Fokker–Planck equations with terminal condition and related McKean probabilistic representation

Abstract

Usually Fokker–Planck type partial differential equations (PDEs) are well-posed if the initial condition is specified. In this paper, alternatively, we consider the inverse problem which consists in prescribing final data: in particular we give sufficient conditions for uniqueness. In the second part of the paper we provide a probabilistic representation of those PDEs in the form of a solution of a McKean type equation corresponding to the time-reversal dynamics of a diffusion process.

Appendix

For ease of reading the paper, we have postponed some technical results in this appendix. Sections A.1 and A.2 link the well-posedness of the PDE (1.1) to the well-posedness of the McKean SDE (1.3). In particular Proposition A.2 (resp. Corollary A.5) links the existence (resp. uniqueness) of the PDE (1.2) with the SDE (1.3). Sections A.3 and A.4 give the proofs of two technical Lemma (Lemma 3.11 and 4.6).

1.1 PDE with terminal condition and existence for the McKean SDE

We suppose that Property 1 is in force for a fixed $\mathcal {C} \subseteq \mathcal {P}\left( {\mathbb {R}}^d\right)$ and consider the Property 2 with respect to ${{\mathcal {C}}}$ and Properties 3 and 4 related to a given function $\mathbf{u}: [0,T] \rightarrow {{\mathcal {M}}}_+({\mathbb {R}}^d)$.

Property 2

• 1. $\mathbf{u}\left( 0\right)$ belongs to $\mathcal {C}$.

• 2. $\forall t \in ]0,T[$, $\mathbf{u}\left( t\right)$ admits a density with respect to the Lebesgue measure on ${\mathbb {R}}^d$ (denoted by $u\left( t,\cdot \right)$) and for all $t_0 > 0$ and all compact $K \subset {\mathbb {R}}^d$

  $$\begin{aligned} \int ^{T}_{t_0}\int _{K} \left| u\left( t,x\right) \right| ^2 + \sum ^{d}_{i=1}\sum ^{d}_{j=1}\left| \sigma _{ij}\left( t,x\right) \partial _{i}u\left( t,x\right) \right| ^2dxdt < \infty . \end{aligned}$$

  (1.1)

Remark A.1

Suppose Assumption 1 holds and let $\mathbf{u}$ be a measure-valued function verifying Property 2. Then (1.1) implies that the family of densities $u\left( T-t,\cdot \right) , t \in ]0,T[$ verifies condition (4.1) appearing in Definition 4.1. To show this, it suffices to check that for all $t_0 > 0$, all compact $K \subset {\mathbb {R}}^d$ and all $\left( i,j,k\right) \in [\![1,d]\!]^2\times [\![1,d]\!]$

$$\begin{aligned} \int ^{T}_{t_0}\int _{K}\left| \partial _j\left( \sigma _{ik}\left( s,y\right) \sigma _{jk}\left( s,y\right) u\left( s,y\right) \right) \right| dyds < \infty . \end{aligned}$$

(1.2)

The integrand appearing in (1.2) is well-defined. Indeed, in the sense of distributions we have

$$\begin{aligned} \partial _j\left( \sigma _{ik}\sigma _{jk}u\right) = \sigma _{ik}\sigma _{jk}\partial _ju + u\left( \sigma _{ik}\partial _j\sigma _{jk} + \sigma _{jk}\partial _j\sigma _{ik}\right) ; \end{aligned}$$

(1.3)

moreover the components of $\sigma$ are Lipschitz, so they are (together with their space derivatives) locally bounded. Also u and $\sigma _{jk}\partial _j u$ are square integrable by (1.1), which implies (1.2).

We introduce two other properties possibly fulfilled by a function $\mathbf{u}: [0,T] \rightarrow {{\mathcal {M}}}_+\left( {\mathbb {R}}^d\right)$.

Property 3

$\mathbf{u}(T)$ admits a density and belongs to ${{\mathcal {A}}}_1$.

Property 4

$\mathbf{u}(T)$ admits a density and belongs to ${{\mathcal {A}}}_2$.

We remark that Property 4 implies 3.

Proposition A.2

Suppose the validity of Assumptions 1 and Property 1 with respect to ${{\mathcal {C}}}$. We also suppose that the backward PDE (1.1) with terminal condition $\mu$ admits at least an ${{\mathcal {M}}}_+\left( {\mathbb {R}}^d\right)$-valued solution $\mathbf{u}$ in the sense of Definition 3.1, fulfilling Property 2 with respect to ${{\mathcal {C}}}$. Then (1.3) admits existence in law in ${{\mathcal {A}}}_{{{\mathcal {C}}}}$.

Moreover, if $\mathbf{u}$ fulfills Property 3 (resp. 4) then (1.3) admits existence in law in ${{\mathcal {A}}}_{{{\mathcal {C}}}}\cap {{\mathcal {A}}}_1$ (resp. strong existence in ${{\mathcal {A}}}_{{{\mathcal {C}}}}\cap {{\mathcal {A}}}_2$).

Proof

Let $\mathbf{u}$ be the function of the statement fulfilling Property 2, in particular $\mathbf{u}\left( 0\right)$ belongs to $\mathcal {C}$. We consider now a filtered probability space $\left( \Omega , {{\mathcal {F}}}, \right. \left. \left( \mathcal {F}_t\right) _{t\in [0,T]}, {\mathbb {P}}\right)$ equipped with an $\left( \mathcal {F}_t\right) _{t\in [0,T]}$-Brownian motion W. Let $X_0$ be a r.v. distributed according to $\mathbf{u}(0)$. Under Assumption 1, it is well-known that there is a solution X to

$$\begin{aligned} X_t = X_0 + \int ^{t}_{0}b\left( s,X_s\right) ds + \int ^{t}_{0}\sigma \left( s,X_s\right) dW_s, \ t \in [0,T]. \end{aligned}$$

(1.4)

Now, by Proposition 3.2, $t \mapsto \mathcal {L}\left( X_t\right)$ is a ${{\mathcal {P}}}\left( {\mathbb {R}}^d\right)$-valued solution of the PDE (3.2) in the sense of (3.3) with initial value $\mathbf{u}\left( 0\right) \in \mathcal {C}$. Then Property 1 for u implies

$$\begin{aligned} \mathcal {L}\left( X_t\right) = \mathbf{u}\left( t\right) , \ t \in [0,T], \end{aligned}$$

(1.5)

since $\mathbf{u}$ solves also the PDE (3.2) with initial value $\mathbf{u}\left( 0\right) \in \mathcal {C}$. This implies in particular that $\mathbf{u}$ is probability valued and that for all $t\in ]0,T[$, $X_t$ has $u\left( t,\cdot \right)$ as a density fulfilling condition (1.1).

Combining this observation with Assumption 1, Theorem 2.1 in [11] states that there exists a filtered probability space $\left( \Omega ,{{\mathcal {G}}}, ({{\mathcal {G}}}_t)_{t\in [0,T]}, {\mathbb {Q}}\right)$ equipped with some Brownian motion $\beta$ and a copy of ${\hat{X}}$ (still denoted by the same letter) such that ${\widehat{X}}$ fulfills the first line of the SDE (1.3) with $\beta$ and

$$\begin{aligned} \mathbf{p}\left( t\right) := \mathbf{u}\left( T-t\right) , \ t \in ]0,T[. \end{aligned}$$

(1.6)

Finally, existence in law for the SDE (1.3) in the sense of Definition 4.1 holds since $({\widehat{X}}, \mathbf{u}\left( T-\cdot \right) )$ is a solution of (1.3) on the same filtered probability space and the same Brownian motion above. This occurs in ${{\mathcal {A}}}_{{{\mathcal {C}}}}$ since $\mathcal {L}\left( {\widehat{X}}_T\right) \in {{\mathcal {C}}}$ thanks to equality (1.5) for $t = T$.

We discuss rapidly the moreover point.

• Suppose that u fulfills Property 3. Then $\mathbf{u}\left( T-\cdot \right)$ belongs to ${{\mathcal {A}}}_{{\mathcal {C}}}\cap {{\mathcal {A}}}_1$ and we also have existence in law in ${{\mathcal {A}}}_{{{\mathcal {C}}}}\cap {{\mathcal {A}}}_1$.

• Suppose that u fulfills Property 4. Then, taking into account (1.6), strong existence and pathwise uniqueness for the first line of (1.3) holds by classical arguments since the coefficients are locally Lipschitz with linear growth, see [24] Exercise (2.10), and Chapter IX.2 and [24], Th. 12. section V.12. of [25]. By Yamada-Watanabe theorem this implies uniqueness in law, which shows that $\mathbf{u}\left( T-\cdot \right)$ constitutes the marginal laws of the considered strong solutions. This concludes the proof of strong existence in ${{\mathcal {A}}}_{{{\mathcal {C}}}}\cap {{\mathcal {A}}}_2$ since $\mathbf{u}\left( T-\cdot \right)$ belongs to ${{\mathcal {A}}}_{{\mathcal {C}}}\cap {{\mathcal {A}}}_2$, by Property 4.

$\square$

Remark A.3

By (1.6), the second component p of the solution of (1.3) is given by $\mathbf{u}\left( T-\cdot \right) .$

1.2 PDE with terminal condition and uniqueness for the McKean SDE

In this subsection we discuss some questions related to uniqueness for the PDE (1.3). We consider the following Property related to a given subset $\mathcal {C}$ of $\mathcal {P}\left( {\mathbb {R}}^d\right)$.

Property 5

The PDE (1.1) with terminal condition $\mu$ admits at most a ${{\mathcal {P}}}\left( {\mathbb {R}}^d\right)$-valued solution $\mathbf{u}$ in the sense of Definition 3.1 such that $\mathbf{u}\left( 0\right)$ belongs to $\mathcal {C}$.

We recall that Sect. 3.2 provides a typical class ${{\mathcal {C}}}$, where Property 5 holds under some conditions.

Proposition A.4

Suppose the validity of Property 5 with respect to ${{\mathcal {C}}}$ and suppose $b,\sigma$ to be locally bounded.

Let $\left( Y^i, \mathbf{p}^i\right) , \ i \in \{1,2\}$ be two solutions of the SDE (1.3) in the sense of Definition 4.1 such that $\mathbf{p}^1\left( T\right) , \mathbf{p}^2\left( T\right)$ belong to ${{\mathcal {C}}}$. Then,

$$\begin{aligned} \mathbf{p}^1 = \mathbf{p}^2. \end{aligned}$$

Proof

Proposition 4.3 shows that $\mathbf{p}^1\left( T-\cdot \right) , \mathbf{p}^2\left( T-\cdot \right)$ are ${{\mathcal {P}}}\left( {\mathbb {R}}^d\right)$-valued solutions of the PDE (1.1) in the sense of Definition 3.1 with terminal value $\mu$. Property 5 gives the result since $\mathbf{p}^1\left( T\right) , \mathbf{p}^2\left( T\right)$ belong to ${{\mathcal {C}}}$. $\square$

As a corollary, we establish some consequences about uniqueness in law and pathwise uniqueness results for the SDE (1.3) in the classes ${{\mathcal {A}}}_1$ and ${{\mathcal {A}}}_2$.

Corollary A.5

Suppose the validity of Property 5 with respect to ${{\mathcal {C}}}$.

Then, the following results hold.

• 1. If b is locally bounded, $\sigma$ is continuous and if the non-degeneracy Assumption 4 holds then the SDE (1.3) admits uniqueness in law in ${{\mathcal {A}}}_{{{\mathcal {C}}}}\cap {{\mathcal {A}}}_1$.

• 2. If $b,\sigma$ are locally Lipschitz with linear growth in space, then (1.3) admits pathwise uniqueness in ${{\mathcal {A}}}_{{{\mathcal {C}}}}\cap {{\mathcal {A}}}_2$.

Proof

If $\left( Y,\mathbf{p}\right)$ is a solution of the SDE (1.3) and is such that $\mathbf{p}\left( T\right)$ belongs to ${{\mathcal {C}}}$, then by Proposition A.4$\mathbf{p}$ is determined by $\mu = \mathcal {L}\left( Y_0\right)$.

To show that item 1. (resp. 2.) holds, it suffices to show that the classical SDE

$$\begin{aligned} dX_t = \left( b(t,X_t; \mathbf{p}_t) - {\widehat{b}}(t,X_t)\right) dt + {\widehat{\sigma }}\left( t,X_t\right) dW_t, \ t \in [0,T[, \end{aligned}$$

(1.7)

where b was defined in (4.4) and W an m-dimensional Brownian motion, admits uniqueness in law (resp. pathwise uniqueness). The mentioned uniqueness in law is a consequence of Theorem 10.1.3 in [27] and pathwise uniqueness holds by [24] Exercise (2.10), and Chapter IX.2 and [25] Th. 12. Section V.12. $\square$

1.3 Proof of Lemma 3.11

Proof

For a given $\left( x,y\right) \in {\mathbb {R}}^d \times {\mathbb {R}}^d$ we set

$$\begin{aligned} Z^{x,y}_t := X^{y}_t - X^{x} _t, t\in [0,T]. \end{aligned}$$

We have

$$\begin{aligned} Z^{x,y}_t = y-x + \int ^{t}_{0}B^{x,y}_r Z^{x,y}_rdr +\sum ^{m}_{j=1} \int ^{t}_{0}C^{x,y,j}_r Z^{x,y}_rdW^j_r, \ t\in [0,T], \end{aligned}$$

(1.8)

with, for all $r\in [0,T]$

$$\begin{aligned} B^{x,y}_r:= & {} \int ^{1}_{0}Jb\left( r,aX^{y}_r+(1-a)X^{x}_r\right) da,\\ C^{x,y,j}_r:= & {} \int ^{1}_{0}J\sigma _{.j}\left( r,aX^{y}_r+(1-a)X^{x}_r\right) da, \forall \ j \in [\![1,m]\!]. \end{aligned}$$

By the classical existence and uniqueness theorem for SDEs with Lipschitz coefficients we know that

$$\begin{aligned} {\mathbb {E}}(\sup _{s \le T} \left| X^{z}_s\right| ^2) < \infty , \end{aligned}$$

(1.9)

for all $z \in {\mathbb {R}}^d$. This implies

$$\begin{aligned} {\mathbb {E}}(\sup _{t\in [0,T]}\left| Z^{x,y}_t \right| ^2) < \infty . \end{aligned}$$

(1.10)

Now, Itô’s formula gives, for all $t \in [0,T]$

$$\begin{aligned} \left| Z^{x,y}_t\right| ^2= & {} \left| y-x\right| ^2 + 2\int ^{t}_{0}\left<B^{x,y}_rZ^{x,y}_r,Z^{x,y}_r\right>dr \nonumber \\&+ \sum ^{d}_{j=1}\int ^{t}_{0}\left| C^{x,y,j}_rZ^{x,y}_r\right| ^2dr + 2\sum ^{d}_{i=1}M^{x,y,i}_t, \end{aligned}$$

(1.11)

where, for a given $i \in [\![1,d]\!]$, $M^{x,y,i}$ denotes the local martingale $\int ^{\cdot }_{0}Z^{x,y,i}_s\sum ^{d}_{j=1}\left( C^{x,y,j}_sZ^{x,y}_s\right) _idW^j_s$.

Consequently, for all $i \in [\![1,d]\!]$, we have

$$\begin{aligned} \sqrt{\left[ M^{x,y,i}\right] _T}&{}= \sqrt{\sum ^{d}_{j=1} \int ^{T}_{0}\left( Z^{x,y,i}_r\right) ^2\left( C^{x,y,j}_r Z^{x,y}_r\right) ^2_idr}, \nonumber \\&{}\le \sqrt{\sum ^{d}_{j=1} \int ^{T}_{0}\left| C^{x,y,j}_r Z^{x,y}_r\right| ^2\left| Z^{x,y}_r\right| ^2dr}, \nonumber \\&{}\le \sqrt{T\sum ^{d}_{j=1} \left( K^{\sigma ,j}\right) ^2}\sup _{r\in [0,T]}\left| Z^{x,y}_r\right| ^2. \end{aligned}$$

(1.12)

By the latter inequality and (1.10), we know that ${\mathbb {E}}\left( [M^{x,y,i}]_T^{\frac{1}{2}}\right) < \infty$, so for all $i \in [\![1,d]\!]$, $M^{x,y,i}$ is a true martingale. Taking expectation in identity (1.11), we obtain

$$\begin{aligned} {\mathbb {E}}\left( \left| Z^{x,y}_t\right| ^2\right) = \left| y-x\right| ^2 + \int ^{t}_{0}{\mathbb {E}}\left( 2\left<B^{x,y}_rZ^{x,y}_r,Z^{x,y}_r\right> + \sum ^{d}_{k=1}\left| C^{x,y,k}_rZ^{x,y}_r\right| ^2\right) dr. \end{aligned}$$

Hence, thanks to Cauchy-Schwarz inequality and to the definition of $K^b$ and $K^{\sigma ,j}$ for all $j \in [\![1,d]\!]$

$$\begin{aligned} {\mathbb {E}}\left( \left| Z^{x,y}_t\right| ^2\right) \le \left| y-x\right| ^2 + K \int ^{t}_{0}{\mathbb {E}}\left( \left| Z^{x,y}_r\right| ^2\right) dr \end{aligned}$$

and we conclude via Gronwall’s Lemma. $\square$

1.4 Proof of Lemma 4.6

Proof

Let $\nu \in {{\mathcal {P}}}\left( {\mathbb {R}}^d\right)$. For each given $t \in [0,T]$, we denote by $G_t$ the differential operator such that for all $f\in \mathcal {C}^2\left( {\mathbb {R}}^d\right)$

$$\begin{aligned} G_tf = \frac{1}{2}\sum ^{d}_{i,j=1}\partial _{ij}\left( \Sigma _{ij}\left( t,\cdot \right) f\right) - \sum ^{d}_{i=1}\partial _i\left( b_i\left( t,\cdot \right) f\right) . \end{aligned}$$

Assumption 6 implies that for a given $f \in \mathcal {C}^{2}\left( {\mathbb {R}}^d\right)$, $G_tf$ can be rewritten in the two following ways:

$$\begin{aligned} G_tf = \frac{1}{2}\sum ^{d}_{i,j=1}\Sigma _{ij}(t,\cdot )\partial _{ij}f + \sum ^{d}_{i=1}(\sum ^{d}_{j=1}\partial _i\Sigma _{ij}(t,\cdot )- b_i(t, \cdot ))\partial _if + c^1(t,\cdot )f,\nonumber \\ \end{aligned}$$

(1.13)

with

$$\begin{aligned} c^1 : (t,x) \mapsto \frac{1}{2}\sum ^{d}_{i,j=1}\partial _{ij}\Sigma _{ij}(t,x) - \sum ^{d}_{i=1}\partial _ib_i(t,x). \end{aligned}$$

$$\begin{aligned} G_tf = \frac{1}{2}\sum ^{d}_{i,j=1}\partial _j (\partial _i\Sigma _{ij} (t,\cdot )f + \Sigma _{ij}(t,\cdot )\partial _if - \sum ^{d}_{i=1}b_i(t,\cdot )\partial _if) - \sum ^{d}_{i=1}\partial _ib_i(t,\cdot )f.\nonumber \\ \end{aligned}$$

(1.14)

On the one hand, combining identity (1.13) with Assumptions 2, 3, 4 and 6, there exists a fundamental solution $\Gamma$ (in the sense of Definition stated in Section 1. p.3 of [10]) of $\partial _tu = G_tu$, thanks to Theorem 10. Section 6 Chap. 1. in the same reference. Furthermore, there exists $C_1,C_2 > 0$ such that for all $i \in [\![1,d]\!]$, $x,\xi \in {\mathbb {R}}^d$, $\tau \in [0,T]$, $t > \tau$,

$$\begin{aligned} \left| \Gamma \left( x,t,\xi ,\tau \right) \right|\le & {} C_1\left( t-\tau \right) ^{-\frac{d}{2}}\exp \left( -\frac{C_2\left| x-\xi \right| ^2}{4\left( t-\tau \right) }\right) , \end{aligned}$$

(1.15)

$$\begin{aligned} \left| \partial _{x_i}\Gamma \left( x,t,\xi ,\tau \right) \right|\le & {} C_1\left( t-\tau \right) ^{-\frac{d+1}{2}}\exp \left( -\frac{C_2\left| x-\xi \right| ^2}{4\left( t-\tau \right) }\right) , \end{aligned}$$

(1.16)

thanks to identities (6.12), (6.13) in Section 6 Chap. 1 in [10].

On the other hand, combining Identity (1.14) with Assumption 6, there exists a so called weak fundamental solution $\Theta$ of $\partial _tu = G_tu$ thanks to Theorem 5 in [1]. In addition, there exists $K_1,K_2,K_3 > 0$ such that for almost every $x,\xi \in {\mathbb {R}}^d$ , $\tau \in [0,T]$, $t \ge \tau$

$$\begin{aligned}&\frac{1}{K_1}\left( t-\tau \right) ^{-\frac{d}{2}}\exp \left( -\frac{K_2\left| x-\xi \right| ^2}{4\left( t-\tau \right) }\right) \nonumber \\&\quad \le \Theta \left( x,t,\xi ,\tau \right) \le K_1\left( t-\tau \right) ^{-\frac{d}{2}}\exp \left( -\frac{K_3\left| x-\xi \right| ^2}{4\left( t-\tau \right) }\right) , \end{aligned}$$

(1.17)

thanks to point (ii) of Theorem 10 in [1].

Our goal is now to show that $\Gamma$ and $\Theta$ coincide. To this end, we adapt the argument developed at the beginning of Section 7 in [1]. Fix a function H from $[0,T]\times {\mathbb {R}}^d$ belonging to $\mathcal {C}^\infty _c\left( [0,T]\times {\mathbb {R}}^d\right)$. Identity (7.6) in Theorem 12 Chap 1. Section 1. of [10] implies in particular that the function

$$\begin{aligned} u: \left( t,x\right) \mapsto \int ^{t}_{0}\int _{{\mathbb {R}}^d}\Gamma \left( x,t,\xi ,\tau \right) H\left( \tau ,\xi \right) d\xi d\tau , \end{aligned}$$

is continuously differentiable in time, two times continuously differentiable in space and is a solution of the Cauchy problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu\left( t,x\right) = G_tu\left( t,x\right) + H\left( t,x\right) , \ \left( t,x\right) \in ]0,T]\times {\mathbb {R}}^d, \\ u\left( 0,\cdot \right) = 0. \end{array}\right. } \end{aligned}$$

(1.18)

It is consequently also a weak (i.e. distributional) solution of (1.18), which belongs to ${{\mathcal {E}}}^2(]0,T]\times {\mathbb {R}}^d)$ (see definition of that space in [1]) since u is bounded thanks to inequality (1.15) and the fact that H is bounded. Then, point (ii) of Theorem 5 in [1] says that

$$\begin{aligned} (t,x) \mapsto \int ^{t}_{0}\int _{{\mathbb {R}}^d}\Theta \left( x,t,\xi ,\tau \right) H\left( \tau ,\xi \right) d\xi d\tau \end{aligned}$$

is the unique weak solution in ${{\mathcal {E}}}^2(]0,T]\times {\mathbb {R}}^d)$ of (1.18). This implies that for every $(t,x) \in ]0,T] \times {\mathbb {R}}^d$ we have

$$\begin{aligned} \int ^{t}_{0}\int _{{\mathbb {R}}^d}\left( \Gamma - \Theta \right) \left( x,t,\xi ,\tau \right) H\left( \tau ,\xi \right) d\xi d\tau = 0. \end{aligned}$$

Point (i) of Theorem 5 in [1] (resp inequality (1.15)) implies that $\Theta$ (resp. $\Gamma$) belongs to $L^{p}\left( ]0,T]\times {\mathbb {R}}^d\right)$ as a function of $(\xi ,\tau )$, for an arbitrary $p \ge d + 2$. Then, we conclude that for all $\left( t,x\right) \in ]0,T] \times {\mathbb {R}}^d$,

$$\begin{aligned} \Theta \left( x,t,\xi ,\tau \right) = \Gamma \left( x,t,\xi ,\tau \right) , \ d\xi d\tau a.e. \end{aligned}$$

(1.19)

for all $(\tau ,\xi ) \in [0,t[ \times {\mathbb {R}}^d$. This happens by density of $\mathcal {C}^\infty _c\left( [0,T]\times {\mathbb {R}}^d\right)$ in $L^{q}\left( ]0,T]\times {\mathbb {R}}^d\right)$, q being the conjugate of p.

This, together with (1.17) and the fact that $\Gamma$ is continuous in $(\tau ,\xi )$ implies that (1.17) holds for all $(\tau ,\xi ) \in [0,t[ \times {\mathbb {R}}^d$ and therefore

$$\begin{aligned}&\frac{1}{K_1}\left( t-\tau \right) ^{-\frac{d}{2}}\exp \left( -\frac{K_2\left| x-\xi \right| ^2}{4\left( t-\tau \right) }\right) \le \Gamma \left( x,t,\xi ,\tau \right) \nonumber \\&\qquad \le K_1\left( t-\tau \right) ^{-\frac{d}{2}}\exp \left( -\frac{K_3\left| x-\xi \right| ^2}{4\left( t-\tau \right) }\right) . \end{aligned}$$

(1.20)

We introduce

$$\begin{aligned} q_{t} := x \mapsto \int _{{\mathbb {R}}^d} \Gamma \left( x,t,\xi ,0\right) \nu \left( d\xi \right) . \end{aligned}$$

By (1.20), with $\tau = 0$ we get

$$\begin{aligned} q_{t}\left( x\right) \ge \frac{1}{K_1}t^{-\frac{d}{2}}\int _{{\mathbb {R}}^d}\exp \left( -\frac{K_2\left| x-\xi \right| ^2}{4t}\right) \nu \left( d\xi \right) . \end{aligned}$$

(1.21)

We denote now by $\mathbf{v}^\nu$ the measure-valued mapping such that $\mathbf{v}^\nu \left( 0,\cdot \right) = \nu$ and for all $t \in ]0,T]$, $\mathbf{v}^\nu \left( t\right)$ has density $q_{t}$ with respect to the Lebesgue measure on ${\mathbb {R}}^d$. We want to show that $\mathbf{v}^\nu$ is a solution of the PDE (3.2) with initial value $\nu$ to conclude $\mathbf{u}^\nu = \mathbf{v}^\nu$ thanks to the validity of Property 1 because of Lemma 3.4 and Assumptions 2, 3 and 4.

To this end, we remark that the definition of a fundamental solution for $\partial _tu = G_tu$ says that u is a $C^{1,2}$ solution and consequently also a solution in the sense of distributions. In particular for all $\phi \in \mathcal {C}^{\infty }_c\left( {\mathbb {R}}^d\right)$, for all $t \ge \epsilon > 0$

$$\begin{aligned} \int _{{\mathbb {R}}^d}\phi \left( x\right) \mathbf{v}^\nu \left( t\right) \left( dx\right) = \int _{{\mathbb {R}}^d}\phi \left( x\right) \mathbf{v}^\nu \left( \epsilon \right) \left( dx\right) + \int ^{t}_{\epsilon }\int _{{\mathbb {R}}^d}L_s\phi \left( x\right) \mathbf{v}^\nu \left( s\right) \left( dx\right) ds.\nonumber \\ \end{aligned}$$

(1.22)

To conclude, it remains to send $\epsilon$ to $0+$. Theorem 15 section 8. Chap 1. and point (ii) of the definition stated p. 27 in [10] imply in particular that for all $\phi \in \mathcal {C}^\infty _c\left( {\mathbb {R}}^d\right)$, $\xi \in {\mathbb {R}}^d$,

$$\begin{aligned} \int _{{\mathbb {R}}^d}\Gamma \left( x,\epsilon ,\xi ,0\right) \phi \left( x\right) dx \underset{\epsilon \rightarrow 0+}{\longrightarrow } \phi \left( \xi \right) . \end{aligned}$$

Fix now $\phi \in \mathcal {C}^\infty _c\left( {\mathbb {R}}^d\right)$. In particular thanks to Fubini’s theorem, (1.17) and Lebesgue’s dominated convergence theorem we have

$$\begin{aligned} \int _{{\mathbb {R}}^d}\phi \left( x\right) \mathbf{v}^\nu \left( \epsilon \right) \left( dx\right)&{}= \int _{{\mathbb {R}}^d}\phi \left( x\right) \int _{{\mathbb {R}}^d}\Gamma \left( x,\epsilon ,\xi ,0\right) \nu \left( d\xi \right) dx \\&{} = \int _{{\mathbb {R}}^d}\int _{{\mathbb {R}}^d}\Gamma \left( x,\epsilon ,\xi ,0\right) \phi \left( x\right) dx\nu \left( d\xi \right) \\&{} \underset{\epsilon \rightarrow 0+}{\longrightarrow } \int _{{\mathbb {R}}^d}\phi \left( \xi \right) \nu \left( d\xi \right) . \end{aligned}$$

By (1.22) $\mathbf{v}^\nu$ is a solution of the PDE (3.2) and consequently $\mathbf{u}^\nu = \mathbf{v}^\nu$, so that, for every $t \in ]0,T]$, $\mathbf{u}^\nu \left( t\right)$ admits $u^\nu (t,\cdot ) = q_{t}$ for density with respect to the Lebesgue measure on ${\mathbb {R}}^d$. Now, integrating the inequalities (1.15), (1.16) with respect to $\nu$ and combining this with inequality (1.21), we obtain the existence of $K_1,K_2,C_1,C_2 > 0$ such that for all $t \in ]0,T]$, for all $x \in {\mathbb {R}}^d$, for all $i \in [\![1,d]\!]$

$$\begin{aligned} \frac{1}{K_1}t^{-\frac{d}{2}}\int _{{\mathbb {R}}^d} \exp \left( -\frac{K_2\left| x-\xi \right| ^2}{4t}\right) \nu \left( d\xi \right) \le u^\nu \left( t,x\right) \le K_1t^{-\frac{d}{2}}, \end{aligned}$$

$$\begin{aligned} \left| \partial _iu^\nu \left( t,x\right) \right| \le C_1t^{-\frac{d+1}{2}}. \end{aligned}$$

Consequently, the upper bounds in (4.5) and (4.6) hold. Concerning the lower bound in (4.5), let I be a compact subset of ${\mathbb {R}}^d$ such that $\nu (I) > 0$, the result follows since $(t,x,\xi ) \mapsto \exp \left( -\frac{K_2\left| x-\xi \right| ^2}{4t}\right)$ is strictly positive, continuous and therefore lower bounded by a strictly positive constant on $K\times I$ for each compact K of $]0,T]\times {\mathbb {R}}^d$. $\square$