Strong solutions to McKean–Vlasov SDEs with coefficients of Nemytskii type: the time-dependent case

Abstract

We consider a large class of nonlinear FPKEs with coefficients of Nemytskii type depending explicitly on time and space, for which it is known that there exists a sufficiently Sobolev-regular Schwartz-distributional solution $u\in L^1\cap L^\infty$. We show that there exists a unique strong solution to the associated McKean–Vlasov SDE with time marginal law densities u. In particular, every weak solution of this equation with time marginal law densities u can be written as a functional of the driving Brownian motion. Moreover, plugging any Brownian motion into this very functional produces a weak solution with time marginal law densities u.

1 Introduction

In this paper, we will consider the following McKean–Vlasov stochastic differential equation (abbreviated by McKean–Vlasov SDE or MVSDE) in ${\mathbb {R}^d}$, $d\in \mathbb {N}$, with coefficients of Nemytskii type, which in our case is of the form

where $t\in [0,T]$, $T\in (0,\infty )$, $\mathbb {1}_{d\times d}$ is the d-dimensional unit matrix, $(W_t)_{t\in [0,T]}$ is a standard d-dimensional $(\mathcal {F}_t)$-Brownian motion and $\xi$ an $\mathcal {F}_0$-measurable function on some stochastic basis $(\Omega ,\mathcal {F},\mathbb {P};(\mathcal {F}_t)_{t\in [0,T]})$, i.e. a complete, filtered probability space, where $(\mathcal {F}_t)_{t\in [0,T]}$ is a normal filtration, and ${\mathcal {L}}_{X(t)}:= \mathbb {P}\circ (X(t))^{-1}, t\in [0,T]$. Here, we assume that the drift and diffusion coefficient are given through Borel-measurable functions

$$\begin{aligned}&b : [0,T]\times {\mathbb {R}^d}\times \mathbb {R}\rightarrow {\mathbb {R}^d},\end{aligned}$$

(1)

$$\begin{aligned}&a: [0,T]\times {\mathbb {R}^d}\times \mathbb {R}\rightarrow \mathbb {R}. \end{aligned}$$

(2)

Further, let us define $\beta (\cdot ,r):= a(\cdot ,r)r, b^*(\cdot ,r):=b(\cdot ,r)r, r\in \mathbb {R}$. As in [6], we impose the following conditions on the coefficients of (MVSDE).

• (H1) $a \in C^1([0,T]\times {\mathbb {R}^d}\times \mathbb {R}), a, \partial _r a$ are bounded and, for all $s,t \in [0,T], x\in {\mathbb {R}^d}, r,\bar{r} \in \mathbb {R}$,

  $$\begin{aligned} (\beta (t,x,r)-\beta (t,x,\bar{r}))(r-\bar{r})&\ge \gamma _0 |r-\bar{r}|^2,\end{aligned}$$

  (3)
  $$\begin{aligned} |\partial _r\beta (t,x,r)-\partial _r\beta (s,x,r)|&\le h(x)|t-s|\partial _r\beta (t,x,r),\end{aligned}$$

  (4)
  $$\begin{aligned} |\beta (t,x,r)-\beta (s,x,r)| + |\nabla _x \beta (t,x,r)-\nabla _x\beta (s,x,r)|&\le h(x)|t-s|(1+|r|), \end{aligned}$$

  (5)
  $$\begin{aligned} |\partial _t\beta (t,x,r)|+|\nabla _x \beta (t,x,r)|&\le h(x)|r|. \end{aligned}$$

  (6)

• (H2) For each $(t,x) \in [0,T]\times {\mathbb {R}^d}$, $b(t,x,\cdot ) \in C^1(\mathbb {R})$, $b, (t,x,r)\mapsto r\partial _r b(t,x,r)$ are bounded and for all $s,t \in [0,T]$, $x\in {\mathbb {R}^d}$, $r \in \mathbb {R}$,

  $$\begin{aligned} |b^*(t,x,r)-b^*(s,x,r)| \le h(x)|t-s|(1+|b^*(t,x,r)|), \end{aligned}$$

  and

  $$\begin{aligned} |b^*(t,x,r)|\le h(x)|r|, \end{aligned}$$

  (7)

where $\gamma _0 >0$, $h \in L^\infty ({\mathbb {R}^d}) \cap L^2({\mathbb {R}^d}), h \ge 0$. We note that the derivative $\partial _z$ denotes the derivative with respect to a scalar z-coordinate. Further, ${{\,\mathrm{\textrm{div}}\,}}={{\,\mathrm{\textrm{div}}\,}}_x$, $\nabla = \nabla _x$, $\Delta = \Delta _x$ and $D=D_x$ denote the divergence, gradient, Laplacian and Jacobian with respect to the spatial x-coordinate. All the derivatives are supposed to be understood in the sense of Schwartz-distributions. Here, we would like to point out that condition (3) and the continuity of a in the r-variable imply

$$\begin{aligned} a\ge \gamma _0> 0, \end{aligned}$$

(8)

which means that the diffusion matrix of (MVSDE) is assumed to be non-degenerate.

The McKean–Vlasov SDE (MVSDE) arises from the study of the following type of nonlinear Fokker–Planck–Kolmogorov equation (FPKE)

which describes, as pointed out in [6] (see also the references therein), particle transport in disordered media with time-dependent temperature with particular example $\beta (t,x,r)\equiv \gamma (t,x) \ln (1+|r|)$ serving as a model for the classical boson dynamics. In [3] also the case of general non-diagonal, explicitly x-dependent diffusion matrix is studied, but for simplicity we consider nonlinear Fokker–Planck–Kolmogorov equations of type (FPKE). In general, (FPKE) is to be understood in the Schwartz-distributional sense. We will say that a family $u=(u_t)_{t\in [0,T]}=(u(t,\cdot ))_{t\in [0,T]}$ of $L^1({\mathbb {R}^d})$-functions is a Schwartz-distributional solution to (FPKE) if $t \mapsto u_t(x)dx$ is narrowly continuous and

$$\begin{aligned} \int _{\mathbb {R}^d} \varphi (x) u(t,x)dx =&\int _{\mathbb {R}^d} \varphi (x) v(x)dx + \int _0^t \int _{\mathbb {R}^d} b^i(s,x,u(s,x)) \partial _i \varphi (x) u(s,x)dxds \nonumber \\&+ \int _0^t\int _{\mathbb {R}^d}a(s,x,u(s,x))\Delta \varphi (x) u(s,x) dxds\ \ \forall t\in [0,T], \end{aligned}$$

(9)

for each $\varphi \in C^{\infty }_c(\mathbb {R}^d)$ (using Einstein summation convention), where $b=(b^i)_{i=1}^d$. If, additionally, $u_t \in \mathcal {P}_0({\mathbb {R}^d})$, i.e. $u_t$ is a probability density on ${\mathbb {R}^d}$, $t \in [0,T]$, then u is simply called a probability solution to (FPKE).

Recently, (FPKE) has been investigated in [6] under the conditions (H1) and (H2). In their work, the authors found that, under their conditions, (FPKE) even has a (unique) analytically strong solution u in $H^{-1}$ with initial condition $v \in D_0$, where $D_0:=\{f \in L^2: \beta (0,\cdot ,f) \in H^1\}$. If, additionally, v is a probability density, then the solution u is a curve of probability densities and is therefore also a probability solution in the above sense. This enabled the authors to construct a weak solution to (MVSDE) via a superposition principle procedure for McKean–Vlasov SDEs presented in [3, Section 2] (see also [2, Section 2]) based upon Trevisan’s superposition principle for SDEs (see [16, Theorem 2.5], generalising [9]; see also [7] for a recent improvement of both results). [6] is subsequent to several papers of the same authors in the study of (FPKE) with time-homogenous coefficients (cf. [2,3,4,5] (and references therein)). Let us note that in (H1) and (H2) a number of the assumptions are always fulfilled in the time-homogeneous setting. However, in the time-dependent case these assumptions are owed to the additional technical challenges, which come along with it.

We recall that [6] follows the approach developed in [2] and [3] by first finding a probability solution to (FPKE) and then associating a weak solution to (MVSDE) such that its time marginal law densities are given by such a probability solution. This constitutes a vital part in the realisation of McKean’s original idea, proposed in [12], to associate Markov processes to certain nonlinear PDEs in a way that the process’s transition probabilities solve the PDE. McKean’s example included the case of the viscous Burgers’ equation and the classical porous medium equation in one dimension. We refer to Rehmeier and Röckner’s article [14] for a full realisation of McKean’s vision covering the previously mentioned equations and providing new classes of examples including the generalised porous medium equation perturbed by a nonlinear transport term in multiple dimensions and also nonlocal analogues. In particular, they pinned down a suitable notion for a nonlinear Markov process in the spirit of McKean. It is also interesting to study (FPKE) and (MVSDE) in order to obtain comparable examples for Markov processes when the coefficients depend explicitly on time.

The aim of this paper is to use the above-mentioned connection between probability solutions to (FPKE) and weak solutions to (MVSDE) in the setup of [6] and show that under the conditions (H1), (H2), and additionally (H3) (see p. 9) there even exists a (probabilistically) strong solution to (MVSDE) which is pathwise unique among all solutions with time marginal law densities u, where u is the probability solution to (FPKE) with initial condition $v \in \mathcal {P}_0({\mathbb {R}^d})\cap L^\infty ({\mathbb {R}^d})$ provided by [6] (for the exact formulation, see Theorem 3.9). In particular, all weak solutions to (MVSDE) whose time marginal law densities coincide with u have the same law in $\mathcal {P}(C([0,T];{\mathbb {R}^d}))$.

We achieve this result starting from the procedure developed in [10] which builds upon a restricted Yamada–Watanabe theorem for SDEs (see [10, Section 2+3] or Theorem 3.8). As we show in this article, the Ansatz from [10] can be extended to the fully explicit time- and space-dependent case. In order to apply the restricted Yamada–Watanabe theorem, the key point is to show pathwise uniqueness for (MVSDE) among all solutions to (MVSDE) with time marginal law densities u where u is as constructed by Barbu and Röckner in [6]. It turns out that for this we can apply the pathwise uniqueness result by Röckner and Zhang [15], see [10, Theorem 4.4], by using a modification of the fine chain rule for compositions of a Lipschitz function with a vector-valued Sobolev function which we employ from the estimated work [1], see Theorem 3.4 and Corollary 3.6. This is in contrast to the previous work [10] where we used an argument involving the simple product rule to deal with the explicitly x-dependent drift vector of tensor type.

Within this work, we consider (MVSDE) as a McKean–Vlasov SDE of the form

$$\begin{aligned} dX(t) \nonumber =&\ \varvec{b}(t,X(t),{\mathcal {L}}_{X(t)})dt + \varvec{\sigma }(t,X(t),{\mathcal {L}}_{X(t)}) dW(t),\ \ t\in [0,T], \nonumber \\ X(0)=&\ \xi , \end{aligned}$$

with coefficients

$$\begin{aligned}{}[0,T]\times {\mathbb {R}^d}\times \mathcal {P}({\mathbb {R}^d})\ni (t,x,\nu )&\mapsto \varvec{b}(t,x,\nu ):= b(t,x,v_a(x)),\\ [0,T]\times {\mathbb {R}^d}\times \mathcal {P}({\mathbb {R}^d})\ni (t,x,\nu )&\mapsto \varvec{\sigma }(t,x,\nu ):=\sqrt{2a(t,x,v_a(x))}\mathbb {1}_{d\times d}, \end{aligned}$$

where $\mathcal {P}({\mathbb {R}^d})$ is the set of all Borel probability measures on ${\mathbb {R}^d}$ equipped with the weak topology, $v_a$ denotes the version of the density of the absolutely continuous part of $\nu$ both with respect to Lebesgue measure which is obtained by setting $v_a(x)= 0$ if $\lim _{R\rightarrow 0}\nicefrac {\nu (B_R(x))}{\lambda ^d(B_R(0))}$ does not exist in $\mathbb {R}$. By the Besicovitch derivation theorem, this makes the map ${\mathbb {R}^d}\times \mathcal {P}({\mathbb {R}^d})\ni (x,\nu )\mapsto v_a(x)$ and therefore also $\varvec{b}$ and $\varvec{\sigma }$ Borel-measurable (for details see [11, Section 4.2]). The dependence of $\varvec{b}$ and $\varvec{\sigma }$ on $\nu$ in terms of $v_a$ evaluated at a fixed point x excludes the continuity of $\varvec{b}$ and $\varvec{\sigma }$ in their measure-component with respect to the topology of weak convergence of probability measures, Wasserstein distance or bounded variation norm. These types of continuity assumptions are made in the major part of the literature (see, for example, [8]).

We emphasise that large parts of this article are taken from the author’s thesis [11].

This paper is structured as follows. First, we will fix some frequently used notation and, afterwards, recall the notation of a $P^{(u_t)}$-solution to and $P^{(u_t)}$-uniqueness for (MVSDE), where u is a probability solution to (FPKE). Second, in Sect. 2, we will provide the reader with a two-step procedure on how to obtain a unique strong solution to (MVSDE) with time marginal law densities u, where u is the probability solution to (FPKE) with initial condition $v \in \mathcal {P}_0({\mathbb {R}^d})\cap L^\infty ({\mathbb {R}^d})$ provided by [6], based on the procedure developed in [10]. This procedure will then be carried out in the last section, Sect. 3, which is divided into three subsections. Section 3.1 is devoted to gathering the results on the existence and the regularity of u and the existence of a weak solution to (MVSDE) with time marginal law densities u. In Sect. 3.2, we will give a pathwise uniqueness result for (MVSDE) among all weak solutions with time marginal law densities u. In Sect. 3.3, we will apply the restricted version of the Yamada–Watanabe theorem for SDEs from [10] to (MVSDE) and combine the results of Sects. 3.1 and 3.2 in order to obtain a strong solution to (MVSDE), which is pathwise unique given the time marginal law densities u.

2 Notation

Within this paper, we will use the following notation, which is essentially taken from [10]. For a topological space $({\textbf {T}},\tau )$, $\mathcal {B}({\textbf {T}})$ shall denote the Borel $\sigma$-algebra on $({\textbf {T}},\tau )$. Throughout this article, d, n, k, m denote natural numbers.

On $\mathbb {R}^n$, we will always consider the usual n-dimensional Lebesgue measure $\lambda ^n$ if not said any differently. If there is no risk for confusion, we will just say that some property for elements in $\mathbb {R}^n$ holds almost everywhere (or $\textit{a.e.}$) if and only if it holds $\lambda ^n$-almost everywhere. Furthermore, on $\mathbb {R}^n$, $|\cdot |_{\mathbb {R}^n}$ denotes the Euclidean norm. If there is no risk for confusion, we will just write $|\cdot |=|\cdot |_{\mathbb {R}^n}$. By $B_R(x)$, we will denote the usual open ball with centre $x\in \mathbb {R}^n$ and radius $R>0$.

Let $(S,\mathscr {S},\eta )$ be a measure space. For $1\le p \le \infty$, $(L^p(S;E), \left\Vert \cdot \right\Vert _{{{L^p(S;E)}}})$ symbolises the usual Bochner space of strongly measurable E-valued functions f on S for which $\left\Vert f\right\Vert _{{E}}^p$ is integrable. If $S=\mathbb {R}^n$ and $E=\mathbb {R}$, we just write $L^p(\mathbb {R}^n;\mathbb {R})=L^p(\mathbb {R}^n)$. The set of strongly measurable functions on $\mathbb {R}^n$ with values in E which are locally p-integrable in norm on E will be denoted by $L^p_{loc}(\mathbb {R}^n;E)$. Moreover, $(W^{1,p}(\mathbb {R}^n), \left\Vert \cdot \right\Vert _{{{W^{1,p}(S;E)}}})$ denotes the usual Sobolev-space, containing all $L^p(\mathbb {R}^n)$-functions, whose first-order distributional derivatives can be represented by elements in $L^p(\mathbb {R}^n)$. In the case $p=2$, we set $H^1(\mathbb {R}^n):=W^{1,p}(\mathbb {R}^n)$ and its continuous dual space shall be denoted by $H^{-1}(\mathbb {R}^n)$. Accordingly, E-valued first-order Sobolev functions on $\mathbb {R}^n$ will be denoted by $W^{1,p}(\mathbb {R}^n;E)$.

Let (M, d) be a metric space. Then, $\mathcal {P}(M)$ denotes the set of all Borel probability measures on (M, d). We will consider $\mathcal {P}(M)$ as a topological space with respect to the topology of weak convergence of probability measures. A curve of probability measures $(\nu _t)_{t\in [0,T]}\subset \mathcal {P}(M)$ is called narrowly continuous if $[0,T]\ni t\mapsto \int \varphi (x)\nu _t(dx)$ is continuous for all $\varphi \in C_b(M)$. By $\mathcal {P}_0(\mathbb {R}^n)$, we will denote the set of all probability densities with respect to Lebesgue measure, i.e.

$$\begin{aligned} \mathcal {P}_0(\mathbb {R}^n) = \left\{ \rho \in L^1(\mathbb {R}^n)\ |\ \rho \ge 0 \text { a.e.}, \int _{\mathbb {R}^n} \rho (x) dx =1\right\} . \end{aligned}$$

Let $(\Omega ,\mathcal {F},\mathbb {P})$ be a probability space and $(S,\mathscr {S})$ a measurable space. If $X:\Omega \rightarrow S$ is an $\mathcal {F}/\mathscr {S}$-measurable function, then we say that ${\mathcal {L}}_{X}:=\mathbb {P}\circ X^{-1}$ is the law of X, whenever there is no risk for confusion about the underlying probability measure $\mathbb {P}$.

By $C_c^\infty (\mathbb {R}^n)$, we denote the set of all infinitely differentiable functions with compact support. Let $(E,\left\Vert \cdot \right\Vert _{{E}})$ be a Banach space. The set of continuous functions on the interval [0, T] with values in E is denoted by C([0, T]; E) and is considered with respect to the usual supremum’s norm. Further, we define

$$\begin{aligned}C([0,T];E)_0:= \{w \in C([0,T];E): w(0)=0\}.\end{aligned}$$

For $t\in [0,T]$, $\pi _t: C([0,T];E) \rightarrow E$ denotes the canonical evaluation map at time t, i.e. $\pi _t(w):=w(t), w\in C([0,T];E)$. Further, we set $\mathcal {B}_t(C([0,T];E)):= \sigma (\pi _s: s\in [0,t])$ and, correspondingly, $\mathcal {B}_t(C([0,T];E)_0):= \sigma (\pi _s: s\in [0,t])\cap C([0,T];E)_0$. Moreover, $\mathbb {P}_W$ denotes the Wiener measure on $(C([0,T];{\mathbb {R}^d})_0,\mathcal {B}(C([0,T];{\mathbb {R}^d})_0))$.

3 $P^{(u_t)}$-solutions to (MVSDE)

Let us briefly recall the solution concepts for (MVSDE) from [10] in order to make the steps in Sect. 2 and the application of the restricted Yamada–Watanabe theorem in Sect. 3.3 conceptually more feasible. If u is a probability solution to (FPKE), we set

$$\begin{aligned} P^{(u_t)}:=\{Q \in \mathcal {P}(C([0,T];{\mathbb {R}^d})): Q \circ \pi _t^{-1}= u_t(x)dx\ \forall t\in [0,T]\}. \end{aligned}$$

A $P^{(u_t)}$-weak solution $(X,W,(\Omega ,\mathcal {F},\mathbb {P};(\mathcal {F}_t)_{t\in [0,T]}))$ is a (probabilistically) weak solution $(X,W, (\Omega ,\mathcal {F},\mathbb {P};(\mathcal {F}_t)_{t\in [0,T]}))$ in the usual sense such that ${\mathcal {L}}_{X(t)} = u_t(x)dx$, for all $t\in [0,T]$. For the convenience of the reader, we will just write $(X,W)=(X,W, (\Omega ,\mathcal {F},\mathbb {P};(\mathcal {F}_t)_{t\in [0,T]}))$ in cases in which we do not need to refer explicitly to the underlying stochastic basis $(\Omega ,\mathcal {F},\mathbb {P};(\mathcal {F}_t)_{t\in [0,T]})$.

We will say that (MVSDE) has a $P^{(u_t)}$-strong solution if there exists a function $F: {\mathbb {R}^d}\! \times \! C([0,T];{\mathbb {R}^d})_0\rightarrow C([0,T];{\mathbb {R}^d})$, which is $\overline{\mathcal {B}({\mathbb {R}^d})\!\otimes \! \mathcal {B}(C([0,\!T];{\mathbb {R}^d})_0)}^{v(x)dx\otimes \mathbb {P}_W} /\mathcal {B}(C([0,T];{\mathbb {R}^d}))$-measurable, such that, for v(x)dx-a.e. $x\in {\mathbb {R}^d}$, $F(x,\cdot )$ is $\overline{\mathcal {B}_t(C([0,T];{\mathbb {R}^d})_0)}^{\mathbb {P}_W}/\mathcal {B}_t(C([0,T];{\mathbb {R}^d}))$-measurable for all $t\in [0,T]$ and, whenever $\xi$ is an $\mathcal {F}_0$-measurable function with ${\mathcal {L}}_{\xi } = v(x)dx$ and W is a standard d-dimensional $(\mathcal {F}_t)$-Brownian motion on some stochastic basis $(\Omega ,\mathcal {F},\mathbb {P};(\mathcal {F}_t)_{t\in [0,T]})$, $(F(\xi ,W),W, (\Omega ,\mathcal {F},\mathbb {P};(\mathcal {F}_t)_{t\in [0,T]}))$ is a $P^{(u_t)}$-weak solution to (MVSDE). Here, $\overline{\mathcal {B}({\mathbb {R}^d})\otimes \mathcal {B}(C([0,T];{\mathbb {R}^d})_0)}^{v(x)dx\!\otimes \! \mathbb {P}_W}$ denotes the completion of $\mathcal {B}({\mathbb {R}^d})\otimes \mathcal {B}(C([0,\!T]; {\mathbb {R}^d})_0)$ with respect to the measure $v(x)dx\otimes \mathbb {P}_W$, and $\overline{\mathcal {B}_t(C([0,T];{\mathbb {R}^d})_0)}^{\mathbb {P}_W}$ denotes the completion of $\mathcal {B}_t(C([0,T];{\mathbb {R}^d})_0)$ with respect to $\mathbb {P}_W$ on $(C([0,T];{\mathbb {R}^d})_0,\mathcal {B}(C([0,T]; {\mathbb {R}^d})_0))$.

Moreover, $P^{(u_t)}$-pathwise uniqueness holds for (MVSDE) if for every two $P^{(u_t)}$-weak solutions $(X,W, (\Omega ,\mathcal {F},\mathbb {P};(\mathcal {F}_t)_{t\in [0,T]}))$, $(Y,W, (\Omega ,\mathcal {F},\mathbb {P};(\mathcal {F}_t)_{t\in [0,T]}))$ (with respect to the same Brownian motion on the same stochastic basis) with $X(0)=Y(0)$ $\mathbb {P}$-a.s., one has

${\sup _{t\in [0,T]}|X(t) - Y(t)|}=0$ $\mathbb {P}$-a.s.

We say that there exists a unique $P^{(u_t)}$-strong solution to (MVSDE) if there exists a $P^{(u_t)}$-strong solution to (MVSDE) with functional F as above, and every $P^{(u_t)}$-weak solution (X, W) is of the form $X = F(X(0),W)$ almost surely with respect to the underlying probability measure.

4 The procedure

We will essentially follow the same procedure as proposed in [10].

Our overall goal is to apply a restricted Yamada–Watanabe theorem for SDEs to (MVSDE) (see Theorem 3.8), which will enable us to show that there exists a unique $P^{(u_t)}$-strong solution to (MVSDE) under the conditions (H1), (H2) and (H3) (see below), where u is a probability solution (FPKE) with initial condition $v\in \mathcal {P}_0({\mathbb {R}^d}) \cap L^\infty ({\mathbb {R}^d})$ obtained by [6] (see Theorem 3.1 and Corollary 3.2). In order to achieve this, we will need the following ingredients.

• 1. A $P^{(u_t)}$-weak solution to (MVSDE),

• 2. $P^{(u_t)}$-pathwise uniqueness holds for (MVSDE). In order to show this, we will use that ${u \in L^2([0,T];H^1({\mathbb {R}^d}))\cap L^\infty ([0,T]\times {\mathbb {R}^d})}$ (see Theorem 3.1).

The ingredients will be gathered in Sects. 3.1 and 3.2 and combined via the previously mentioned restricted Yamada–Watanabe theorem in Sect. 3.3.

5 The main result

5.1 Ingredient 1: A $P^{(u_t)}$-weak solution to (MVSDE)

In [6], Barbu and Röckner studied (FPKE) as an evolution equation in $H^{-1}$ in the analytically strong sense. The following result can be found in [6, Theorem 2.1], and we will, thus, omit the proof.

Theorem 3.1

Let $D_0:= \{f \in L^2({\mathbb {R}^d}): \beta (0,\cdot ,f) \in H^1({\mathbb {R}^d})\}$. Assume that (H1) and (H2) hold. Then, for each $v \in L^1({\mathbb {R}^d})\cap D_0$ there exists a Schwartz-distributional solution $u=(u_t)_{t\in [0,T]}$ to (FPKE) with $\left. u\right| _{t=0}=v$ such that

$$\begin{aligned} u&\in C([0,T];L^2({\mathbb {R}^d}))\cap W^{1,2}([0,T]; H^{-1}({\mathbb {R}^d})) \cap L^\infty ([0,T];L^1({\mathbb {R}^d})),\\ u, \beta (\cdot ,u)&\in L^2([0,T];H^1({\mathbb {R}^d})), \nonumber \end{aligned}$$

(10)

and $\left\Vert u(t)\right\Vert _{{L^1({\mathbb {R}^d})}} \le \left\Vert v\right\Vert _{{L^1({\mathbb {R}^d})}}$ for all $t\in [0,T]$. If $v \in \mathcal {P}_0({\mathbb {R}^d}) \cap D_0$, then u is a probability solution to (FPKE).

Finally, assume that the condition (7) is replaced by the stronger condition

$$\begin{aligned} |b^*(t,x,r)-b^*(t,x,\bar{r})|\le h(x)|r-\bar{r}|\ \ \forall t\in [0,T], x\in {\mathbb {R}^d},\ r,\bar{r} \in \mathbb {R}, \end{aligned}$$

(11)

where $h \in (L^2\cap L^\infty )({\mathbb {R}^d}), h\ge 0$ is the function introduced in (H1) and (H2). Then, for all $v, \bar{v} \in L^1({\mathbb {R}^d})\cap D_0$, the corresponding solutions $u(t,v), u(t,\bar{v})$ to (FPKE) satisfy

$$\begin{aligned} \left\Vert u(t,v)- u(t,\bar{v})\right\Vert _{{L^1({\mathbb {R}^d})}} \le \left\Vert v-\bar{v}\right\Vert _{{L^1({\mathbb {R}^d})}}\ \ \forall t\in [0,T]. \end{aligned}$$

(12)

Moreover, if $D_x b\in L^1_{loc}([0,T]\times {\mathbb {R}^d}\times \mathbb {R};\mathbb {R}^{d\times d}),\Delta _x \beta \in L^1_{loc}([0,T]\times {\mathbb {R}^d}\times \mathbb {R})$, and

$$\begin{aligned} \Lambda (b,\beta ):&={{\,\mathrm{ess~sup}\,}}\{|D_x b(t,x,r) r| + |\Delta _x\beta (t,x,r)| : t\in [0,T], x\in {\mathbb {R}^d}, r\in \mathbb {R}\}\\&<\infty , \end{aligned}$$

then, for all $v \in L^1({\mathbb {R}^d})\cap D_0\cap L^\infty ({\mathbb {R}^d})$, $u \in L^\infty ([0,T]\times {\mathbb {R}^d})$ with

$$\begin{aligned} \left\Vert u\right\Vert _{{{L^\infty ([0,T]\times {\mathbb {R}^d})}}} \le \Lambda (b,\beta )T + \left\Vert v\right\Vert _{{L^\infty ({\mathbb {R}^d})}}. \end{aligned}$$

(13)

In the preceding theorem, the condition $v\in D_0$ is indispensable when regarding the existence and uniqueness of an analytically strong solutions to (FPKE) in $H^{-1}({\mathbb {R}^d})$. However, additionally assuming condition (11), it is possible to conclude from the previous result that for all $v \in L^1({\mathbb {R}^d})$ there exists a Schwartz-distributional solution u to (FPKE) with $\left. u\right| _{t=0}=v$. This was already remarked and the proof sketched in [6, Remark 2.2, Remark 3.3]. For the sake of completeness, we will give the proof. Moreover, we show that such u is bounded in space and time and has spatial Sobolev regularity if $v\in L^1({\mathbb {R}^d})\cap L^\infty ({\mathbb {R}^d})$. This will be used in Sect. 3.2.

Corollary 3.2

Assume that (H1), (H2), and (11) hold. Let $v \in L^1({\mathbb {R}^d})$. Then, there exists a Schwartz-distributional solution u to (FPKE) with $\left. u\right| _{t=0}=v$.

Furthermore, if $v \in L^1({\mathbb {R}^d})\cap L^\infty ({\mathbb {R}^d})$ and, in addition to the previous assumptions, $\Lambda (b,\beta )<\infty$ (see Theorem 3.1 for its definition) holds, then

$$\begin{aligned} u \in L^\infty ([0,T]\times {\mathbb {R}^d})\cap L^2([0,T];H^1({\mathbb {R}^d})) \end{aligned}$$

(14)

with $\left\Vert u\right\Vert _{{{L^\infty ([0,T]\times {\mathbb {R}^d})}}}\le \Lambda (b,\beta )T + \left\Vert v\right\Vert _{{L^\infty ({\mathbb {R}^d})}}$.

Proof

Let $v \in L^1({\mathbb {R}^d})$. Let $v^n \in C_c^\infty ({\mathbb {R}^d}) (\subset L^1({\mathbb {R}^d})\cap D_0)$, $n\in \mathbb {N}$, such that $v^n \rightarrow v$ in $L^1({\mathbb {R}^d})$, as $n\rightarrow \infty$. Then, due to (12)

$$\begin{aligned} \sup _{t\in [0,T]}\left\Vert u(t,v^n)-u(t,v^m)\right\Vert _{{L^1({\mathbb {R}^d})}} \le \left\Vert v^n-v^m\right\Vert _{{L^1({\mathbb {R}^d})}} \rightarrow 0,\ \ \text { as } n,m \rightarrow \infty . \end{aligned}$$

(15)

Consequently, for each $t\in [0,T]$ there exists $u(t) \in L^1({\mathbb {R}^d})$ such that $u(t,v^n) \rightarrow u(t)$ in $L^1({\mathbb {R}^d})$ with $u \in L^\infty ([0,T];L^1({\mathbb {R}^d}))$. Since $(u(t,v^n))_{n\in \mathbb {N}}$ is Cauchy in $L^1({\mathbb {R}^d})$ uniformly in $t\in [0,T]$, we can choose a subsequence $(n_k)_{k\in \mathbb {N}}\subset \mathbb {N}$ such that for all $t\in [0,T]$

$$\begin{aligned} u(t,v^{n_k}) \rightarrow u(t) \text { a.e. in } {\mathbb {R}^d}. \end{aligned}$$

Abbreviating $u^{k}(t):= u(t,v^{n_k})$, $t\in [0,T], k\in \mathbb {N}$, we have

$$\begin{aligned} \int _{\mathbb {R}^d}&u^{k}(t,x)\varphi (x)dx -\int _{\mathbb {R}^d}v^{k}(x) \varphi (x)dx\\&- \int _0^T \int _{\mathbb {R}^d}a(t,x,u^{k}(t,x))\Delta \varphi (x)u^{k}(x) dxdt \\&- \int _0^T \int _{\mathbb {R}^d}(b(t,x,u^{k}(x))\cdot \nabla \varphi (x))u^{k}(t,x)dxdt \\ \rightarrow \int _{\mathbb {R}^d}&u(t,x)\varphi (x)dx - \int _{\mathbb {R}^d}v(x) \varphi (x)dx \\&- \int _0^t \int _{\mathbb {R}^d}a(t,x,u(t,x))\Delta \varphi (x)u(t,x) dxdt \\&- \int _0^t \int _{\mathbb {R}^d}(b(t,x,u(t,x))\cdot \nabla \varphi (x))u(t,x)dxdt, \end{aligned}$$

as $k\rightarrow \infty$ for $\varphi \in C_c^\infty ({\mathbb {R}^d})$. Indeed, the i-th summand on the left-hand side convergence to the i-th summand on the right-hand side for the following reasons. Due to (15) and the choice of $v^n, n\in \mathbb {N}$, the convergence of the first and second summand is immediate. As a and b are bounded and, for t, x fixed, the functions $r\mapsto a(t,x,r)$ and $r\mapsto b(t,x,r)$ are continuous, the convergence of the third and fourth summand on the left-hand side can easily be seen by the generalised Lebesgue’s dominated convergence theorem. The existence of a narrowly continuous version of $(u_t)_{t\in [0,T]}$ is guaranteed by [13, Lemma 2.3]. This finishes the first part of the assertion.

Let $v \in L^1({\mathbb {R}^d})\cap L^\infty ({\mathbb {R}^d})$ from now on. Let $v^n, n\in \mathbb {N},$ as above with the additional property that $\left\Vert v^n\right\Vert _{{L^2({\mathbb {R}^d})}}\le \left\Vert v\right\Vert _{{L^2({\mathbb {R}^d})}}$, for all $n\in \mathbb {N}$. Let us first prove that $u \in L^2([0,T];H^1({\mathbb {R}^d}))$. For this, we recall that by [6, pp. 13+16] there exists a constant $C=C(\left\Vert a\right\Vert _{{L^\infty }},\left\Vert b\right\Vert _{{L^\infty }}, \gamma _0, T)>0$, such that for the above approximations $u(\cdot , v^n)$, $n\in \mathbb {N}$, of u in $C([0,T];L^1({\mathbb {R}^d}))$ we have

$$\begin{aligned}&\sup _{t\in [0,T]}\left\Vert u(t,v^n)\right\Vert _{{L^2({\mathbb {R}^d})}}^2 +\int _0^t \left\Vert \nabla _x u(s,v^n)\right\Vert _{{L^2({\mathbb {R}^d};{\mathbb {R}^d})}}^2ds\\&\quad \le C\left\Vert v^n\right\Vert _{{L^2({\mathbb {R}^d})}}^2\le C\left\Vert v\right\Vert _{{L^2({\mathbb {R}^d})}}^2. \end{aligned}$$

Hence, there exists $\bar{u} \in L^2([0,T];H^1({\mathbb {R}^d}))$ such that for a non-relabelled subsequence $u(\cdot , v^n) \rightarrow \bar{u}$ weakly in $L^2([0,T];H^1({\mathbb {R}^d}))$, as $n\rightarrow \infty$. Since $u(\cdot ,v^n) \rightarrow u$ in $C([0,T];L^1({\mathbb {R}^d}))$, $\bar{u}$ coincides with u.

Let us prove $u \in L^\infty ([0,T]\times {\mathbb {R}^d})$. Obviously, we can choose $v^n$, $n \in \mathbb {N}$, as above with the additional property $\left\Vert v^n\right\Vert _{{L^\infty ({\mathbb {R}^d})}} \le \left\Vert v\right\Vert _{{L^\infty ({\mathbb {R}^d})}}$. Consequently, we obtain from (13)

$$\begin{aligned} \left\Vert u(t,v^n)\right\Vert _{{{L^\infty ([0,T]\times {\mathbb {R}^d})}}} \le \Lambda (b,\beta )T + \left\Vert v\right\Vert _{{L^\infty ({\mathbb {R}^d})}}, \end{aligned}$$

for all $n\in \mathbb {N}$. By the Banach–Alaoglu theorem, $(u(t,v^n))_{n\in \mathbb {N}}$ has a weak-* convergent subsequence in $L^\infty ([0,T]\times {\mathbb {R}^d})$. With a similar argument as above, the limit necessarily coincides with u. This completes the second part of the assertion and thus the proof. $\square$

By Corollary 3.2, the same proof as for [6, Corollary 2.3], which is based on the superposition principle procedure from [3, Section 2], yields the following theorem.

Theorem 3.3

($P^{(u_t)}$-weak solution) Assume that (H1), (H2), and (11) hold. Let $v \in \mathcal {P}_0({\mathbb {R}^d})$. Then, there exists a $P^{(u_t)}$-weak solution (X, W) to (MVSDE) where u is the probability solution to (FPKE) with $\left. u\right| _{t=0}=v$ provided by Theorem 3.1.

5.2 Ingredient 2: $P^{(u_t)}$-pathwise uniqueness for (MVSDE)

Let u be the probability solution to (FPKE) with initial condition $v \in \mathcal {P}_0({\mathbb {R}^d})$ provided by Corollary 3.2. Recall that, by definition, $P^{(u_t)}$-weak solutions to (MVSDE) all have the same time marginal laws. In particular, all these solutions fulfil the following ordinary SDE

where $b^u(t,x):=b(t,x,u_t(x))$ and $a^u(t,x):=a(t,x,u_t(x))$, $(t,x)\in [0,T]\times {\mathbb {R}^d}$.

Our aim is to show $P^{(u_t)}$-pathwise uniqueness for (MVSDE), which is obviously equivalent to show $P^{(u_t)}$-pathwise uniqueness for (MVSDE). We will do so via a pathwise uniqueness result for SDEs extracted from the proof of [15, Theorem 1.1]. In view of the explicit time- and space-dependence of the coefficients in (MVSDE), we need to employ a chain rule for the composition of a Lipschitz function with a vector-valued Sobolev function. The following theorem is a simple consequence of [1, Corollary 3.2], for the proof see [11, Corollary 4.4.2].

Theorem 3.4

Let $k,m,n \in \mathbb {N}$ and let $O \subset \mathbb {R}^n$ be an open set. Let $p \in [1,\infty ]$, $g \in W^{1,p}(O;\mathbb {R}^m)\cap L^\infty (O;\mathbb {R}^m)$, and let $f:\mathbb {R}^m \rightarrow \mathbb {R}^k$ be a locally Lipschitz continuous function. Then $f\circ g \in L^\infty (O;\mathbb {R}^k)$, $D(f\circ g) \in L^p(O;\mathbb {R}^{m\times n})$ and, for a.e. $x\in O$, the restriction of the function f to the affine space

$$\begin{aligned} T^g_x := \{y \in \mathbb {R}^m : y=g(x) + D g(x)\cdot z, \text { for some } z\in \mathbb {R}^n\} \end{aligned}$$

is differentiable at g(x), and

$$\begin{aligned} D (f\circ g)(x) = D (\left. f\right| _{T^g_x})(g(x))D g (x) \text { for a.e. } x\in O. \end{aligned}$$

(16)

Remark 3.5

We would like to emphasise that in the situation of Theorem 3.4 it is clear that one can always estimate $|D (\left. {f}\right| _{T^g_x})(g(x))| \le CL$ for a.e. $x\in O$ for some constant $C=C(m,n)>0$, and therefore

$$\begin{aligned} |D(f\circ g)| \le CL |Dg| \text { a.e. in } O, \end{aligned}$$

where $L\ge 0$ is the Lipschitz constant of f restricted to the essential range of g.

Using the proof of Theorem 3.4 and Remark 3.5, we obtain the following corollary.

Corollary 3.6

Let $k,m,n \in \mathbb {N}$. Let $B \subset {\mathbb {R}^d}$ and $O\subset \mathbb {R}^n$ be open sets. Let $p,q \in [1,\infty )$ such that $p\le q$. Let $f:B\times \mathbb {R}^m \rightarrow \mathbb {R}^k$ such that for every bounded open set $U\subset \mathbb {R}^m$ there exists $h \in L^{\frac{qp}{q-p}}(B)$ (with the convention $\frac{qp}{q-p}=\infty$ if $p=q$) such that $\left. f(r,\cdot )\right| _{U}$ is Lipschitz continuous with Lipschitz constant $h(r)\ge 0$, for a.e. $r\in B$. Assume that $g\in L^q(B;W^{1,p}(O;\mathbb {R}^m))\cap L^\infty (B\times O;\mathbb {R}^m)$. Then, $(r\mapsto Df(r,g(r,\cdot ))) \in L^p(B;L^p(O;\mathbb {R}^m))$.

Proof

Let $U\subset {\mathbb {R}^d}$ be a bounded open set containing the smallest closed set K such that $g(\cdot ,\cdot )\in K$ a.e. Furthermore, let $h \in L^{\frac{qp}{q-p}}(B)$ and $N\in \mathcal {B}(B)$ with $\lambda ^d(N)=0$ such that for all $r\in N^\complement$, $\left. f(r,\cdot )\right| _{U}$ is Lipschitz continuous with Lipschitz constant $h(r)\ge 0$ and $g(r,\cdot ) \in W^{1,p}(O;\mathbb {R}^m)\cap L^\infty (O;\mathbb {R}^m)$. Then, according to the proof of Corollary 3.4 and Remark 3.5, for all $r\in N^\complement$, $f(r,g(r,\cdot ))$ is weakly differentiable and there exists a constant $C=C(d,m,n)>0$ such that

$$\begin{aligned} |Df(r,g(r,\cdot ))(x)| \le Ch(r)|Dg(r,\cdot )(x)| \text { for a.e. } x\in O. \end{aligned}$$

Hence, we may calculate

$$\begin{aligned} \int _B\int _O|Df(r,g(r,\cdot ))(x)|^p dxdr&\le \int _B (Ch(r))^p \left( \int _O |Dg(r,\cdot )(x)|^p dx\right) dr \\&\le C^p\left\Vert h\right\Vert _{{L^{\frac{qp}{q-p}}(B)}}^p\left\Vert Dg\right\Vert _{{{L^{q}(B;L^p(O;\mathbb {R}^m))}}}^p. \end{aligned}$$

$\square$

In order to prove Theorem 3.7, we need the following additional set of conditions.

• (H3) Assume that (11) holds, $\Lambda (b,\beta ) < \infty$ (see Theorem 3.1 for its definition). Furthermore, assume that for every bounded open set $U\subset {\mathbb {R}^d}\times \mathbb {R}$ there exists a nonnegative function $g \in L^2([0,T])$ such that for a.e. $t \in [0,T]$,

  $$\begin{aligned} |b(t,z)-b(t,\bar{z})| \le g(t) |z-\bar{z}|\ \ \ \forall z,\bar{z} \in U. \end{aligned}$$

  (17)

The following theorem provides the main result of this subsection.

Theorem 3.7

($P^{(u_t)}$-pathwise uniqueness) Assume that (H1), (H2) and (H3) hold. Let $v \in \mathcal {P}_0({\mathbb {R}^d})\cap L^\infty ({\mathbb {R}^d})$ and let u denote the corresponding probability solution to (FPKE) provided by Corollary 3.2. Let (X, W), (Y, W) be two $P^{(u_t)}$-weak solutions on a common stochastic basis $(\Omega ,\mathcal {F},\mathbb {P};(\mathcal {F}_t)_{t\in [0,T]})$ with respect to the same standard d-dimensional $(\mathcal {F}_t)$-Brownian motion W.

Then, $\sup _{t\in [0,T]} |X(t)-Y(t)| = 0\ \mathbb {P}\text {-a.s.}$

Proof

As explained before, we exactly need to show that $P^{(u_t)}$-pathwise uniqueness holds for (MVSDE). Therefore, we will check the assumptions of [10, Theorem 4.4] (extracted from the proof of [15, Theorem 2.5]). These will be implied by the following conditions together with (14).

• (a) $b^u \in L^\infty ([0,T]\times {\mathbb {R}^d};{\mathbb {R}^d})$, $a^u \in L^\infty ([0,T]\times {\mathbb {R}^d})$,

• (b) $D_x b^u \in L^2_{loc}([0,T]\times {\mathbb {R}^d};\mathbb {R}^{d\times d})$,

• (c) $\nabla _x \sqrt{a^u} \in L^2_{loc}([0,T]\times {\mathbb {R}^d};{\mathbb {R}^d})$.

Clearly, 3.2 is satisfied since a, b are bounded. Let us now consider 3.2. Fix an arbitrary $R>0$. By (14), we have

$$\begin{aligned} (t,x) \!\mapsto \! (x,u_t(x)) \!\in \! L^2([0,T];H^1(B_R(0);\mathbb {R}^d\!\times \!\mathbb {R}))\!\cap \! L^\infty ([0,T]\!\times \! B_R(0);\mathbb {R}^d\!\times \!\mathbb {R}). \end{aligned}$$

From (H3) and Corollary 3.6, it follows that $D_x b^u \in L^2_{loc}([0,T]\times B_R(0);\mathbb {R}^{d\times d})$. Let us now turn to 3.2. Fix an arbitrary $t\in [0,T]$. By (H1) and the usual chain rule for Sobolev functions, $a^u(t,\cdot )$ has a weak gradient $\nabla _x a^u(t,\cdot )$, with

$$\begin{aligned} \nabla _x a^u(t,x)=(\nabla _x a)(t,x,u(t,x))+(\partial _{r}a)(t,x,u(t,x))\nabla _{x}u(t,x), \end{aligned}$$

for almost every $x\in \mathbb {R}^d$. It is easy to extend the restricted square-root function $\sqrt{\ \cdot \ }_{|[\gamma _0, \infty )}$ to a function $h_{\sqrt{\ }}\in C^1(\mathbb {R})$ with $h_{\sqrt{\ }}(x) = \sqrt{x}, x\in [\gamma _0,\infty )$. Now, employing (8), we may calculate again with the help of the usual chain rule for Sobolev functions

$$\begin{aligned} \nabla _x \sqrt{a^u(t,x)}&=\nabla _x h_{\sqrt{\ }}(a^u(t,x)) =h_{\sqrt{\ }}'(a^u(t,x))\nabla _x(a^u(t,x)) \\&=\frac{(\nabla _x a)(t,x,u(t,x))+(\partial _{r}a)(t,x,u(t,x))\nabla _{x}u(t,x)}{2\sqrt{a(t,x,u(t,x))}}, \end{aligned}$$

for almost every $x\in \mathbb {R}^d$. Since $a \in C^1([0,T]\times {\mathbb {R}^d}\times \mathbb {R})$ and due to (14), for each $R>0$, we can find a constant $C>0$ such that

$$\begin{aligned}&\int _0^T\int _{B_R(0)}|\nabla _{x}\sqrt{a^u(t,x)}|^2dxdt \\&\qquad \le \frac{C}{4\gamma _0}\left( T\lambda ^d(B_R(0)) + \int _0^T\int |\nabla _x u(t,x)|^2dxdt \right) <\infty . \end{aligned}$$

This concludes the proof. $\square$

5.3 Application of the restricted Yamada–Watanabe theorem to (MVSDE)

Now, let us combine the ingredients from the previous two subsections and apply the restricted Yamada–Watanabe theorem for SDEs obtained in [10] to (MVSDE) in order to obtain the unique $P^{(u_t)}$-strong solution to (MVSDE). For the terminology of $P^{(u_t)}$-solutions and uniqueness, please consult page 5.

The following theorem can essentially be found in [10, Theorem 3.3].

Theorem 3.8

Let $u=(u_t)_{t\in [0,T]}$ be a probability solution to (FPKE). The following statements regarding (MVSDE) are equivalent.

• 1. There exists a $P^{(u_t)}$-weak solution and $P^{(u_t)}$-pathwise uniqueness holds.

• 2. There exists a unique $P^{(u_t)}$-strong solution to (MVSDE).

Gathering the results of Sects. 3.1 and 3.2, we conclude the main result of this article.

Theorem 3.9

(unique $P^{(u_t)}$-strong solution) Assume that (H1), (H2) and (H3) hold. Let $v \in \mathcal {P}_0({\mathbb {R}^d})\cap L^\infty ({\mathbb {R}^d})$, and let u denote the probability solution to (FPKE) with $\left. u\right| _{t=0}=v$ provided by Theorem 3.1. Then, there exists a unique $P^{(u_t)}$-strong solution to (MVSDE).