,X_s^{\tau _n})+(c\sigma )^{ij}(s,X_s^{\tau _n})+C^{ij}_s)\mathrm{d}s. \end{aligned}$$

(163)

Note that if $\eta _m>0$ then $1_{s\le \eta _m}\Vert a_s\Vert \le m$, $1_{s\le \eta _m}\Vert c_s\Vert \le m$ and if $\eta _m=0$ then the integrals involving $1_{s\le \eta _m}$ are zero. Therefore

$$\begin{aligned}&V(t\wedge \tau _n,X_t^{\tau _n})-V(0,X_{0})\nonumber \\&\quad \le \int _{0}^{t\wedge \tau _n} M(s)\mathrm{d}s+\int _{0}^{t\wedge \tau _n}C(s)V(s,X_s^{\tau _n})\mathrm{d}s\nonumber \\&\qquad +\int _{0}^{t\wedge \tau _n} \partial _{x^i}V(s,X^{\tau _n}_s)(1_{s\le \eta _m}(c_s)^i_j+\sigma ^i_j(s,X^{\tau _n}_s))dB^j_s\nonumber \\&\qquad +\frac{1}{2}\int _{0}^{t\wedge \tau _n}\Vert D^2V(s,X^{\tau _n}_s)\Vert (2m\Vert \sigma (s,X_s^{\tau _n})\Vert +m^2)\mathrm{d}s\nonumber \\&\qquad +\int _{0}^{t\wedge \tau _n} m \Vert \nabla V(s,X^{\tau _n}_s)\Vert \mathrm{d}s\nonumber \\&\quad \le \int _{0}^{t\wedge \tau _n} M(s)\mathrm{d}s+\int _{0}^{t\wedge \tau _n}C(s)V(s,X_s^{\tau _n})\mathrm{d}s\nonumber \\&\qquad +\int _{0}^{t\wedge \tau _n} \partial _{x^i}V(s,X^{\tau _n}_s)(1_{s\le \eta _m}(c_s)^i_j+\sigma ^i_j(s,X^{\tau _n}_s))\mathrm{d}W^j_s, \end{aligned}$$

(164)

where we have absorbed constants into M(s) and C(s).

$X^{\tau _n}_s$ is valued in $U_n$, a precompact subset of U. Therefore continuity of V and $\partial _{x^i}V$ imply all of these terms have finite expectations. Also

$$\begin{aligned} E\left[ \int _{0}^t |1_{s\le t\wedge \tau _n}\partial _{x^i}V(s,X^{\tau _n}_s)(1_{s\le \eta _m} (c_s)^i_j+\sigma ^i_j(s,X^{\tau _n}_s)|^2\mathrm{d}s\right] <\infty \end{aligned}$$

(165)

for all t, implying the stochastic integral is a martingale. Therefore

$$\begin{aligned}&E[V(t\wedge \tau _n,X_t^{\tau _n})]\nonumber \\&\quad \le E[V(0,X_{0})]+\int _{0}^tM(s)\mathrm{d}s+\int _{0}^t C(s)E[V(s\wedge \tau _n,X^{\tau _n}_s)]\mathrm{d}s. \end{aligned}$$

(166)

The integrands are in $L^1$, hence Gronwall’s inequality implies

$$\begin{aligned} E[V(t\wedge \tau _n,X_t^{\tau _n})]\le \left( E[V(0,X_{0})]+\int _{0}^tM(s)\mathrm{d}s\right) \exp \left( \int _{0}^t C(s)\mathrm{d}s\right) \end{aligned}$$

(167)

for all $t\ge 0$.

Taking $n\ge t$ and using Fatou’s lemma gives

$$\begin{aligned} \left( E[V(0,X_{0})]+\int _{0}^tM(s)\mathrm{d}s\right) \exp \left( \int _{0}^t C(s)\mathrm{d}s\right) \ge E[\liminf _{n\rightarrow \infty }V(\tau _n,X_{\tau _n}) 1_{e<t}]. \end{aligned}$$

(168)

Now take $\omega \in \Omega$ with $e(\omega )<t$. Given $R>0$ we have a compact $C\subset U$ and a $\delta >0$ such that $V\ge R$ on $[e(\omega )-\delta ,e(\omega )]\times C^c$. Noting that $\tau _n(\omega )\nearrow e(\omega )$ we can take N large enough that for $n\ge N$ we have $\tau _n(\omega )\in [e(\omega )-\delta ,e(\omega )]$ and $C\subset U_n$. Therefore $V(\tau _n(\omega ),X_{\tau _n}(\omega ))\ge R$ for $n\ge N$. So $\liminf _{n\rightarrow \infty }V(\tau _n(\omega ),X_{\tau _n}(\omega ))\ge R$ i.e. $\liminf _{n\rightarrow \infty }V(\tau _n,X_{\tau _n}) 1_{e<t}=\infty 1_{e<t}$. But we have a finite upper bound Eq. (168) so we must have $P(e<t)=0$. $t\ge 0$ was arbitrary and so $e=\infty$ a.s. $\square$

Appendix E: Proof of Theorem 4.1

In this section we provide a proof of Theorem 4.1, which extends Theorem 3.8 to unbounded forces, at the cost of weakening the convergence mode to convergence in probability. Recall that here, we are working under Assumption 4.1. First, we require several lemmas:

Assumption 4.1 is sufficient to ensure non-explosion of solutions to the Langevin equation. This can be shown by constructing Lyapunov functions:

Lemma E1

Given Assumption 4.1, there exist unique global in time solutions $(q_t^m,u_t^m)$ to Eqs. (31)–(32) and $q_t$ to Eq. (10).

Proof

Despite the slightly different assumptions made here, the proof in Appendix C of [9] goes through essentially unchanged. We omit the details. $\square$

For y a continuous semimartingale, we define $z^m_t[y]$ and $R^{m}_t[y]$ as in Definition 3.7. The following two properties will be needed:

Lemma E2

If $\eta$ is a stopping time and $y,\tilde{y}$ are continuous semimartingales that satisfy $y^\eta _t=\tilde{y}^\eta _t$ then

$$\begin{aligned} R_{t\wedge \eta }^m[y]=R_{t\wedge \eta }^{m}[\tilde{y}] \end{aligned}$$

(169)

for all $t\ge 0$, P-a.s.

Proof

The proof is a straightforward use of the formulas in Definition 3.7. $\square$

Lemma E3

Define $\tilde{Y}$ to be the set of continuous semimartingales of the form

$$\begin{aligned} y_t=y_0+\int _0^t a_s \mathrm{d}s+\int _0^t c_s\mathrm{d}W_s \end{aligned}$$

(170)

where $y_0$ is $\mathcal {F}_0$-measurable and $a:[0,\infty )\times \Omega \rightarrow \mathbb {R}^n$, $c:[0,\infty )\times \Omega \rightarrow \mathbb {R}^{n\times k}$ are pathwise continuous, adapted processes.

If $y\in \tilde{Y}$ then $z^m_t[y]\in \tilde{Y}$ and $R^{m}_t[y]\in \tilde{Y}$.

Proof

The set of semimartingales of the form Eq. (170) is a vector space and, using integration by parts, one can see that is closed under multiplication by $\mathbb {R}$-valued processes of the form Eq. (170) (i.e., with $n=1$), and contains $z_t^m[y]$ for any continuous semimartingale y.

The result then follows for $R^m_t[y]$ by noting that Assumption 4.1 implies all of the integrands are pathwise continuous, adapted, and that $\tilde{\gamma }^{-1}(t,q)$, $Q^{ikl}(t,q)$, and $G^{a,b}_{kl}(t,q)$ are $C^2$. The latter allows Itô’s Lemma to be applied to $\tilde{\gamma }^{-1}(t,y_t)$ etc., yielding terms in $\tilde{Y}$, provided that $y\in \tilde{Y}$. $\square$

We also need to know that solutions to the SDE defining the hierarchy exist under the current weakened assumptions:

Lemma E4

Under Assumption 4.1, for any $y\in \tilde{Y}$ (defined in Lemma E3) there is a unique continuous semimartingale, $x_t$, defined for all $t\ge 0$ that solves

$$\begin{aligned} x_t=\,&q_0+\int _0^t\tilde{\gamma }^{-1}(s,x_s)F(s,x_s)\mathrm{d}s+\int _0^tS(s,x_s)\mathrm{d}s\nonumber \\&+\int _0^t\tilde{\gamma }^{-1}(s,x_s)\sigma (s,x_s) \mathrm{d}W_s +\sqrt{m}R^{m}_t[y]. \end{aligned}$$

(171)

We also have $x\in \tilde{Y}$.

Proof

$\tilde{\gamma }^{-1}F+S$ and $\tilde{\gamma }^{-1}\sigma$ are continuous and satisfy the local Lipschitz property, Eq. (155). Lemma E3 implies $R^m[y]$ is a continuous semimartingale (in fact, $R^m[y]\in \tilde{Y}$). Therefore Theorem D1 shows a unique maximal solution exists up to explosion time.

One can check that the function

$$\begin{aligned} (t,q)\rightarrow \Vert q\Vert ^2+\tilde{V}(t,q), \end{aligned}$$

(172)

where $\tilde{V}$ was defined in Eq. (122), satisfies all the conditions required by Theorem Eq. (D3) to make it a Lyapunov function for the SDE Eq. (171), thereby proving $x_t$ has explosion time $e=\infty$. $R^m[y]\in \tilde{Y}$ together with Eq. (171) shows that $x\in \tilde{Y}$ as well. $\square$

We are now ready to prove Theorem 4.1:

Proof

By Lemma E1, there exist unique global in time solutions $(q_t^m,u_t^m)$ to Eqs. (31)–(32) and $q_t\in \tilde{Y}$ ($\tilde{Y}$ was defined in Lemma E3) to Eq. (10), and by induction, Lemma E4 gives globally defined continuous semimartingale solutions to the approximation hierarchy, Eq. (171).

Let $\chi :\mathbb {R}^n\rightarrow [0,1]$ be a $C^\infty$ bump function, equal to 1 on $\overline{B_1(0)}\equiv \{\Vert q\Vert \le 1\}$ and zero outside $\overline{B_2(0)}$. Given $r>0$ let $\chi _r(q)=\chi (q/r)$. Define

$$\begin{aligned} V_r(t,q)&=\chi _r(q)V(t,q), \quad \tilde{F}_r(t,q)=\chi _r(q) \tilde{F}(t,q), \quad \psi _r(t,q)=\chi _r(q)\psi (t,q),\nonumber \\ \gamma _r(t,q)&=\chi _r(q)\gamma (t,q)+(1-\chi _r(q))\lambda I. \end{aligned}$$

(173)

For each $r>0$, replacing V with $V_r$, F with $F_r$ etc., we arrive at an SDE satisfying the hypotheses of Theorem 3.8. We will call these the cutoff systems.

Let $R^{r,m}_t[y]$ denote Eq. (94), with V replaced by $V_r$, etc. All of these objects and their derivatives agree on $\overline{B_r(0)}$, so for any continuous semimartingale, y, if we let $\eta _r^y=\inf \{t:\Vert y_t\Vert \ge r\}$, we have

$$\begin{aligned} R_{t\wedge \eta _r^y}^m[y]=R_{t\wedge \eta _r^y}^{r,m}[y] \end{aligned}$$

(174)

for all $t\ge 0$, P-a.s.

Let $(q_t^{r,m},u_t^{r,m})$ be the solutions to the cutoff system, $q_t^r$ the solution to the corresponding homogenized equation, and $q_t^{r,\ell ,m}$ the solutions to the corresponding approximating hierarchy, all using the same initial conditions as the system without the cutoff.

For each $r>R$ define the stopping times

$$\begin{aligned}&\eta ^m_r=\inf \{t:\Vert q_t^m\Vert \ge r\},\,\, \eta ^{\ell ,m}_r=\inf \{t:\Vert q^{\ell ,m}_t\Vert \ge r\},\nonumber \\&\eta ^{r,\ell ,m}_r=\inf \{t:\Vert q^{r,\ell ,m}_t\Vert \ge r\}, \end{aligned}$$

(175)

and

$$\begin{aligned} \tau ^{\ell ,m}_r=\eta ^{\ell ,m}_r\wedge \eta ^{\ell -1,m}_r\wedge ...\wedge \eta ^{1,m}_r,\,\, \tau ^{r,\ell ,m}_r=\eta ^{r,\ell ,m}_r\wedge \eta ^{r,\ell -1,m}_r\wedge ...\wedge \eta ^{r,1,m}_r. \end{aligned}$$

(176)

Note that $\eta ^{1,m}_r=\inf \{t:\Vert q_t\Vert \ge r\}\equiv \eta _r$ is independent of m. Finally, define $\sigma ^{\ell ,m}_r=\tau ^{\ell ,m}_r\wedge \eta ^m_r$, the first exit time for any of the position processes up to level $\ell$ of the hierarchy.

The drifts and diffusions of the modified and unmodified SDEs agree on the ball $\{\Vert q\Vert \le r\}$. Therefore, using induction on $\ell$, Lemma E2, Eq. (174), and pathwise uniqueness of solutions, we see that the driving semimartingales of the hierarchy up to $\ell$ for both the original and cutoff systems agree up to the stopping time $\tau ^{\ell ,m}_r$ and

$$\begin{aligned} q^m_{t\wedge \eta ^m_r}&=q^{r,m}_{t\wedge \eta ^m_r} \quad \text { for all } t\ge 0\text { a.s., } \end{aligned}$$

(177)

$$\begin{aligned} \tau ^{\ell ,m}_r&=\tau ^{r,\ell ,m}_r a.s., \,\,\text { and }\,\, q^{\ell ,m}_{t\wedge \tau ^{\ell ,m}_r}= q^{r,\ell ,m}_{t\wedge \tau ^{\ell ,m}_ r} \text { for all } t\ge 0 \text { a.s.} \end{aligned}$$

(178)

Fixing $r>0$ and using Eqs. (177) and (178), for any $T>0$, $\delta >0$, $\epsilon >0$, $\ell \in \mathbb {Z}^+$ we can calculate

$$\begin{aligned}&P\left( \frac{\sup _{t\in [0,T]}\Vert q_t^m-q^{\ell ,m}_t\Vert }{m^{\ell /2-\epsilon }}>\delta \right) \nonumber \\&\quad =P\left( \sigma ^{\ell ,m}_r> T,\frac{\sup _{t\in [0,T]}\Vert q_{t\wedge \eta ^m_r}^{r,m}-q^{r,\ell ,m}_{t\wedge \tau ^{\ell ,m}_r}\Vert }{m^{\ell /2-\epsilon }}>\delta \right) \nonumber \\&\qquad +P\left( \sigma ^{\ell ,m}_r\le T,\frac{\sup _{t\in [0,T]}\Vert q_t^m-q^{\ell ,m}_t\Vert }{m^{\ell /2-\epsilon }}>\delta \right) \nonumber \\&\quad \le P\left( \frac{\sup _{t\in [0,T]}\Vert q_{t}^{r,m}-q^{r,\ell ,m}_{t}\Vert }{m^{\ell /2-\epsilon }}>\delta \right) +P\left( \sigma ^{\ell ,m}_r\le T\right) . \end{aligned}$$

(179)

The first term, involving the cutoff system, converges to zero as $m\rightarrow 0$ by Markov’s inequality and the convergence result for bounded forces, Eq. (98). Hence we focus on the second term. We note that the only essential difference between the argument below and the similar computation in the proof of Theorem 6.1 from [9] is the need to consider all processes in the hierarchy up to level $\ell$, and not just the processes, $q_t^m$ and $q_t^{\ell ,m}$, that were being compared in Eq. (179). This is due to the iterative construction of each level in the hierarchy from the solution at the previous level. The second term can be bounded as follows:

$$\begin{aligned}&P\left( \sigma ^{\ell ,m}_r\le T\right) \nonumber \\&\quad \le P(\eta _r\le T)+P\left( \eta ^{m}_{r}\le T,\eta _r>T, \sup _{t\in [0,T]}\Vert q_t^r-q_t^{r,m}\Vert \le 1\right) \nonumber \\&\qquad +\sum _{k=2}^\ell P\left( \tau ^{k-1,m}_r>T,\eta ^{k,m}_r\le T, \sup _{t\in [0,T]}\Vert q_t^r-q_t^{r,k,m}\Vert \le 1\right) \nonumber \\&\qquad +P\left( \sup _{t\in [0,T]}\Vert q_t^r-q_t^{r,m}\Vert>1\right) +\sum _{k=2}^\ell P\left( \sup _{t\in [0,T]}\Vert q_t^r-q_t^{r,k,m}\Vert > 1\right) \nonumber \\&\quad \le P\left( \sup _{t\in [0,T]}\Vert q_t\Vert \ge r\right) +P\left( \eta ^{m}_{r}\le T, \Vert q_{T\wedge \eta _r^m}-q_{T\wedge \eta _r^m}^{m}\Vert \le 1\right) \nonumber \\&\qquad +\sum _{k=2}^\ell P\left( \eta ^{k,m}_r\le T, \Vert q_{T\wedge \eta ^{k,m}_r}-q_{T\wedge \eta ^{k,m}_r}^{k,m}\Vert \le 1\right) \nonumber \\&\qquad +E\left[ \sup _{t\in [0,T]}\Vert q_t^r-q_t^{r,m}\Vert \right] +\sum _{k=2}^\ell E\left[ \sup _{t\in [0,T]}\Vert q_t^r-q_t^{r,k,m}\Vert \right] , \end{aligned}$$

(180)

where we again used the uniqueness results, Eqs. (177)–(178). The terms in the last line go to zero as $m\rightarrow 0$, as seen from the triangle inequality and Eq. (98).

On the event where $\eta _r^m\le T$ and $\Vert q_{T\wedge \eta _r^m}-q^m_{T\wedge \eta _r^m}\Vert \le 1$ we have $\Vert q^m_{\eta ^m_r}\Vert \ge r$ and

$$\begin{aligned} \Vert q_{\eta _r^m}\Vert \ge \Vert q^m_{T\wedge \eta ^m_r}\Vert -\Vert q_{T\wedge \eta _r^m}-q^m_{T\wedge \eta _r^m}\Vert \ge r-1. \end{aligned}$$

(181)

Hence $\sup _{t\in [0,T]}\Vert q_t\Vert \ge r-1$ on this event. Similarly,

$$\begin{aligned} \left\{ \eta ^{k,m}_r\le T, \Vert q_{T\wedge \eta ^{k,m}_r}-q_{T\wedge \eta ^{k,m}_r}^{k,m}\Vert \le 1\right\} \subset \left\{ \sup _{t\in [0,T]}\Vert q_t\Vert \ge r-1\right\} . \end{aligned}$$

(182)

Therefore we obtain

$$\begin{aligned}&\limsup _{m\rightarrow 0}P\left( \frac{\sup _{t\in [0,T]}\Vert q_t^m-q^{\ell ,m}_t\Vert }{m^{\ell /2-\epsilon }}>\delta \right) \nonumber \\&\quad \le P\left( \sup _{t\in [0,T]}\Vert q_t\Vert \ge r\right) +\ell P\left( \sup _{t\in [0,T]}\Vert q_t\Vert \ge r-1\right) \nonumber \\&\quad \le (\ell +1)P\left( \sup _{t\in [0,T]}\Vert q_t\Vert \ge r-1\right) . \end{aligned}$$

(183)

This holds for all $r>0$ and non-explosion of $q_t$ implies that

$$\begin{aligned} P\left( \sup _{t\in [0,T]}\Vert q_t\Vert \ge r-1\right) \rightarrow 0 \end{aligned}$$

(184)

as $r\rightarrow \infty$, hence we have proven the claimed result. $\square$