Coupled Quantum Harmonic Oscillators and Feynman–Kac path integrals for Linear Diffusive Particles

Abstract

We propose a new solvable class of multidimensional quantum harmonic oscillators for a linear diffusive particle and a quadratic energy absorbing well associated with a semi-definite positive matrix force. Under natural and easily checked controllability conditions, for non necessarily reversible models with possibly transient free particle diffusions, the ground state and the zero-point energy are explicitly computed in terms of the unique positive fixed point of a continuous time algebraic Riccati matrix equation. We also present an explicit solution of normalised and time dependent Feynman–Kac measures on path spaces in terms of a time varying linear dynamical system coupled with a differential Riccati matrix equation. A refined non asymptotic analysis of the stability of these models is developed based on a recently developed Floquet-type representation of time varying exponential semigroups of Riccati matrices. We provide explicit and non asymptotic estimates of the exponential decays to equilibrium of Feynman–Kac semigroups in terms of Wasserstein distances or Boltzmann-relative entropy. For reversible models we develop a series of functional inequalities including de Bruijn identity, Fisher’s information decays, log-Sobolev inequalities, and entropy contraction estimates. In this context, we also provide a complete and explicit description of all the spectrum and the excited states of the Hamiltonian, yielding what seems to be the first result of this type for this class of models. We illustrate these formulae with the traditional harmonic oscillator associated with real time Brownian particles and Mehler’s formula. The analysis developed in this article can also be extended to solve time dependent Schrodinger equations equipped with time varying linear diffusions and quadratic potential functions.

Appendices

A Riccati Matrix Differential Equations

In this section, we discuss some of the theory behind Riccati matrix differential equations and present some results that are of use throughout the paper. This section is mainly taken from [12].

1.1 A.1 Gians fixed point formulae

Recall the the positive and negative fixed points $(P^-_{\infty },P_{\infty })$ of the Riccati equation (28). The difference between the two

$$\begin{aligned} P_{\infty }-P^{-}_{\infty }=\Delta _{\infty }^{-1} \end{aligned}$$

(97)

is defined via the Gramian matrices

$$\begin{aligned} \Delta _t:= & {} \int _0^{t}\, e^{s(A-P_{\infty }S)^{\prime }} S~e^{s(A-P_{\infty }S)}\,ds\longrightarrow _{t\rightarrow \infty }\Delta _{\infty }\nonumber \\:= & {} \int _0^{\infty }\, e^{s(A-P_{\infty }S)^{\prime }} S~e^{s(A-P_{\infty }S)}\,ds \in {{\mathcal {S}} }^+_r. \end{aligned}$$

(98)

Consider now the linear matrix functional

$$\begin{aligned} \mathbb {F}_t~:~P\in {{\mathcal {S}} }_r^0\mapsto \mathbb {F}_t(P):=\left[ (\Delta _t^{-1}-\Delta ^{-1}_{\infty })+(P-P_{\infty }^-)\right] ~\Delta _t~\in {{\mathcal {G}} }l_r. \end{aligned}$$

(99)

Rearranging and using (98) implies that

$$\begin{aligned} \mathbb {F}_t(P)=I+(P-P_{\infty })\,\Delta _t \quad \text { and } \quad \mathbb {F}_t(P_{\infty })=I. \end{aligned}$$

(100)

Recall that $Q_{\infty }$ is defined as $P_{\infty }$ by replacing (A, R, S) by $(A^{\prime },S,R)$. In the same vein, $\Delta _t^h$ is defined as $\Delta _t$ by replacing (A, R, S) and $P_{\infty }$ by $(A^{\prime },S,R)$ and $Q_{\infty }$. Thus, by symmetry arguments and (30), we also have

$$\begin{aligned} (\Delta _{\infty }^h)^{-1}=P^{-1}_{\infty }-(P^-_{\infty })^{-1}=Q_{\infty }-Q_{\infty }^-\quad \hbox { and}\quad \left( \phi ^h_{t}(0),P^h_{\infty }\right) =\left( \Delta _t^h,\Delta _{\infty }^h\right) ,\nonumber \\ \end{aligned}$$

(101)

with the Gramian matrices

$$\begin{aligned} \Delta _t^h:= & {} \int _0^t e^{(A-RQ_{\infty })s}~R~e^{(A-RQ_{\infty })^{\prime }s}~ds\longrightarrow _{t\rightarrow \infty }\Delta _{\infty }^h\\:= & {} \int _0^{\infty }\, e^{(A-RQ_{\infty })s}~R~e^{(A-RQ_{\infty })^{\prime }s}~ds\in {{\mathcal {S}} }^+_r. \end{aligned}$$

The following lemma proves the second equality on the right hand side of (39).

Lemma A.1

For any (A, R, S) satisfying the rank condition (4), we have

$$\begin{aligned} \hbox { Tr}(SP_{\infty })=2~\hbox { Tr}(A)-\hbox { Tr}(SP^{-}_{\infty })=\hbox { Tr}(Q_{\infty }R). \end{aligned}$$

(102)

Proof

The Gramian $\Delta _{\infty }$ satisfies the Sylvester equations given by

$$\begin{aligned}{} & {} (A-P_{\infty }S)^{\prime }\Delta _{\infty }+\Delta _{\infty }(A-P_{\infty }S)+S=0=\Delta _{\infty }^{-1}(A-P_{\infty }S)^{\prime }+(A-P_{\infty }S)\Delta _{\infty }^{-1}\\{} & {} \quad +\Delta _{\infty }^{-1} S\,\Delta _{\infty }^{-1}. \end{aligned}$$

It then follows that

$$\begin{aligned} \begin{array}{l} \displaystyle \hbox { Tr}\left( \Delta _{\infty }\left( \Delta _{\infty }^{-1}(A-P_{\infty }S)^{\prime }+(A-P_{\infty }S)\Delta _{\infty }^{-1}+\Delta _{\infty }^{-1}S\,\Delta _{\infty }^{-1} \right) \right) =0\\ \\ \Longrightarrow 2\hbox { Tr}(SP_{\infty })=2\hbox { Tr}(A)+\hbox { Tr}(S\,\Delta _{\infty }^{-1})=2\hbox { Tr}(A)+\hbox { Tr}(SP_{\infty })-\hbox { Tr}(SP^{-}_{\infty }), \end{array} \end{aligned}$$

where we have used (97) to obtain the final equality. From this we obtain

$$\begin{aligned} \hbox { Tr}(SP^{-}_{\infty })=2\hbox { Tr}(A)-\hbox { Tr}(SP_{\infty }). \end{aligned}$$

(103)

In the same vein, we have

$$\begin{aligned} \hbox { Tr}\left( P^{-1}_{\infty }\hbox { Ricc}(P_{\infty })\right) =0&\Longrightarrow&\hbox { Tr}(SP_{\infty })= 2\hbox { Tr}(A)+\hbox { Tr}(P^{-1}_{\infty }R)\\ \hbox { Tr}\left( (P^-_{\infty })^{-1}\hbox { Ricc}(P^-_{\infty })\right) =0&\Longrightarrow&\hbox { Tr}(SP^-_{\infty })= 2\hbox { Tr}(A)+\hbox { Tr}((P^-_{\infty })^{-1}R). \end{aligned}$$

Combining the last assertion with (103) we conclude that

$$\begin{aligned} \hbox { Tr}(P_{\infty }S)=- \hbox { Tr}((P^-_{\infty })^{-1}R)=\hbox { Tr}(Q_{\infty }R), \end{aligned}$$

as required. $\square$

1.2 A.2 A Floquet-type representation

For any $P\in {{\mathcal {S}} }^0_r$ and $\delta >0$ set

$$\begin{aligned} \chi (P):= \Vert P_{\infty }^-\Vert ^{-1} \left[ \Vert P_{\infty }-P_{\infty }^-\Vert +\Vert P-P_{\infty }\Vert \right] \quad \hbox {and}\quad \chi _{\delta }:=\left[ {\lambda _{ min}\left( \Delta _{\delta }\right) \lambda _{ min}\left( -P_{\infty }^-\right) }\right] ^{-1}. \end{aligned}$$

The next theorem provides an explicit description of ${{\mathcal {E}} }_t(P)$, defined in (33), in terms of the matrices $(A,S,P_{\infty })$. In what follows, $\Vert \cdot \Vert$ stands for the spectral norm of matrices.

Theorem A.2

[Floquet-type representation [12]]. For any time horizon $t\ge 0$ and any $P\in {{\mathcal {S}} }_r^0$ we have Riccati exponential semigroup formula

$$\begin{aligned} {{\mathcal {E}} }_{t}(P)= e^{t (A-P_{\infty }S) }~\mathbb {F}_t(P)^{-1} = {{\mathcal {E}} }_t(P_\infty )\mathbb {F}_t(P)^{-1}, \end{aligned}$$

(104)

where $\mathbb {F}_t(P)$ was defined in (99). For any $t\ge \delta >0$ we have the uniform estimates

$$\begin{aligned} {\Vert \mathbb {F}_t(P)^{-1}\Vert } \,\le \chi _{\delta }&\hbox {and}&\Vert {{\mathcal {E}} }_t(P)\Vert \,\le ~\chi _{\delta }\,\Vert {{\mathcal {E}} }_t(P_{\infty })\Vert . \end{aligned}$$

(105)

In addition, for any $t\ge 0$ we have the exponential estimates

$$\begin{aligned} \forall t\ge \delta >0, \quad 0<\Pi _{-,\delta }\le \phi _t(P)\le \Pi _{+,\delta } \quad \hbox { and}\quad \forall t\ge 0,\quad \Vert {{\mathcal {E}} }_{t}(P_{\infty }) \Vert \,\le \, \alpha \,e^{-\beta t}, \end{aligned}$$

(106)

for some positive matrices $\Pi _{-,\delta },\Pi _{+,\delta }$ and some $\alpha ,\beta >0$, all of which depend on the model parameters (A, R, S). Finally, we have the bounds

$$\begin{aligned} {\Vert \mathbb {F}_t(P)^{-1}\Vert }\,\le \chi (P)&\hbox {and}&~~ \quad \Vert {{\mathcal {E}} }_t(P)\Vert \,\le ~\chi (P)\,\Vert {{\mathcal {E}} }_t(P_{\infty })\Vert . \end{aligned}$$

(107)

From this theorem, we can develop some useful identities. Indeed, using the decomposition

$$\begin{aligned} \hbox { Ricc}(Q_1)-\hbox { Ricc}(Q_2) \,=\,(A-Q_1S)(Q_1-Q_2)+(Q_1-Q_2)(A-Q_2S)^{\prime }, \end{aligned}$$

(108)

for $Q_1, Q_2 \in {{\mathcal {S}} }_r^0$, applying (104) we have the closed form Lipschitz type matrix formula

$$\begin{aligned} \phi _t(Q_1)-\phi _t(Q_2)= {{\mathcal {E}} }_t(P_{\infty })~\mathbb {F}_t(Q_1)^{-1}(Q_1-Q_2)~\left( {{\mathcal {E}} }_t(P_{\infty })~\mathbb {F}_t(Q_2)^{-1}\right) ^{\prime }. \end{aligned}$$

(109)

Applying (109) with $Q_2=P_{\infty }$ and using (100), we recover the Bernstein-Prach-Tekinalp formula [71, 72] given by

$$\begin{aligned} \phi _t(P)=P_{\infty }+ {{\mathcal {E}} }_t(P_{\infty })~\mathbb {F}_t(P)^{-1}(P-P_{\infty })~ {{\mathcal {E}} }_t(P_{\infty })^{\prime }. \end{aligned}$$

(110)

1.3 A.3 Lipschitz inequalities

Combining Theorem A.2, (106) and (109) we easily obtain the following result.

Theorem A.3

For any time horizon $t\ge \delta >0$ and any $Q_1,Q_2\in {{\mathcal {S}} }_r^0$ we have the Lipschitz estimate

$$\begin{aligned} \Vert \phi _t(Q_1)-\phi _t(Q_2)\Vert \le (\alpha \chi _{\delta })^2\,e^{-2\beta t}~\Vert Q_1-Q_2\Vert \end{aligned}$$

with the parameters $(\alpha ,\beta , \chi _{\delta })$ defined in (106) and Theorem A.2. In addition, for any $t\ge 0$ we have the local Lipschitz estimate

$$\begin{aligned} \Vert \phi _t(Q_1)-\phi _t(Q_2)\Vert \le \alpha ^2\chi (P_1)\chi (P_2)~e^{-2\beta t}~\Vert Q_1-Q_2\Vert , \end{aligned}$$

with the parameters $\chi (P_i)$ defined in Theorem A.2.

Noting that

$$\begin{aligned} {{\mathcal {E}} }_t(Q_1)- {{\mathcal {E}} }_t(Q_2)= & {} {{\mathcal {E}} }_t(P_{\infty })~\mathbb {F}_t(Q_1)^{-1}~\left[ \mathbb {F}_t(Q_2)-\mathbb {F}_t(Q_1)\right] ~\mathbb {F}_t(Q_2)^{-1}\\= & {} {{\mathcal {E}} }_t(P_{\infty })~\mathbb {F}_t(Q_1)^{-1}~(Q_2-Q_1)~\Delta _t~\mathbb {F}_t(Q_2)^{-1}, \end{aligned}$$

where $\Delta _t$ was defined in (98), we also obtain the following corollary.

Corollary A.4

For any time horizon $t\ge \delta >0$ and any $Q_1,Q_2\in {{\mathcal {S}} }_r^0$ we have the Lipschitz estimate

$$\begin{aligned} \Vert {{\mathcal {E}} }_t(Q_1)- {{\mathcal {E}} }_t(Q_2)\Vert \le \alpha \,\chi _{\delta }^2\,\Vert \Delta _{\infty }\Vert \,e^{-\beta t}~\Vert Q_1-Q_2\Vert \end{aligned}$$

with the parameters $(\alpha ,\beta , \chi _{\delta })$ defined in (106) and Theorem A.2. In addition, for any $t\ge 0$ we have local Lipschitz estimate

$$\begin{aligned} \Vert {{\mathcal {E}} }_t(Q_1)- {{\mathcal {E}} }_t(Q_2)\Vert \le \alpha \Vert \Delta _{\infty }\Vert \,\chi (Q_1)\chi (Q_2)\,e^{-\beta t}~\Vert Q_1-Q_2\Vert \end{aligned}$$

with the parameter $\chi (Q_i)$ defined in Theorem A.2.

The first coordinate of the evolution semigroup (32) can be written as

$$\begin{aligned} \widehat{X}_t(x,P_0)= {{\mathcal {E}} }_t(P_0) x \end{aligned}$$

Using the decomposition

$$\begin{aligned} \widehat{X}_t(x_1,Q_1)-\widehat{X}_t(x_2,Q_2)=( {{\mathcal {E}} }_t(Q_1)- {{\mathcal {E}} }_t(Q_2)) x_1+ {{\mathcal {E}} }_t(Q_2) (x_1-x_2), \end{aligned}$$

we readily check the following theorem.

Theorem A.5

For any time horizon $t\ge \delta >0$ and any $Q_1,Q_2\in {{\mathcal {S}} }_r^0$ we have the estimate

$$\begin{aligned} \Vert \widehat{X}_t(x_1,Q_1)-\widehat{X}_t(x_2,Q_2)\Vert \le \alpha \chi _{\delta }\,e^{-\beta t}~ \left( \chi _{\delta }\Vert \Delta _{\infty }\Vert \,\,\Vert x_1\Vert ~\Vert Q_1-Q_2\Vert +\Vert x_1-x_2\Vert \right) \end{aligned}$$

with the parameters $(\alpha ,\beta , \chi _{\delta })$ defined in (106) and Theorem A.2. In addition, for any $t\ge 0$ we have the estimate

$$\begin{aligned} \Vert \widehat{X}_t(x_1,Q_1)-\widehat{X}_t(x_2,Q_2)\Vert \le \ \alpha \,\chi (Q_2)\,e^{-\beta t}~ \left( \chi (Q_1)\Vert \Delta _{\infty }\Vert \,\,\Vert x_1\Vert ~\Vert Q_1-Q_2\Vert +\Vert x_1-x_2\Vert \right) \end{aligned}$$

with the parameter $\chi (Q_i)$ defined in Theorem A.2.

B Path Integral Formulations

In this section, we discuss in further detail the h-process and its link to the absorbed particle process $(X_t^c)_{t \ge 0}$. We also discuss how one may extend the methodology used in this article to non-linear diffusions.

1.1 B.1 Backward h-processes

For a fixed time horizon $t\ge 0$, we let $\overline{X}_t$ be a random sample from ${{\mathcal {N}} }(\widehat{X}_t,P_t)$. We also denote by $X^h_{t,s}(x)$, with $s\in [0,t]$, be the backward diffusion defined by

$$\begin{aligned} dX^h_{t,s}(x)=\left( A X^h_{t,s}(x)+RP_s^{-1}( X^h_{t,s}(x)-\widehat{X}_s)\right) ~ds+BdW_s, \end{aligned}$$

starting at $X^h_{t,t}(x)=x$ at time $s=t$. In the above display, $P_s$ stands for the solution of the Riccati matrix differential equation defined in (27). The backward semigroup property is given for any $u\le s \in [0,t]$ by the mapping composition formulae

$$\begin{aligned} X^h_{t,u}=X^h_{s,u}\circ X^h_{t,s}\quad \hbox { and}\quad X^h_{t,t}=I, ~~\hbox { the identity function.} \end{aligned}$$

We assume that $\overline{X}_t$ and $(W_s)_{s\le t}$ are independent.

Rewritten in terms of the density $g_s$ of the Gaussian distribution ${{\mathcal {N}} }(\widehat{X}_s,P_s)$, we have

$$\begin{aligned} \overline{X}_{t,s}^h:=X^{h}_{t,s}(\overline{X}_t) \Longrightarrow d \overline{X}_{t,s}^h=\left( A \overline{X}_{t,s}^h-R\,\nabla \log {g_s( \overline{X}_{t,s}^h)}\right) \,ds+ B\,dW_s. \end{aligned}$$

(111)

The following theorem, taken from [8] links the non-absorbed particle process with the above backward diffision

Theorem B.1

[ [8]]. Assume that $X_0\sim {{\mathcal {N}} }(\widehat{X}_0,P_0)$. In this situation, for any $t\ge 0$ we have the backward formulation of the Feynman–Kac path integral

$$\begin{aligned} \begin{array}{l} \displaystyle \mathbb {E}\left( F\left( (X_s^c)_{s\in [0,t]}\right) ~|~\tau ^c\ge t\right) =\mathbb {E}\left( F\left( (\overline{X}^h_{t,s})_ {s\in [0,t]}\right) \right) .\end{array} \end{aligned}$$

The random state $\overline{X}_{t,s}^h$ is a Gaussian variable with a mean $\widehat{X}^h_{t,s}$ and covariance matrix $P^h_{t,s}$ satisfying the backward equations

$$\begin{aligned} \left\{ \begin{array}{rcl} \partial _s\widehat{X}^h_{t,s}&{}=&{}A \widehat{X}^h_{t,s}+RP_s^{-1}(\widehat{X}^h_{t,s}-\widehat{X}_s)\\ &{}&{}\\ \partial _{s}P_{t,s}^h&{}=&{}(A+RP^{-1}_s)P^h_{t,s}+P^h_{t,s}(A+RP^{-1}_s)^{\prime }-R \end{array}\right. \end{aligned}$$

with the terminal condition $(\widehat{X}^h_{t,t},P_{t,t}^h)=(\widehat{X}_t,P_t)$, where $(\widehat{X}_s,P_s)$ is the solution to the forward equations described in (27).

1.2 B.2 Extensions to non-linear diffusions

The h-process methodology can be extended to more general generators ${{\mathcal {L}} }$ and other choices of the potential function V. We now assume that ${{\mathcal {L}} }$ is the generator of the diffusion equation

$$\begin{aligned} dX_t=A(X_t)dt+B(X_t)dW_t \end{aligned}$$

(112)

for some drift function A(x) and some diffusion matrix valued function B(x) with appropriate dimensions. We also assume there exists some ground state $h_0$ associated with some energy $\lambda _0$; that is, we have that

$$\begin{aligned} h_0^{-1} {{\mathcal {L}} }(h_0)(x)=V(x)-\lambda _0 \end{aligned}$$

In this situation, the h-process $X^{h}_{t}$ is a diffusion with generator defined by

$$\begin{aligned} {{\mathcal {L}} }^{h}(f)= {{\mathcal {L}} }(f)+h_0^{-1}\Gamma _{ {{\mathcal {L}} }}(h_0,f) \end{aligned}$$

with the carré-du-champ operator

$$\begin{aligned} \Gamma _{ {{\mathcal {L}} }}(h_0,f )(x):=\left( B(x)\nabla h_0(x)\right) ^{\prime }\left( B(x)\nabla f(x)\right) = \left( R(x)\nabla h_0(x)\right) ^{\prime }\nabla f(x), \end{aligned}$$

where we have defined $R(x):=B(x)B(x)^{\prime }$. Equivalently, the h-process is defined by the diffusion

$$\begin{aligned} dX^{h}_{t}=\left( A(X^{h}_{t})+R(X^{h}_{t})\nabla \log {h_0(X^{h}_{t})}\right) dt+B(X^{h}_{t})dW_t \end{aligned}$$

Let $\overline{X}_t$ a random sample from the Feynman–Kac probability measures $\eta _t$ defined as in (21) for some potential function V.

Whenever it exists, let $g_s$ be the density of the normalised or unnormalised Feynman–Kac measures $\eta _s$ or $\gamma _s$. In this situation, following the analysis developed in [8], the assertion of Theorem B.1 remains valid with the backward diffusion

$$\begin{aligned} d \overline{X}_{t,s}^h=\left( A(\overline{X}_{t,s}^h)-\hbox { div}_{R}\log {g_s( \overline{X}_{t,s}^h)}\right) \,ds+ B(\overline{X}_{t,s}^h)\,dW_s \end{aligned}$$

(113)

with the terminal condition $\overline{X}_{t,t}^h=\overline{X}_{t}$ and the R-divergence m-column vector operator with j-th entry given by the formula

$$\begin{aligned} \hbox { div}_{R}(f)(x)^j:=~\sum _{1\le i\le r}~\partial _{x_i}\left( R_{i,j}(x)~f(x)\right) . \end{aligned}$$

C McKean–Vlasov Interpretations

In this section, we discuss several classes of McKean-Vlasov interpretations of the distribution of a non-absorbed particle. These probabilistic models and their mean field simulation are defined in terms of a nonlinear Markov process that depends on the distribution of the random states so that the flow of distributions of all random states coincides with the conditional distribution of a non-absorbed particle. To the best of our knowledge, these particle models have not been applied to the models considered in this article, despite being applicable in a wide variety of situations. In particular, these methods can be applied in high dimension and even when A is unstable.

1.1 C.1 Interacting jump processes

Let $\overline{X}_t$ be a nonlinear jump diffusion process with generator

$$\begin{aligned} {{\mathcal {L}} }_{\overline{\eta }_t}(f)(x)= {{\mathcal {L}} }(f)(x)+V(x)~\int ~(f(y)-f(x))~\overline{\eta }_t(dy)\quad \hbox { where}\quad \overline{\eta }_t:= \hbox { Law}(\overline{X}_t). \end{aligned}$$

The process starts at $\overline{X}_0=X_0$. Recall that the second order differential kinetic energy operator ${{\mathcal {L}} }$ defined in (2) coincides with the generator of the process $X_t$ arising in the Feynman–Kac representation (8). Between the jumps the process $\overline{X}_t$ evolves as $X_t$. At rate $V(\overline{X}_t)$ the process jumps onto a new location randomly selected according to the distribution $\overline{\eta }_t$. Observe that

$$\begin{aligned} \partial _t\overline{\eta }_t(f)= & {} \overline{\eta }_t\left( {{\mathcal {L}} }_{\overline{\eta }_t}(f)\right) =\overline{\eta }_t( {{\mathcal {L}} }(f))-\overline{\eta }_t(fV)+\overline{\eta }_t(f)\overline{\eta }_t(V). \end{aligned}$$

This shows that $\overline{\eta }_t$ satisfies the same evolution equation as the one satisfied by $\eta _t$ given in (26). Thus, for any choice of the generator ${{\mathcal {L}} }$ and any choice of the potential function V we have that

$$\begin{aligned} \overline{\eta }_t(dx)=\eta _t(dx):=\mathbb {P}(X^c_t\in dx~|~\tau ^c>t). \end{aligned}$$

The mean field particle interpretation of the nonlinear process $\overline{X}_t$ is defined by a system of N walkers, $\xi ^i_t$, evolving independently as $X_t$ with a spatial jump rate $V_t$, for $1\le i\le N$. At each jump time, the particle $\xi ^{i}_t$ jumps onto a particle uniformly chosen in the pool. The occupation measure of system is given by the empirical measure

$$\begin{aligned} \eta ^N_t=\frac{1}{N}\sum _{1\le i\le N}\delta _{\xi ^i_t}\longrightarrow _{N\rightarrow \infty }\eta _t\longrightarrow _{t\rightarrow \infty }\eta _{\infty }. \end{aligned}$$

(114)

Mimicking (23) we also define the normalising constant approximations

$$\begin{aligned} \frac{1}{t}\int _0^t \eta ^N_s(V) ds:=-\frac{1}{t}\log {\gamma ^N_t(1)}\longrightarrow _{N\rightarrow \infty }-\frac{1}{t}\log {\gamma _t(1)}\longrightarrow _{t\rightarrow \infty }\lambda _0=\eta _{\infty }(V).\nonumber \\ \end{aligned}$$

(115)

Observe that the N ancestral lines $\zeta _t^i:=(\xi ^i_{s,t})_{0\le s\le t}$ of length t of the above genetic-type process can also be seen as a system of N path-valued particles evolving independently as the historical process $Y_t=(X_{s})_{0\le s\le t}$ of $X_t$, with a jump rate $V_t(X_t)$ that only depends on the terminal state $X_t$ of the ancestral line $Y_t$.

The interacting particle system discussed above belongs to the class of diffusion Monte Carlo algorithms, see for instance the series of articles [17,18,19,20], as well as the articles [21, 38, 61, 62, 74, 75] in the context of ground state calculations and the series of articles [3, 34, 36,37,38] in the context of a general class of continuous time Feyman-Kac path integral formulae. The discrete time version of the interacting jump processes discussed above coincides with the genetic-type samplers and the Monte Carlo reconfiguration/selection methodologies discussed in [33, 36, 38], see also the pioneering articles in the mid-1980s by Hetherington [51] and further extended by Caffarel and his co-authors in the series of articles [17,18,19], see also Buonaura-Sorella [7] as well as the pedagogical introduction to quantum Monte Carlo by Caffarel-Assaraf [20].

1.2 C.2 Interacting diffusions

For any probability measure $\eta$ on $\mathbb {R}^r$ we let ${{\mathcal {P}} }_{\eta }$ denote the $\eta$-covariance

$$\begin{aligned} \eta \mapsto {{\mathcal {P}} }_{\eta }:=\eta \left( [e-\eta (e)][e-\eta (e)]'\right) \end{aligned}$$

(116)

where $e(x):=x$ is the identity function and $\eta (f)$ is a column vector whose i-th entry is given by $\eta (f^i)$ for some measurable function $f:\mathbb {R}^r\rightarrow \mathbb {R}^r$.

We now consider three different nonlinear McKean-Vlasov-type diffusion process,

$$\begin{aligned} \begin{array}{lrcl} (1)&{} d\overline{X}_t~&{}=&{}~(A- {{\mathcal {P}} }_{\overline{\eta }_t}S)\,\overline{X}_t~dt+ {{\mathcal {P}} }_{\overline{\eta }_t}~S^{1/2}~ d {{\mathcal {W}} }_{t}+B\,d\overline{ {{\mathcal {W}} }}_t,\\ &{}&{}&{}\\ (2)&{} \displaystyle d\overline{X}_t~&{}=&{}\displaystyle ~\left( A\,\overline{X}_t~-\frac{1}{2}~ {{\mathcal {P}} }_{\overline{\eta }_t}\,S\left( \overline{X}_t+ \overline{\eta }_t(e)\right) \right) dt+B\,d\overline{ {{\mathcal {W}} }}_t,\\ &{}&{}&{}\\ (3)&{} \partial _t\overline{X}_t~&{}=&{}\displaystyle A\,\overline{X}_t~-\frac{1}{2}~ {{\mathcal {P}} }_{\overline{\eta }_t}\,S\left( \overline{X}_t+ \overline{\eta }_t(e)\right) + (R+M_t)\, {{\mathcal {P}} }_{\overline{\eta }_t}^{-1}\left( \overline{X}_t-\overline{\eta }_t(e)\right) , \end{array} \end{aligned}$$

(117)

for any skew symmetric matrix $M^\prime _t=-M_t$ that may also depend $\overline{\eta }_t$. In all three cases $( {{\mathcal {W}} }_t,\overline{ {{\mathcal {W}} }}_t)$ are independent copies of $W_t$; and $\overline{X}_0$ is an independent copies of $X_0$. We also assume that $( {{\mathcal {W}} }_t, {{\mathcal {W}} }_t,\overline{X}_0)$ are independent. In all three cases in (117), $\overline{\eta }_t$ stands for the probability distribution of $\overline{X}_t$; that is, we have that

$$\begin{aligned} \overline{\eta }_t:= \hbox { Law}(\overline{X}_t). \end{aligned}$$

(118)

Observe that, in all three cases the stochastic processes discussed above depend in some nonlinear fashion on the law of the diffusion process itself.

Theorem C.1

In all the three cases presented in (117), for any $t\ge 0$ we have the Gaussian preserving property

$$\begin{aligned} \eta _0= {{\mathcal {N}} }(\widehat{X}_0,P_0)=\overline{\eta }_0\Longrightarrow \overline{\eta }_t= {{\mathcal {N}} }(\widehat{X}_t,P_t)=\eta _t. \end{aligned}$$

Proof

Let $\overline{X}_t$ be the process defined as in (1) by replacing ${{\mathcal {P}} }_{\overline{\eta }_t}$ by $P_t$. In this case, we have

$$\begin{aligned} d\left( \overline{X}_t-\mathbb {E}(\overline{X}_t)\right) ~=~(A-P_tS)\,\left( \overline{X}_t-\mathbb {E}(\overline{X}_t)\right) ~dt+P_t~S^{1/2}~d {{\mathcal {W}} }_{t}+B\,d\overline{ {{\mathcal {W}} }}_t.\ \end{aligned}$$

Applying Ito’s formula and taking expectations we obtain

$$\begin{aligned} \partial _t {{\mathcal {P}} }_{\overline{\eta }_t}=(A-P_tS) {{\mathcal {P}} }_{\overline{\eta }_t}+ {{\mathcal {P}} }_{\overline{\eta }_t}(A-P_tS)^{\prime }+P_tSP_t+R. \end{aligned}$$

This yields the linear system

$$\begin{aligned} \partial _t\left( {{\mathcal {P}} }_{\overline{\eta }_t}-P_t\right) =(A-P_tS)( {{\mathcal {P}} }_{\overline{\eta }_t}-P_t)+ {{\mathcal {P}} }_{\overline{\eta }_t}(A-P_tS)^{\prime }( {{\mathcal {P}} }_{\overline{\eta }_t}-P_t)\Longrightarrow P_t= {{\mathcal {P}} }_{\overline{\eta }_t}. \end{aligned}$$

We conclude that $\overline{X}_t$ is a linear diffusion with mean $\widehat{X}_t$ and covariance matrix $P_t$. The proof for the other two cases follows the same lines of arguments, thus we leave the details to the reader. $\square$

The mean-field particle interpretation of the first nonlinear diffusion process in (117) is given by the Mckean–Vlasov type interacting diffusion process

$$\begin{aligned} \begin{array}{rcl} d\xi ^i_t= & {} (A-P^N_tS)\,\overline{X}_t~dt+P^N_t~S^{1/2}~ d {{\mathcal {W}} }_{t}^i+B\,d\overline{ {{\mathcal {W}} }}^i_t, \quad i=1,\ldots ,N, \end{array} \end{aligned}$$

(119)

where $( {{\mathcal {W}} }_{t}^i,\overline{ {{\mathcal {W}} }}^i_t,\xi ^i_0)_{1\le i\le N}$ are N independent copies of $( {{\mathcal {W}} }_{t},\overline{ {{\mathcal {W}} }}_t\overline{X}_0)$. In the above display, the $P^N_t$ are the rescaled empirical covariance matrices given by the formulae

$$\begin{aligned} P^N_t:=\left( 1-\frac{1}{N}\right) ^{-1}~ {{\mathcal {P}} }_{\eta ^{N}_t}=\frac{1}{N-1}\sum _{1\le i\le N}\left( \xi ^i_t-m^N_t\right) \left( \xi ^i_t-m^N_t\right) ^{\prime }, \end{aligned}$$

(120)

with the empirical measures

$$\begin{aligned} \eta ^{N}_t:=\frac{1}{N}\sum _{1\le i\le N}\delta _{\xi ^i_t} \quad \hbox { and the sample mean}\quad m^N_t:=\frac{1}{N}\sum _{1\le i\le N}\xi _t^i. \end{aligned}$$

Note that (119) is a set of N stochastic differential equations coupled via the empirical covariance matrix $P_t^N$. The mean-field particle interpretation of the second and third nonlinear diffusion processes in (117) are defined as above by replacing ${{\mathcal {P}} }_{\eta _t}$ by the sample covariance matrices $P^N_t$. The quasi-invariant measure $\eta _{\infty }$ and the parameter $\lambda _0$ are computed using the limiting formulae (114) and (115).

The interacting diffusions discussed above belong to the class of Ensemble Kalman filters, see for instance the pioneering article by Evensen [47], the series of articles [9,10,11], as well as [40, 41] and the references therein.

In contrast with the interacting jump process discussed in “Appendix C.1” none of the nonlinear diffusions discussed in (117) can be extended to more general generators ${{\mathcal {L}} }$ and other choices of the potential function V.

We end this section with an application of the seminal feedback particle filter methodology recently developed by Mehta and Meyn and their co-authors [81,82,83,84,85] to Feynman–Kac models. Consider the diffusion

$$\begin{aligned} d\overline{X}_t=\left( A(\overline{X}_t)+U_t(\overline{X}_t)\right) dt+B(\overline{X}_t)dW_t, \end{aligned}$$

where $U_t(x)$ is the solution of the Poisson equation

$$\begin{aligned} \sum _{1\le i\le r}\frac{1}{g_t(x)}\partial _{x_i}\left( U^i_t(x)~g_t(x)\right) =(V(x)-\eta _t(V)), \quad t \ge 0. \end{aligned}$$

In the above display $g_t(x)$ stands for the density of the distribution $\overline{\eta }_t$ of the random state $\overline{X}_t$. The generator ${{\mathcal {L}} }_{\overline{\eta }_t}$ of the above time varying diffusion satisfies the equation

$$\begin{aligned} \overline{\eta }_t( {{\mathcal {L}} }_{\overline{\eta }_t}(f))=\overline{\eta }_t( {{\mathcal {L}} }(f))+\sum _{1\le i\le r}\int ~U^i_t(x) \partial _{x_i}f(x)~g_t(x)~dx. \end{aligned}$$

Integrating by part the last term we obtain the formula

$$\begin{aligned} \overline{\eta }_t( {{\mathcal {L}} }_{\overline{\eta }_t}(f))=\overline{\eta }_t(L(f))-\int ~f(x)~(V(x)-\overline{\eta }_t(V))~\overline{\eta }_t(dx), \end{aligned}$$

from which we conclude that

$$\begin{aligned} \overline{\eta }_t( {{\mathcal {L}} }_{\overline{\eta }_t}(f))=\overline{\eta }_t(L(f))-\overline{\eta }_t(fV)+\overline{\eta }_t(f)\overline{\eta }_t(V). \end{aligned}$$

This shows that $\overline{\eta }_t=\hbox { Law}(\overline{X}_t)=\eta _t$ coincides with the normalised Feynman–Kac measures.

For linear-Gaussian models we have $\eta _t= {{\mathcal {N}} }(\widehat{X}_t,P_t)$. Thus, the Poisson equation resumes to the formula

$$\begin{aligned} \sum _{1\le i\le r}\partial _{x_i}U^i_t(x)-(x-\widehat{X}_t)^{\prime }P_t^{-1}U_t(x)= & {} \frac{1}{2}(x^{\prime }Sx-\widehat{X}_t^{\prime }S\widehat{X}_t-\hbox { Tr}(SP_t))\\= & {} \frac{1}{2}~(x-\widehat{X}_t)^{\prime }S(x-\widehat{X}_t). \end{aligned}$$

The solution of the above equation is clearly given by

$$\begin{aligned} U_t(x)=-\frac{1}{2}~P_tS(x+\widehat{X}_t)\Longrightarrow \sum _{1\le i\le r}\partial _{x_i}U^i_t(x)=\hbox { Tr}(P_tS). \end{aligned}$$

The resulting diffusion coincides with the second case in (117).