Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails

Abstract

We study the non-negative solution $u=u(x,t)$ to the Cauchy problem for the parabolic equation $\partial _t u={\varDelta }u+\xi u$ on $\mathbb Z^d\times [0,\infty )$ with initial data $u(x,0)=\mathsf{1}_0(x)$. Here ${\varDelta }$ is the discrete Laplacian on $\mathbb Z^d$ and $\xi =(\xi (z))_{z\in \mathbb Z^d}$ is an i.i.d. random field with doubly-exponential upper tails. We prove that, for large t and with large probability, most of the total mass $U(t):=\sum _x u(x,t)$ of the solution resides in a bounded neighborhood of a site $Z_t$ that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian ${\varDelta }+\xi$ and the distance to the origin. The processes $t\mapsto Z_t$ and $t \mapsto \tfrac{1}{t} \log U(t)$ are shown to converge in distribution under suitable scaling of space and time. Aging results for $Z_t$, as well as for the solution to the parabolic problem, are also established. The proof uses the characterization of eigenvalue order statistics for ${\varDelta }+\xi$ in large sets recently proved by the first two authors.

Appendices

Appendix: A tail estimate

In this section, we prove (7.18) for $\widehat{Y}_t$ given by (7.21) using an approach from [7]. We will strongly rely on Assumption 2.1. The first step concerns the asymptotic for the upper tail of $\xi$.

Lemma 12.1

For any $\varepsilon >0$, there exists $t_0>0$ such that, for all $t\ge t_0$,

$$\begin{aligned} t^d \,\text{ Prob } \left( \xi (0) > \widehat{a}_t + s d_t \right) \le \mathrm{e}^{-s (1- \varepsilon )} \qquad \forall s \ge 0. \end{aligned}$$

(12.1)

Proof

Recall the definition of F in (2.1). Note that $t^d = \exp (\mathrm{e}^{F(\widehat{a}_t)})$ to write

$$\begin{aligned}&-\ln \left\{ t^d \text{ Prob } \left( \xi (0) > \widehat{a}_t + s d_t \right) \right\} \nonumber \\&\quad = \mathrm{e}^{F(\widehat{a}_t)} \left( \mathrm{e}^{F(\widehat{a}_t + s d_t)- F(\widehat{a}_t)} - 1 \right) \nonumber \\&\quad \ge \mathrm{e}^{F(\widehat{a}_t)} \left\{ F(\widehat{a}_t + s d_t)- F(\widehat{a}_t) \right\} , \end{aligned}$$

(12.2)

where in the last inequality we used $\mathrm{e}^x-1 \ge x$. Using (2.2) and the Mean Value Theorem, we obtain $F(\widehat{a}_t + sd_t)-F(\widehat{a}_t) \ge s d_t (1-\varepsilon ) / \rho$ for all $s \ge 0$ if t is large enough. Since $d_t = \rho \mathrm{e}^{-F(\widehat{a}_t)}$, (12.1) follows from (12.2). $\square$

Lemma 12.1 will allow us to reduce the sum in (7.18) to $|x| \le 6 d \theta t / d_t$.

Corollary 2

For any $\eta \in \mathbb R$, $\theta \in (0,\infty )$,

(12.3)

Proof

Recall that $\max _{x \in B_{\widehat{N}_t}} \xi (x) \ge \lambda ^{{\scriptscriptstyle {({1}})}}_{B_{\widehat{N}_t}}$ by (5.5). Using $a_t = \widehat{a}_t - \chi + o(1)$ and $\chi \le 2d$, we obtain, for each $L \in \mathbb N$,

(12.4)

by Lemma 12.1 and (2.6). Since the last integral converges to 0 as $L \rightarrow \infty$, the claim (12.3) follows. $\square$

To control the sum in (7.18) with $|x| \le t 6 d \theta / d_t$, we will use the following lemma.

Lemma 12.2

There exist $c_0, \varepsilon >0$ such that, for all large enough t and all $s \ge 0$,

$$\begin{aligned} \frac{t^d}{(2\widehat{N}_t)^d} \text{ Prob } \left( \widehat{Y}_t(0) > s \right) \le 4 \, \mathrm{e}^{-c_0 s} + t^{-\varepsilon }. \end{aligned}$$

(12.5)

Before we prove Lemma 12.2, let us finish the proof of (7.18).

Proof of (7.18)

By Corollary 2, we may restrict the sum over $|x| \le t 6 d \theta / d_t$. Fix $\eta \in \mathbb R$. Taking $n\ge \theta |\eta |$, if $|x| \ge nt$ then $|x|/(\theta t) + \eta \ge 0$. Thus we may bound, by Lemma 12.2,

(12.6)

for a constant $c_2>0$ and all large enough t. To conclude (7.17), note that the right-hand side of (12.6) converges as $t \rightarrow \infty$ to

$$\begin{aligned} 4 \int _{|z| \ge n} \mathrm{e}^{-c_0 \left( \frac{|z|}{\theta } + \eta \right) } \mathrm{d}\,z, \end{aligned}$$

(12.7)

which converges itself to 0 as $n \rightarrow \infty$. $\square$

The remainder of this section is dedicated to the proof of Lemma 12.2. Note that, by Assumption 2.1, $\xi (0)$ has a density f with respect to Lebesgue measure given by

$$\begin{aligned} f(r) = \left\{ \begin{array}{ll} F'(r) \exp \left\{ F(r)-\mathrm{e}^{F(r)} \right\} , &{} r > {\text {essinf}}\,\xi (0),\\ 0 &{} \text {otherwise.} \end{array}\right. \end{aligned}$$

(12.8)

The following bound holds for f.

Lemma 12.3

Fix a finite ${\varLambda }\subset \mathbb Z^d$ and two functions $\alpha , \varphi :{\varLambda }\rightarrow \mathbb R$. Then, as $t \rightarrow \infty$,

$$\begin{aligned}&\prod _{x \in {\varLambda }} \frac{f(\widehat{a}_t + \varphi (x) + \alpha (x) d_t)}{f(\widehat{a}_t + \varphi (x))} \nonumber \\&\quad \le \exp \left\{ -(1+ o(1))\sum _{x \in {\varLambda }} \alpha (x) \mathrm{e}^{\frac{\varphi (x)}{\rho }} + o(1) \mathcal L_{\varLambda }(\varphi )\right\} \end{aligned}$$

(12.9)

where $\mathcal L_{\varLambda }(\varphi )$ is as in (5.9). If $\alpha (x) \ge 0$ and $|\varphi (x)| \le M$, then o(1) only depends on M. If $|\alpha (x)| \vee |\varphi (x)| \le M$, then equality holds in (12.9) with o(1) only depending on M.

Proof

This follows by the arguments in the proof of [7, Lemma 7.5]. $\square$

Fix now $c_0 := \tfrac{1}{4} \mathrm{e}^{-2(d+1)/\rho }$; this will the constant appearing in (12.2). The following corollary is a convenient rephrasing of (12.9):

Corollary 3

There exists $t_0 >0$ such that, for all $t \ge t_0$, $s \ge 0$, ${\varLambda }\subset \mathbb Z^d$ and all $\alpha , \varphi :{\varLambda }\rightarrow \mathbb R$ with $\alpha (x) \ge 0$, $-2(d+1) \le \varphi (x) \le 1$,

$$\begin{aligned} \prod _{x \in {\varLambda }} \frac{f(\widehat{a}_t + \varphi (x) + s \alpha (x)d_t)}{f(\widehat{a}_t + \varphi (x))} \le \exp \left\{ - 2 c_0 s \sum _{x \in {\varLambda }} \alpha (x) + \mathcal L_{{\varLambda }}(\varphi ) \right\} . \end{aligned}$$

(12.10)

We can now prove Lemma 12.2:

Proof of Lemma 12.2

For $t >0$ such that $a_t > {\text {essinf}}\,\xi (0)+1$, define the continuous map

$$\begin{aligned} \mathcal F_{t,s}(r) := \left\{ \begin{array}{ll} r &{}\quad \text { if } r \le a_t-1,\\ r - s d_t &{}\quad \text { if } r \ge a_t + s d_t,\\ \text {linear, } &{}\quad \text { otherwise.} \end{array}\right. \end{aligned}$$

(12.11)

Then $\mathcal F_{t,s}$ is bijective with inverse

$$\begin{aligned} \mathcal F^{-1}_{t,s}(r) := \left\{ \begin{array}{ll} r &{}\quad \text { if } r \le a_t-1,\\ r + s d_t &{}\quad \text { if } r \ge a_t,\\ \text {linear, } &{}\quad \text { otherwise.} \end{array}\right. \end{aligned}$$

(12.12)

Let $\xi _{t,s}(x) := \mathcal F_{t,s}(\xi (x))$. Then $\xi _{t,s}(x)$ has a density with respect to $\xi (x)$ given by

$$\begin{aligned} \frac{\mathrm{d}\,\xi _{t,s}(x)}{\mathrm{d}\,\xi (x)}(r) = \left\{ \begin{array}{ll} 1 &{} \text { if } r \le a_t -1, \\ \left( 1 + s d_t \right) ^{\mathsf{1}\{r < a_t\}} \frac{f(\mathcal F^{-1}_{t,s}(r))}{f(r)} &{} \text { otherwise.} \end{array}\right. \end{aligned}$$

(12.13)

Recalling that $\lambda ^{{\scriptscriptstyle {({1}})}}_{B_R}(\xi )$ denotes the principal Dirichlet eigenvalue of ${\varDelta }+ \xi$ in $B_R$, define

$$\begin{aligned} G_{t,s} := \left\{ \xi :\, \lambda ^{{\scriptscriptstyle {({1}})}}_{B_{R_t}}(\xi ) > a_t + s d_t, \, \mathcal L_{B_{R_t}}(\xi -\widehat{a}_t) \le \ln 2, \, \max _{x \in B_{R_t}} \xi (x) \le \widehat{a}_t + 1 \right\} . \end{aligned}$$

(12.14)

Since $\xi (x)-s d_t \le \xi _{t,s}(x) \le \xi (x)$, $\xi \in G_{t,s}$ implies $\xi _{t,s} \in G_{t,0}$. Write

(12.15)

where E denotes expectation with respect to environment law $\text{ Prob }$. Bound the middle term in (12.15) by

$$\begin{aligned} \left( 1 + s d_t \right) ^{\left| B_{R_t}\right| } \le \mathrm{e}^{s d_t (2R_t+1)^d} \le \mathrm{e}^{s c_0} \end{aligned}$$

(12.16)

for large t by (5.11). For the product term, define $\varphi (x) := \xi (x) - \widehat{a}_t \le 1$ on $\mathcal G_{t,0}$, and $\alpha (x) \in [0,1]$ by the equation $\xi (x) + s d_t \alpha (x) = \mathcal F^{-1}_{t,s}(\xi (x))$. Note that, if $\alpha (x) \ne 0$, then $\varphi (x) > a_t - 1 -\widehat{a}_t \ge -2(d+1)$ for large t; thus, by Corollary 3,

$$\begin{aligned} \prod _{x \in B_{R_t} :\, \xi (x)> a_t -1} \frac{f(\mathcal F^{-1}_{t,s}(\xi (x))) }{f(\xi (x))} \le 2 \exp \left\{ -2 c_0 s \sum _{x \in B_{R_t} :\, \xi (x) > a_t -1} \alpha (x) \right\} \end{aligned}$$

(12.17)

since $\mathcal L_{B_{R_t}}( \varphi ) \le \ln 2$ on $G_{t,0}$. Moreover, by (5.5), on $\mathcal G_{t,0}$ we have $\xi (x) > a_t$ for some $x \in B_{R_t}$ and thus also $\alpha (x) = 1$. Noting now that, by (12.1) and Lemma 6.4 of [7],

$$\begin{aligned} \text{ Prob } \left( \lambda ^{{\scriptscriptstyle {({1}})}}_{B_{R_t}}(\xi ) > a_t +s d_t \right) \le \text{ Prob } \left( \xi \in G_{t,s} \right) + o(t^{-(d+\varepsilon _0)}) \end{aligned}$$

(12.18)

for some $\varepsilon _0>0$, we obtain by (12.14–12.18)

$$\begin{aligned} \text {Prob} \left( \lambda ^{{\scriptscriptstyle {({1}})}}_{B_{R_t}}(\xi ) \ge a_t + s d_t \right) \le 2 \mathrm{e}^{-c_0 s} \text {Prob} \left( \lambda ^{{\scriptscriptstyle {({1}})}}_{B_{R_t}}(\xi ) \ge a_t \right) + o(t^{-(d+\varepsilon _0)}). \end{aligned}$$

(12.19)

To pass the estimate to $\lambda ^{{\scriptscriptstyle {({1}})}}_{B_{\widehat{N}_t}}(\xi )$, note first that, by Lemma 7.6 of [7],

$$\begin{aligned} \limsup _{t \rightarrow \infty } \frac{t^d}{(2 R_t)^d}\text {Prob} \left( \lambda ^{{\scriptscriptstyle {({1}})}}_{B_{R_t}}(\xi ) \ge a_t \right) \le 1, \end{aligned}$$

(12.20)

and thus for large t the right-hand side of (12.19) is at most $3 \, \mathrm{e}^{-c_0 s} (2 R_t/t)^d + o(t^{-(d+\varepsilon _0)})$. Moreover, by Lemma 7.7 of [7] applied to $t_L : = a_L-\widehat{a}_L +s d_L$ and $R'_L := (\ln _2 L)^2$,

$$\begin{aligned} \frac{t^d}{(2 \widehat{N}_t)^d} \text {Prob} \left( \lambda ^{{\scriptscriptstyle {({1}})}}_{B_{\widehat{N}_t}}(\xi ) \ge a_t + sd_t\right) \le \widehat{N}^{-d}_t + 4 \, \mathrm{e}^{-c_0 s} + o(t^{-\varepsilon _0}) \end{aligned}$$

(12.21)

for t large enough, noting that $o(L^{-d})$ and o(1) in equation (7.27) of [7] are uniform on the sequence $t_L$. Note that the factor 2 multiplying $R_t$ and $\widehat{N}_t$ here and not in [7] appears since our boxes have side-length $2R+1$ while theirs R. Recalling that $\widehat{N}_t \gg t^{\beta }$ for some $\beta > 0$ and taking $\varepsilon := \varepsilon _0 \wedge (\beta d)$, the lemma is proved. $\square$

Appendix: Compactification

Let $\mathfrak {E} := (\mathbb R\times \mathbb R^d) \cup [0,\infty )$ be equipped with a metric $\mathbf {d}$ defined by setting, for $\theta , \theta ' \in [0,\infty )$ and $(\lambda , z), (\lambda ', z') \in \mathbb R\times \mathbb R^d$,

$$\begin{aligned} \begin{aligned} \mathbf {d}(\theta , \theta ')&:= \left| \theta - \theta '\right| , \qquad \; \mathbf {d}(\theta , (\lambda , z)) := \mathrm{e}^{-\lambda } + \left| \frac{|z|}{1 \vee \lambda }-\theta \right| ,\\ \mathbf {d}((\lambda , z), (\lambda ', z'))&:= \mathrm{e}^{-\lambda \wedge \lambda '} \left( 1 - \mathrm{e}^{-|\lambda - \lambda '| - |z - z'|} \right) + \left| \frac{|z|}{1 \vee \lambda }-\frac{|z'|}{1 \vee \lambda '}\right| .\\ \end{aligned} \end{aligned}$$

(13.1)

One may verify that $\mathbf {d}$ is indeed a metric under which $\mathfrak {E}$ is separable, complete and locally compact. Moreover:

Lemma 13.1

For any $(\theta , \eta ) \in (0,\infty ) \times \mathbb R$, the set $\mathcal H^\theta _\eta \subset \mathfrak {E}$ defined in (7.15) is relatively compact.

Proof

Note that the closure of $\mathcal H^\theta _\eta$ in $\mathfrak {E}$ is given by

$$\begin{aligned} \overline{\mathcal H^\theta _\eta } = \left\{ (\lambda , z) \in \mathbb R\times \mathbb R^d :\, \lambda - \frac{|z|}{\theta } \ge \eta \right\} \cup [0,\theta ]. \end{aligned}$$

(13.2)

Fix a sequence $({\varXi }_n)_{n \in \mathbb N}$ in $\overline{\mathcal H^\theta _\eta }$ and consider the following three cases:

• 1. ${\varXi }_n \in [0, \theta ]$ for infinitely many n;

• 2. There is an infinite subsequence ${\varXi }_{n_j} = (\lambda _j, z_j) \in \mathbb R\times \mathbb R^d$ and $(\lambda _j)_{j\in \mathbb N}$ is bounded, implying that $\{{\varXi }_{n_j} :\, j \in \mathbb N\}$ is contained in a compact subset of $\mathbb R\times \mathbb R^d$;

• 3. There is an infinite subsequence ${\varXi }_{n_j} = (\lambda _j, z_j) \in \mathbb R\times \mathbb R^d$ and $\lim _{j \rightarrow \infty }\lambda _j =\infty$. Note that $\limsup _{j \rightarrow \infty } |z_j|/\lambda _j \le \theta$.

As is directly checked, in each case there exists a subsequence converging in $\mathfrak {E}$ to a point of $\overline{\mathcal H^\theta _\eta }$, thus proving the claim. $\square$

We finish the section with the following important property of $\mathfrak {E}$.

Lemma 13.2

For any compact set $K \subset \mathfrak {E}$, there exist $\theta \in (0,\infty )$ and $\eta \in \mathbb R$ such that $K \cap (\mathbb R\times \mathbb R^d) \subset \mathcal H^{\theta }_\eta$.

Proof

Cover each $x \in K$ with an open set $\mathcal H^{\theta _x}_{\eta _x} \cup [0, \theta _x)$ for some $\theta _x >0, \eta _x \in \mathbb R$. Use compactness to extract a finite subcover corresponding to $x_1, \ldots , x_N$ and set $\theta := \max _{i=1}^N \theta _{x_i}$, $\eta := \min _{i=1}^N \eta _{x_i}$ to obtain the result. $\square$

Appendix: Properties of the cost functional

In this section we prove Lemmas 7.5, 7.6, 7.8 and 7.9.

Proof of Lemma 7.5(i)

Fix $\theta _0 < \theta _1$ and set $(\lambda _i, z_i)={\varXi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)(\theta _i)$, $i=0, 1$. Then

$$\begin{aligned} \theta _0(\lambda _1 - \lambda _0) \le |\vartheta (\lambda _1, z_1)| - |\vartheta (\lambda _0, z_0)| \le \theta _1 (\lambda _1 - \lambda _0) \end{aligned}$$

(14.1)

by the definition of ${\varPsi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)$, so that all three functions are non-decreasing. Now, if $(\lambda _0, z_0)\ne (\lambda _1, z_1)$, then one of the inequalities above is strict, since otherwise $\lambda _1 = \lambda _0$, $|\vartheta (\lambda _1, z_1)|= |\vartheta (\lambda _0, z_0)|$ and we would have $(\lambda _i, z_i) \in \mathfrak {S}^{{\scriptscriptstyle {({1}})}}_{\vartheta }(\mathcal P)(\theta _j)$ for all $i, j \in \{0,1\}$, implying that $(\lambda _1, z_1) = (\lambda _0,z_0)$ by the definition of ${\varXi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)$. This concludes the proof. $\square$

Proof of Lemma 7.5(ii)

We will first consider the case $|{\text {supp}}\,(\mathcal P)|<\infty$. We may assume $|{\text {supp}}\,(\mathcal P)| \ge 2$ since otherwise there is nothing to prove.

Consider first the case $i=1$. ${\varPsi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)$ is continuous as the pointwise maximum of finitely many continuous functions. Lemma 7.5(i) implies that ${\varXi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)$ jumps finitely many times, and thus has left limits; let us to show that it is càdlàg. Fix $\theta _0 > 0$ and let $(\lambda _0, z_0) := {\varXi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)(\theta _0)$. Note first that, if $(\lambda , z) \in \mathfrak {S}^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)(\theta _0)$, then $\psi ^\vartheta _\theta (\lambda ,z) \le \psi ^\vartheta _\theta (\lambda _0,z_0)$ for all $\theta \ge \theta _0$ because $\lambda \le \lambda _0$ by definition. On the other hand, if $(\lambda , z) \notin \mathfrak {S}^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)(\theta _0)$, then there exists $\delta _{\lambda , z}>0$ such that $\psi ^\vartheta _\theta (\lambda , z) < \psi ^\vartheta _\theta (\lambda _0, z_0)$ for all $\theta \in [\theta _0, \theta _0+\delta _{\lambda , z}]$. Setting $\delta >0$ to be the smallest among these, we can see that

$$\begin{aligned} (\lambda _0, z_0) \in \mathfrak {S}^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)(\theta ) \subset \mathfrak {S}^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)(\theta _0) \;\; \forall \, \theta \in \left[ \theta _0, \theta _0+\delta \right] \end{aligned}$$

(14.2)

implying ${\varXi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)(\theta ) = {\varXi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)(\theta _0)$ for all $\theta \in [\theta _0, \theta _0+\delta ]$, i.e., ${\varXi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)$ is right-continuous.

Assume now by induction that the statement of Lemma 7.5(ii) has been proved in the case $|{\text {supp}}\,(\mathcal P)|<\infty$ for all $i \le k-1$, $k \ge 2$. Note that, by the definition of ${\varPhi }^{{\scriptscriptstyle {({k}})}}_\vartheta$,

$$\begin{aligned} {\varPhi }^{{\scriptscriptstyle {({k}})}}_\vartheta (\mathcal P)(\theta )= \sum _{{\varXi }\in {\text {supp}}\,(\mathcal P)} \mathsf{1}_{\left\{ {\varXi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)(\theta )= {\varXi }\right\} } {\varPhi }^{{\scriptscriptstyle {({k-1}})}}_\vartheta (\mathcal P_{{\varXi }})(\theta ) \end{aligned}$$

(14.3)

where $\mathcal P_{{\varXi }}(\cdot ) := \mathcal P( \cdot {\setminus } \{{\varXi }\})$. Since ${\varXi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)$ is càdlàg, it follows from the induction hypothesis that ${\varPhi }^{{\scriptscriptstyle {({k}})}}_\vartheta (\mathcal P)$ is also càdlàg. To prove in addition that ${\varPsi }^{{\scriptscriptstyle {({k}})}}_\vartheta (\mathcal P)$ is continuous, we only need to show that, if ${\varXi }_0:={\varXi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)(\theta -)\ne {\varXi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)(\theta )=:{\varXi }$, then ${\varPsi }^{{\scriptscriptstyle {({k-1}})}}_\vartheta (\mathcal P_{{\varXi }_0})(\theta ) = {\varPsi }^{{\scriptscriptstyle {({k-1}})}}_\vartheta (\mathcal P_{{\varXi }})(\theta )$; but this follows from the definition of ${\varPsi }^{{\scriptscriptstyle {({k-1}})}}_\vartheta$ since, by the continuity of ${\varPsi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)$, $\psi ^\vartheta _\theta ({\varXi }_0) = \psi ^\vartheta _\theta ({\varXi })$. This finishes the proof in the case $|{\text {supp}}\,(\mathcal P)| < \infty$.

The case $|{\text {supp}}\,(\mathcal P)|=\infty$ can be reduced to the previous one as follows. First note that we may substitute $(0,\infty )$ by [a, b] with $0< a< b <\infty$ arbitrary. Fix $i \in \mathbb N$. Since $\mathcal H^a_\eta \uparrow \mathbb R\times \mathbb R^d$ as $\eta \rightarrow -\infty$, $\mathcal H^b_{\eta }$ is relatively compact and $\mathcal P^\vartheta \in \mathscr {M}_{\mathrm{P}}$, there exists an $\eta \in \mathbb R$ such that $i \le |{\text {supp}}\,(\mathcal P^\vartheta ) \cap \mathcal H^a_{\eta }| \le \mathcal P^\vartheta (\mathcal H^b_{\eta }) < \infty$. Noting that, on [a, b], ${\varPhi }^{{\scriptscriptstyle {({i}})}}_\vartheta (\mathcal P) = {\varPhi }^{{\scriptscriptstyle {({i}})}}_\vartheta (\mathcal P')$ where $\mathcal P'(\cdot ):=\mathcal P(\cdot \cap \{(\lambda , z) :(\lambda , \vartheta (\lambda , z)) \in \mathcal H^b_\eta \})$, we fall into the previous case.

For the last statements, note that the proof above shows that ${\varXi }^{{\scriptscriptstyle {({i}})}}_\vartheta (\mathcal P)$ jumps finitely many times in each compact interval $[\theta _1, \theta _2] \subset (0,\infty )$. Moreover, if we have $\vartheta ({\varXi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)(\theta _1)) \ne 0$ and ${\varXi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)$ is constant in $[\theta _1, \theta _2]$, then ${\varPsi }^{{\scriptscriptstyle {({1}})}}_\vartheta (\mathcal P)$ is strictly increasing in $[\theta _1, \theta _2]$. $\square$

Proof of Lemma 7.6

We first consider the case $1 \le |{\text {supp}}\,(\mathcal P)|<\infty$. By Proposition 3.13 of [24], for t large enough there exist bijections $T_t : {\text {supp}}\,(\mathcal P) \rightarrow {\text {supp}}\,(\mathcal P_t)$ such that

$$\begin{aligned} \lim _{t\rightarrow \infty }\sup _{{\varXi }\in {\text {supp}}\,(\mathcal P)} {\text {dist}}\,(T_t({\varXi }),{\varXi }) = 0. \end{aligned}$$

(14.4)

Letting $\mathcal T_t(\lambda , z):=(\lambda , \vartheta _t(\lambda , z))$, by (7.45) and ${\text {supp}}\,(\mathcal P) \cap \mathbb R\times \{0\}=\emptyset$ we also have

$$\begin{aligned} \lim _{t \rightarrow \infty } \sup _{{\varXi }\in {\text {supp}}\,(\mathcal P)} {\text {dist}}\,( \mathcal T_t \circ T_t ({\varXi }), {\varXi }) = 0, \end{aligned}$$

(14.5)

and $\mathcal T_t \circ T_t$ is a bijection onto ${\text {supp}}\,(\mathcal P_t^{\vartheta _t})$. In particular, $\mathcal P_t^{\vartheta _t} \rightarrow \mathcal P$.

To characterize the jump times of our processes, the following definition will be useful: For $\vartheta : \mathbb R\times \mathbb R^d$, ${\varXi }_i = (\lambda _i, z_i) \in \mathbb R\times \mathbb R^d$, $i=0,1$, and $\theta > 0$, let

$$\begin{aligned} \mathcal F^{\vartheta }_\theta ({\varXi }_1, {\varXi }_0) := \left\{ \begin{array}{ll} \frac{\left| \vartheta ({\varXi }_1)\right| - \left| \vartheta ({\varXi }_0)\right| }{\lambda _1 - \lambda _0} &{} \text { if } \lambda _1 > \lambda _0 \text { and } \psi ^\vartheta _\theta ({\varXi }_1) < \psi ^\vartheta _\theta ({\varXi }_0), \\ \infty &{} \text { otherwise.}\end{array} \right. \end{aligned}$$

(14.6)

When $\vartheta (\lambda , z)=z$, we omit it from the notation.

We now proceed with the proof. Let $a_0 := a$ and, recursively for $\ell \in \mathbb N$,

$$\begin{aligned} a_{\ell } := \inf \left\{ \theta > a_{\ell -1} :\, \exists \, 1 \le i \le \left| {\text {supp}}\,(\mathcal P)\right| , {\varXi }^{{\scriptscriptstyle {({i}})}}_\vartheta (\mathcal P)(\theta ) \ne {\varXi }^{{\scriptscriptstyle {({i}})}}_\vartheta (\mathcal P)(a_{\ell -1})\right\} . \end{aligned}$$

(14.7)

Note that ${\varXi }^{{\scriptscriptstyle {({i}})}}(\mathcal P)$ jumps finitely many times. Indeed, for $i=1$ this follows by Lemma 7.5(i), and for $i \ge 2$, by induction using (14.3). Thus $\ell _* = \ell _*(a, \mathcal P):= \inf \{\ell \ge 0 :\, a_{\ell +1}=\infty \} < \infty$.

We proceed by induction on $\ell _*$, starting with $\ell _* = 0$. Since $\mathcal P\in \widetilde{\mathscr {M}}^a_\text {P}$, the values $i \mapsto \psi _a({\varXi }^{{\scriptscriptstyle {({i}})}}(\mathcal P)(a))$ are all distinct, which together with (14.4–14.5) implies that ${\varXi }^{{\scriptscriptstyle {({i}})}}_{\vartheta _t}(\mathcal P_t)(a) = T_t({\varXi }^{{\scriptscriptstyle {({i}})}}(\mathcal P)(a))$ for all i when t is large enough. In particular, (14.4) implies the result in the case $\ell _* =0$. Assume by induction that, for some $L \in \mathbb N$, the statement has been proved for all $a' \in (0, \infty )$ and $\mathcal P' \in \widetilde{\mathscr {M}}^{a'}_{\text {P}}$ satisfying $|{\text {supp}}\,(\mathcal P')| < \infty$ and $\ell _*(a', \mathcal P') \le L-1$, and suppose that $\ell _*=\ell _*(a, \mathcal P)=L$ (in which case necessarily $|{\text {supp}}\,(\mathcal P)|\ge 2$).

Note now that, because $\mathcal P\in \widetilde{\mathscr {M}}^{a}_{\text {P}}$, there exists a unique $i_1$ such that both ${\varXi }^{{\scriptscriptstyle {({i_1}})}}(\mathcal P)$ and ${\varXi }^{{\scriptscriptstyle {({i_1+1}})}}(\mathcal P)$ jump at $a_1$ while ${\varXi }^{{\scriptscriptstyle {({i}})}}(\mathcal P)$ is continuous at $a_1$ for all $i \notin \{i_1, i_1+1\}$. Moreover, ${\varXi }^{{\scriptscriptstyle {({i_1}})}}(\mathcal P)(a_1)$ is the point ${\varXi }\in {\text {supp}}\,(\mathcal P)$ minimizing $\mathcal F_a({\varXi }, {\varXi }^{{\scriptscriptstyle {({i_1}})}}(\mathcal P)(a))$ [cf. (14.6)], while $a_1 = \mathcal F_a({\varXi }^{{\scriptscriptstyle {({i_1}})}}(\mathcal P)(a_1), {\varXi }^{{\scriptscriptstyle {({i_1}})}}(\mathcal P)(a))$ and ${\varXi }^{{\scriptscriptstyle {({i_1+1}})}}(\mathcal P)(a_1) = {\varXi }^{{\scriptscriptstyle {({i_1}})}}(\mathcal P)(a)$.

Let $a^t_\ell$, $\ell ^t_*$ be the analogues of $a_\ell$, $\ell _*$ for ${\varXi }^{{\scriptscriptstyle {({i}})}}_{\vartheta _t}(\mathcal P_t)$, and fix $a' \in (a_1, a_2) \cap \mathbb Q$. By (14.4–14.5) and the previous discussion, when t is large enough, ${\varXi }^{{\scriptscriptstyle {({i}})}}_{\vartheta _t}(\mathcal P_t)$ does not jump in $[a, a']$ for all $i \notin \{i_1, i_1+1\}$, ${\varXi }^{{\scriptscriptstyle {({i_1}})}}_{\vartheta _t}(\mathcal P_t)(a^t_1) = T_t({\varXi }^{{\scriptscriptstyle {({i_1}})}}(\mathcal P)(a_1))$, and ${\varXi }^{{\scriptscriptstyle {({i_1+1}})}}_{\vartheta _t}(\mathcal P_t)(a^t_1) = {\varXi }^{{\scriptscriptstyle {({i_1}})}}_{\vartheta _t}(\mathcal P_t)(a) = T_t({\varXi }^{{\scriptscriptstyle {({i_1}})}}(\mathcal P)(a))$. Moreover, $a^t_2> a' > a^t_1$ and

$$\begin{aligned} a^t_1= & {} \mathcal F_a^{\vartheta _t}( {\varXi }^{{\scriptscriptstyle {({i_1}})}}_{\vartheta _t}(\mathcal P_t)(a^t_1), {\varXi }^{{\scriptscriptstyle {({i_1}})}}_{\vartheta _t})(a) ) \nonumber \\= & {} \mathcal F_a(\mathcal T_t \circ T_t({\varXi }^{{\scriptscriptstyle {({i_1}})}}(\mathcal P)(a_1)), \mathcal T_t \circ T_t({\varXi }^{{\scriptscriptstyle {({i_1}})}}(\mathcal P)(a))), \end{aligned}$$

(14.8)

allowing us to conclude, by (14.5),

(14.9)

Define now a time change $\sigma _t:[a, a']\rightarrow [a, a']$ by setting

$$\begin{aligned} \sigma _t(a)=a, \;\;\; \sigma _t(a_1)=a^t_1, \;\;\; \sigma _t(a')=a' \;\;\; \text { and linear otherwise.} \end{aligned}$$

(14.10)

Then, by the previous discussion together with (14.4), (14.5) and (14.9),

$$\begin{aligned} \lim _{t \rightarrow \infty } \, \sup _{1 \le i \le |{\text {supp}}\,(\mathcal P)|} \, \sup _{\theta \in [a, a']} \, \left| \sigma _t(\theta ) - \theta \right| \vee \left| {\varPhi }^{{\scriptscriptstyle {({i}})}}_{\vartheta _t}(\mathcal P_t)(\sigma _t(\theta )) - {\varPhi }^{{\scriptscriptstyle {({i}})}}(\mathcal P)(\theta )\right| = 0. \end{aligned}$$

(14.11)

Since $\ell _*(a', \mathcal P) = L-1$ and $\mathcal P\in \widetilde{\mathscr {M}}^{a'}_{\text {P}}$, by the induction hypothesis we can extend $\sigma _t$ to $[a, \infty )$ in such a way that (14.11) holds with $[a, a']$ substituted by $[a, \infty )$, finishing the proof in the case $|{\text {supp}}\,(\mathcal P)|<\infty$.

Consider now the case $|{\text {supp}}\,(\mathcal P)| = \infty$. We may assume without loss of generality that $c_*$ in (7.46) is not larger than 1. Let us first show (7.47). Fix $k \in \mathbb N$ and a point $b \in (a, \infty ) \cap \mathbb Q$. Note that, since $\mathcal P\in \widetilde{\mathscr {M}}^{a}_{\text {P}}$, b is a continuity point of ${\varPhi }^{{\scriptscriptstyle {({i}})}}(\mathcal P)$ for all $1 \le i \le k$. Let $\eta \in \mathbb R$ be negative enough such that, for all t large enough,

$$\begin{aligned}&k \le \left| {\text {supp}}\,(\mathcal P) \cap \mathcal H^{a}_\eta \right| \nonumber \\&\quad = \left| {\text {supp}}\,(\mathcal P_t) \cap \mathcal H^{a}_\eta \right| \le \mathcal P_t(\mathcal H^{2b/c_*}_\eta ) = \mathcal P(\mathcal H^{2b/c_*}_\eta ) < \infty , \end{aligned}$$

(14.12)

which is possible because $\mathcal P\in \mathscr {M}_{\mathrm{P}}$ and $\mathcal P_t \rightarrow \mathcal P$. Moreover, since ${\text {supp}}\,(\mathcal P) \cap \mathbb R\times \{0\} = \emptyset$, by (7.45–7.46) we may also assume that

$$\begin{aligned} k \le \left| {\text {supp}}\,(\mathcal P_t^{\vartheta _t}) \cap \mathcal H^a_\eta \right| \end{aligned}$$

(14.13)

and

$$\begin{aligned} {\text {supp}}\,(\mathcal P_t^{\vartheta _t}) \cap \mathcal H^b_\eta \subset \mathcal T_t\left( {\text {supp}}\,(\mathcal P_t) \cap \mathcal H^{2b/c_*}_\eta \right) , \end{aligned}$$

(14.14)

where $\mathcal T_t$ is defined right before (14.5). Now (14.12–14.14) imply that, on [a, b], ${\varPhi }^{{\scriptscriptstyle {({i}})}}(\mathcal P) = {\varPhi }^{{\scriptscriptstyle {({i}})}}(\mathcal P')$ and ${\varPhi }^{{\scriptscriptstyle {({i}})}}_{\vartheta _t}(\mathcal P_t) = {\varPhi }^{{\scriptscriptstyle {({i}})}}_{\vartheta _t}(\mathcal P'_t)$ for all $1 \le i \le k$, where $\mathcal P'(\cdot ) := \mathcal P(\cdot \cap \mathcal H^{2b/c_*}_\eta )$ and analogously for $\mathcal P'_t$. Since $\mathcal P'_t \rightarrow \mathcal P'$, (7.47) follows by the previous case and Theorem 16.2 of [4]. The convergence $\mathcal P_t^{\vartheta _t} \rightarrow \mathcal P$ follows from (14.14), (7.45) and $\mathcal P_t \rightarrow \mathcal P$ (note that b, $\eta$ above can be taken arbitrarily large, respec. negative). $\square$

Proof of Lemma 7.8

By Lemma 7.7, it suffices to show $(4) \Rightarrow (1)$. Arguing as at the end of the proof of Lemma 7.6, we reduce to the case $|{\text {supp}}\,(\mathcal P)| < \infty$. Denote by $\pi :\mathbb R\times \mathbb R^d \rightarrow \mathbb R^d$ the projection on the second coordinate, i.e., the map $\pi (\lambda , z) := z$. For $z \in \pi ({\text {supp}}\,(\mathcal P))$, set $\lambda _z := \max \{\lambda :(\lambda , z) \in {\text {supp}}\,(\mathcal P)\}$ and define $\widehat{\mathcal P} := \sum _{z \in \pi ({\text {supp}}\,(\mathcal P))} \delta _{(\lambda _z, z)}$. Note that $\pi$ is injective over ${\text {supp}}\,(\widehat{\mathcal P})$, and that ${\varXi }^{{\scriptscriptstyle {({1}})}}(\mathcal P) = {\varXi }^{{\scriptscriptstyle {({1}})}}(\widehat{\mathcal P})$. By (14.4–14.5), when t is large enough, $\pi$ is injective over the support of $\widehat{\mathcal P}_t:= \widehat{\mathcal P} \circ T_t^{-1}$, and moreover ${\varXi }^{{\scriptscriptstyle {({1}})}}_{\vartheta _t}(\mathcal P_t)(\theta ) = {\varXi }^{{\scriptscriptstyle {({1}})}}_{\vartheta _t} (\widehat{\mathcal P}_t)(\theta )$ for all $\theta \in [a,b]$. This concludes the proof. $\square$

Proof of Lemma 7.9

For $(\lambda , z) \in \mathbb R\times (\mathbb R^d {\setminus } \{0\})$, let

$$\begin{aligned} \mathcal A(\lambda , z) := \left\{ (\lambda ', z') \in \mathbb R\times \mathbb R^d :\, \begin{array}{l} \psi _a(\lambda ', z')> \psi _a(\lambda , z) \text { or}\\ \psi _a(\lambda ', z') = \psi _a(\lambda , z) \text { and } \lambda '>\lambda \end{array} \right\} . \end{aligned}$$

(14.15)

By the definition of $\mathcal P^\vartheta$, $\mathcal F_a^{\vartheta }(\mathcal P, \lambda , z) = \mathcal P^\vartheta \left\{ \mathcal A(\lambda , \vartheta (\lambda , z)) \right\}$. Since $\vartheta _t(\lambda _t, z_t) \rightarrow z_*$ by (7.45) and $\mathcal P_t^{\vartheta _t} \rightarrow \mathcal P$ by Lemma 7.6, we may assume that $\vartheta _t(\lambda , z) = z$ for all $(\lambda ,z) \in \mathbb R\times \mathbb R^d$.

Now, since $\mathcal P\in \widetilde{\mathscr {M}}^a_{\text {P}}$, $\mathcal F_a(\mathcal P,\lambda _*, z_*) = \mathcal P\left\{ \mathcal H^a_{\psi _a(\lambda _*, z_*)} \right\}$ and there exists a $\delta >0$ such that

$$\begin{aligned} \mathcal P\left\{ \overline{\mathcal H^{a}_{\psi _a(\lambda _*, z_*) - \delta }} \right\} = 1 + \mathcal P\left\{ \overline{\mathcal H^a_{\psi _a(\lambda _*, z_*) + \delta }} \right\} . \end{aligned}$$

(14.16)

On the other hand, since $\mathcal P_t \rightarrow \mathcal P$ and $(\lambda _t, z_t) \rightarrow (\lambda _*, z_*)$, when t is large we also have

$$\begin{aligned} \mathcal P_t \left\{ \overline{\mathcal H^a_{\psi _a(\lambda _*, z_*)\pm \delta }} \right\} = \mathcal P\left\{ \overline{\mathcal H^a_{\psi _a(\lambda _*, z_*)\pm \delta }} \right\} \end{aligned}$$

(14.17)

and

$$\begin{aligned} (\lambda _t, z_t) \in \mathcal H^{a}_{\psi _b(\lambda _*, z_*)-\delta } {\setminus } \mathcal H^a_{\psi _a(\lambda _*, z_*)+\delta }. \end{aligned}$$

(14.18)

In particular, for all t large enough,

$$\begin{aligned} \begin{aligned} \mathcal P_t \left\{ \mathcal A(\lambda _t, z_t) \right\}&= \mathcal P_t \left\{ \overline{\mathcal H^a_{\psi _a(\lambda _*, z_*)+\delta }} \right\} \\&= \mathcal P\left\{ \overline{\mathcal H^a_{\psi _a(\lambda _*, z_*)+\delta }} \right\} = \mathcal P\left\{ \mathcal H^a_{\psi _a(\lambda _*, z_*)} \right\} , \end{aligned} \end{aligned}$$

(14.19)

concluding the proof. $\square$