Strong laws for growth-fragmentation processes with bounded cell size

Emma Horton

Alex Watson

IECL seminar, 6 May 2021

A model of growth-fragmentation

(Diagram of cell division. Left to right: a single cell of size x(0) = x. Cell grows to x(t), with arrow x.(t) = tau(x(t)). Cell divides to cells of size px(t), (1-p)x(t), with label 'rate B(x(t))' and 'chosen by kappa(x(t),dp)'. The cell of size px(t) continues to grow and divide. The cell of size (1-p)x(t) dies, with label 'rate D(.)'.)

  • • List sizes at time \(t\): \((Z_u(t): u \in U)\)

  • • \(\Zb (t) = \sum _{u\in U} \delta _{Z_u(t)}\)

Equilibrium behaviour

(-tikz- Diagram of simulated cell division. Top row, left to right: single cell; arrow; collection of coloured cells with 't=4.71, 100 cells'; arrow; collection of cells with 't=6.32, 500 cells'; arrow; collection of cells with 't=7.02, 1000 cells'. Middle row: histograms beneath the cell collections, with overlaid density estimate. The histograms appear to be converging loosely to a density. Bottom left: plot of log <Z(t),1> against t, roughly linear, with linear fit line overplotted. Bottom right: '<Z(t),f> ~ exp(lambda t)<Z(0),h><nu, f>'. )

Homogeneous model

  • • Exponential growth and size-independent rates: \(\tau (x) = ax\), \(B(x) = B\), \(D(x) = \mathtt {k}\), \(\kappa (x,\cdot ) = \kappa (\cdot )\)

  • • No equilibrium behaviour

  • • Underlying Lévy process: \(\EE _x \ip {\Zb (t)}{f} = e^{at}\EE _x[e^{-\chi _t}f(e^{\chi _t})]\)

Perturbations

  • • ‘Refracted process’:

    \[ \tau (x) = \protect \begin {cases} ax, & 0 < x < c, \\ a'x, & x > c, \protect \end {cases} \]

    where \(a>a'\).

    Cavalli (2020, Acta Appl. Math.): equilibrium behaviour of mean measures

  • • Our interest: ‘reflected process’.

    \[ \tau (x) = \protect \begin {cases} ax, & 0 < x < c, \\ 0, & x > c. \protect \end {cases} \]

  • • Underlying both: perturbed Lévy processes

Growth-fragmentation process

  • • Cell labels in \(U = \bigcup _{n\ge 0} \{0,1\}^n\)

  • • Initial cell \(\varnothing \). Births \(\varnothing \to 0,1\); \(0 \to 00, 01\); etc.

  • • Cells grow exponentially, with cap at \(c\)

  • • Cells die at rate \(\mathtt {k}\) and divide at rate \(B\)

  • • At a division time \(d_u\), offspring have initial size \(\xi Z_u(d_u-)\) and \((1-\xi )Z_u(d_u-)\), where \(\xi \sim \kappa \).

  • • \(Z_u(t) = {}\)size of cell \(u\) at \(t\), if present

  • • \(\Zb (t) = \sum _{u\in U} \delta _{Z_u(t)}\Indic {u\text { alive at }t}\)

Growth-fragmentation process – schematic

(Schematic diagram of cell sizes plotted against time. On y-axis, c is labelled. A plot line starting from time zero and labelled varnothing grows exponentially until hitting c, when it remains flat. At time labelled d-varthing = b-0 = b-1, a jump happens and two new lines start, labelled 0 and 1. The line labelled 0 grows exponentially, capped at c, and a similar jump occurs with the new lines labelled 00 and 01; these lines also grow exponentially. The line labelled 1 grows exponentially, a similar jump occurs with lines labelled 10 and 11. These lines grow exponentially and end in an 'x' at time labelled d-10 and d-11 respectively.)

Mean measures

  • • Interested in \(\Zb (t)\) as \(t\to \infty \)

  • • Start with \(\mu _t = \EE _x \Zb (t)\), i.e. \(\ip {\mu _t}{f} = \EE _x\ip {\Zb (t)}{f}\)

\begin{align*} \partial _t \langle \mu _t,f\rangle &= \langle \mu _t,\symcal {A}f\rangle \\ \symcal {A}f(x) &= axf'(x) + 2B \int _0^1 f(xp) \, \tilde {\kappa }(\mathrm {d}p) - Bf(x) - \mathtt {k}f(x) \\ \symcal {D}(\symcal {A}) &\supset \{ f\in C^1_{\protect \text {c}}(0,c] : f'(c) = 0 \} \end{align*} ...where \(\tilde {\kappa }(\dd p) = \frac {\kappa (\dd p)+\kappa (1-\dd p)}{2}\).

Theorem 1(a)

If

\[ a > 2B \int _0^1 (-\log p) \tilde {\kappa }(\mathrm {d}p), \]

then for \(f\) continuous and bounded,

\[ \symbb {E}_x\langle \mathbf {Z}(t),f\rangle = \langle \mu _t,f\rangle \sim e^{\lambda t} h(x) \langle \nu ,f\rangle , \qquad t\to \infty , \]

where \(\ip {\nu }{h} = 1\) and

\begin{align*} \symcal {A}h &= \lambda h \\ \nu \symcal {A}&= \lambda \nu . \end{align*}

Proof ideas

  • Lemma 1 (Many-to-one). Let \(\eta \) be a Lévy process with Lévy measure \(2B \tilde {\kappa } \circ \log ^{-1}\), drift \(a\), and reflection above at \(b=\log c\). Call \(\eta \) the spine. Then, \(\ip {\mu _t}{f} = e^{(B-\mathtt {k})t} \EE \bigl [ f(e^{\eta _t}) \mid \eta _0 = \log x\bigr ]\).

(A repeat of the previous 'schematic' diagram. This time, the labels are omitted, and one line is drawn bold throughout. The initial line is bold; at its jump time, the first of the two new lines is bold; at this line's jump time, the second of the new lines is bold.)

Idea: conditional on living to \(t\), follow offspring uniformly

Proof ideas

  • Lemma 2. \(\eta \) is positive recurrent if and only if \(\tilde {\eta }\), the unreflected Lévy process, drifts to \(+\infty \).

  • Lemma 3. \(\eta \) is positive recurrent if and only if \(a > 2B\int _0^1 (-\log p) \tilde {\kappa }(\dd p)\).

    The invariant distribution \(m\) satisfies

    \[ \int _{-\infty }^b e^{-(b-x)q} m(\mathrm {d}x) = \protect \text {cst}\cdot \protect \frac {q}{aq + 2B\int _0^1 (p^q-1) \, \tilde {\kappa }(\mathrm {d}p)}. \]

  • • \(\ip {\mu _t}{f} = e^{(B-\mathtt {k})t} \EE f(e^{\eta _t}) \sim e^{(B-\mathtt {k})t} \ip {m}{f(e^{\cdot })}\).

  • • We get Theorem 1(a), with \(\lambda = B-\mathtt {k}\), \(h = 1\) and \(\nu = m\circ \exp ^{-1}\).

Theorem 1(b)

If

\[ a > 2B \int _0^1 (-\log p) \tilde {\kappa }(\mathrm {d}p) \qquad \protect \text {\alert {and}} \qquad \int _0^1 p^{-(r+\epsilon )} \tilde {\kappa }(\mathrm {d}p) < \infty , \quad \protect \text {some }r,\epsilon >0, \]

then there exist \(C,k>0\) such that for \(f\) continuous,

\[ \protect \bigl \lvert e^{-\lambda t}\langle \mu _t,f\rangle - h(x)\langle \nu ,f\rangle \protect \bigr \rvert \le \lVert f_r\rVert _{\infty } \protect \bigl ( (c/x)^r+C\protect \bigr ) e^{-kt}, \]

where \(f_r(x) = (x/c)^r f(x)\).

  • • Similar proof: exponential recurrence of reflected Lévy process

Theorem 2

If

\[ a > 2B \int _0^1 (-\log p) \tilde {\kappa }(\mathrm {d}p), \int _0^1 p^{-(r+\epsilon )} \tilde {\kappa }(\mathrm {d}p) < \infty \qquad \protect \text {\alert {and}} \qquad B > \mathtt {k}, \]

then, for continuous bounded \(f\) and \(x\in (0,c]\),

\[ e^{-\lambda t} \langle \mathbf {Z}(t),f\rangle \to M_\infty \langle \nu ,f\rangle \qquad \symbb {P}_x\protect \text {-a.s. and in } L^1(\symbb {P}_x), \]

where \(M_\infty \) is a random variable with \(\EE _xM_\infty = h(x) = 1\).

  • • Start with case \(\mathtt {k} = 0\): no killing

  • • \(M_t = e^{-\lambda t} \ip {\Zb (t)}{1}\) is a UI martingale – even \(L^2\)-bounded

  • • \(M_\infty \) is its (a.s. or \(L^1\)) limit

Proof ideas

(Left to right: a collection of green cells; arrow labelled 't'; a collection of blue cells, with one labelled 'Zu(t)'; arrow labelled 's'; a collection of red cells with a dashed line around it, labelled 'Du' and 'Descendants of Zu(t)'; a collection of yellow cells labelled 'other cells'.)

\begin{align*} e^{-\lambda (t+s)} \mathbf {Z}(t+s) & = {\color {TolDarkBlueAcc} e^{-\lambda t} \protect \sum _{u}} {\color {TolDarkPink} e^{-\lambda s}\protect \sum _{v\in D_u}\delta _{Z_v(t+s)}} \\ & = {\color {TolDarkBlueAcc} e^{-\lambda t} \protect \sum _{u}} \biggl ( {\color {TolDarkPink} e^{-\lambda s}\protect \sum _{v\in D_u}\delta _{Z_v(t+s)}} - \symbb {E}_x \biggl [ \protect \sum _{v\in D_u} e^{-\lambda s} \delta _{Z_v(t+s)} \bigg \vert \symcal {F}_t \biggr ] \biggr ) \\ & \quad {} + e^{-\lambda t} \protect \sum _{u} \biggl ( \symbb {E}_x \biggl [ \protect \sum _{v\in D_u} e^{-\lambda s} \delta _{Z_v(t+s)} \bigg \vert \symcal {F}_t \biggr ] - \nu \biggr ) \\ & \quad {} + M_t \nu \end{align*}

Term 1 \(\to \) 0 by comparison with \(M\). Term 3 \(\to M_\infty \nu \). Term 2?

Proof ideas

\begin{align*} \lvert \langle \protect \text {Term 2},f\rangle \rvert &= \biggl \lvert e^{-\lambda t} \protect \sum \nolimits _{u} \biggl ( \symbb {E}_x\biggl [ \protect \sum \nolimits _{v\in D_u} e^{-\lambda s} f(Z_v(t+s)) \bigg \vert \symcal {F}_t \bigg ] - \langle \nu ,f\rangle \biggr ) \biggr \rvert \\ & = \left \lvert e^{-\lambda t} \protect \sum \nolimits _u \left ( \symbb {E}_{Z_u(t)}\protect \bigl [ e^{-\lambda s} \langle \mathbf {Z}(s),f\rangle \protect \bigr ] - \langle \nu ,f\rangle \right ) \right \rvert \\ & \le e^{-\lambda t} \protect \sum \nolimits _u \protect \Bigl \lvert \symbb {E}_{Z_u(t)} \protect \bigl [e^{-\lambda s}\langle \mathbf {Z}(s),f\rangle \protect \bigr ] - \langle \nu ,f\rangle \protect \Bigr \rvert =: R_{s,t} \end{align*}

  • • Use exponential rate asymptotics:

    \[ \protect \bigl \lvert \symbb {E}_{Z_u(t)}\protect \bigl [ e^{-\lambda s} \langle \mathbf {Z}(s),f\rangle \protect \bigr ] - \langle \nu ,f\rangle \protect \bigr \rvert \le e^{-ks} \lVert f_r \rVert _\infty ((c/Z_u(t))^r + C) \]

  • • ...so:

    \[ \symbb {E}_x R_{s,t} \le e^{-ks} \lVert f_r \rVert _{\infty } \symbb {E}_x\protect \bigl [{\textstyle \protect \sum _u} e^{-\lambda t} ((c/Z_u(t))^r+C)\protect \bigr ] \le e^{-ks} \lVert f_r\rVert _\infty \symbb {E}_x\protect \bigl [ e^{r(b-\eta _t)} + C\protect \bigr ] \]

    ...and the RHS is bounded in \(t\).

  • • Thus \(\EE _x R_{\delta n, \delta n}\) is summable, and Borel-Cantelli yields a.s. convergence along \(\delta n\).

  • • This is the core of the proof.

Proof ideas

  • • What if \(\texttt {k} > 0\), i.e., cells may be killed?

  • • Colour blue lines of descent which live forever, others red

  • • Blue tree satisfies \(\mathtt {k} = 0\) and only differs in offspring distribution: skeleton decomposition

  • • Show that red tree has negligible contribution to limit

(A binary tree drawn with the root on the left. Nodes all of whose ancestors die before they reach the right are coloured red. Other nodes are coloured blue.)

Reflection on proof

  • • Nice aspects: explicit \(h\), \(\nu \) and \(\lambda \), fairly transparent proofs

  • • Used extensively that \(h=1\)

    • – Without this, \(M\) becomes \(M_t = e^{-\lambda t} \ip {\Zb (t)}{h}\); need estimates on \(h\) and \(1/h\) for uniform integrability

  • • Used precise \(x\)-dependence of \(\lvert e^{-\lambda t}\EE _x\ip {\Zb (t)}{f} - \ip {\nu }{f}\rvert \)

  • • Reflected Lévy arguments allow control over \(\Zb (t)(\dd x)\)-integrals

    • – More generally, need to understand precisely asymptotics of another spine process.

  • • Require \(B < \infty \), both for \(\lambda < \infty \) to hold and for passage from \(\Zb (\delta n)\) to \(\Zb (t)\).

    • – Could imagine more general Lévy processes appearing, but \(h\) will not be \(1\)

Theorem 3: transient case

If

\[ B > \mathtt {k}, \quad a < 2B\int _0^1 (-\log p) \tilde {\kappa }(\mathrm {d}p) \quad \protect \text { (plus extra condition)} \]

there exist \(\lambda _0<\lambda \), \(q_0\in \RR \) such that for \(f\) continuous with \(f(x) = O(x^{q_0})\) as \(x\to 0\),

\[ e^{-\lambda _0 t}\langle \mathbf {Z}(t),f\rangle \to 0. \]

  • • Cell sizes decay to zero too fast to be seen by \(f\)

  • • Analysis involves nice reflected Lévy process with killing at reflection boundary

Further reading

  • [1]  E. Horton and A. R. Watson Strong laws of large numbers for a growth-fragmentation process with bounded cell size arXiv:2012.03273 [math.PR]

Thank you!