Large Deviations of Jump Process Fluxes

Abstract

We study a general class of systems of interacting particles that randomly interact to form new or different particles. In addition to the distribution of particles we consider the fluxes, defined as the rescaled number of jumps of each type that take place in a time interval. We prove a dynamic large deviations principle for the fluxes under general assumptions that include mass-action chemical kinetics. This result immediately implies a dynamic large deviations principle for the particle distribution.

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Acknowledgements

This research has been funded by the Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 “Scaling Cascades in Complex Systems”, Project C08.

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Correspondence to D. R. Michiel Renger.

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Appendix: A change-of-measure result for linear test functionals on jump processes

Appendix: A change-of-measure result for linear test functionals on jump processes

Changes of measure are central to the proof of the large deviations principle presented in this work. This appendix arose out of the need to clarify under exactly what technical conditions [26, Appendix 1, Prop. 7.3] could be adapted to the setting of the present work, in particular so that functions of the form xζx could be used since these are not bounded functions (although they are bounded linear operators). This boundedness restriction is avoided in [38], but functions used in the change of measure are no longer time dependent and the conditions are less explicit. Here the aim is to include unbounded, time dependent functions in the change of measure formula, but to give relatively explicit, sufficient conditions that can easily be checked using the model assumptions from the main part of the paper. In this endeavour the results are restricted to pure jump processes.

Let \(\mathcal {X}\) be a Banach space, \(T \in (0,\infty ]\) and \(\left ({\Omega }, \mathcal {F}, (\mathcal {F}_{t})_{t\in [0,T)}\right )\) be a filtered probability space with canonical random variable X : Ω →Ω, where

  • Ω is a subset of the càdlàg functions \([0,T) \rightarrow \mathcal {X}\), with the convention f(T) := f(T−) if \(T<\infty \),

  • \(\mathcal {F}\) is the Borel σ-algebra generated by a separable topology on Ω and equal to the σ-algebra generated by the time evaluation functions XX(t).

Note that \(T=\infty \) is allowed for now. The application in this paper is to the case \({\Omega } = \text {BV}(0,T, \mathbb {R}^{\mathcal {Y}} \times \mathbb {R}^{\mathcal {R}})\) with the hybrid topology, but this is not a necessary assumption.

We define the jump process through a given family of jump kernels \((\alpha _{t}(x,\cdot )_{t\in [0,T),x\in \mathcal {X}}\) where αt(x,A) is the instantaneous jump rate at time t from \(x\in \mathcal {X}\) into a measurable set \(A\subset \mathcal {X}\), together with a given initial distribution μ. Let \(\mathbb {P}\) be the law of this process, a probability measure on \(({\Omega }, \mathcal {F})\) and \(\mathbb {E}\) the associated expectation operator.

We now define a class of test functions for which the associated propagators (a two-paramater semigroup of linear operators) are well-defined. To construct this set we will assume that there exists a family of measurable (not necessarily compact or bounded) subsets (Kx)x of \(\mathcal {X}\) such that for all \(x\in \mathcal {X}\):

  • xKx and \(\bigcup _{y\in K_{x}}K_{y} = K_{x}\),

  • \({{\int \nolimits }_{0}^{T}}\sup _{y \in K_{x}}\alpha _{t}(y,\mathcal {X})\mathrm {d} t < \infty \),

  • \(\sup _{t\in [0,T)}\sup _{y \in K_{x}}\alpha _{t}(y,\mathcal {X} \setminus K_{x}) = 0\).

This expresses the idea that the process started from x can neither explode nor leave Kx. Then the propagators \((P_{s,t}f)(x) := \mathbb {E}\left [f(X(t)) \middle \vert X(s)= x \right ]\) preserve the set

$$ B_{\mathrm{K}}(\mathcal{X}) := \left\{f :\mathcal{X}\rightarrow \mathbb{R} \text{ measurable, such that } \forall x \in \mathcal{X} \sup_{y\in K_{x}}|{f(y)}|< \infty \right\}, $$

and satisfy \(\frac {\mathrm {d}}{\mathrm {d} s}(P_{s,t}f)(x) = -(\mathcal {Q}_{s} P_{s,t}f)(x)\) with (time-dependent) generator

$$ \begin{array}{@{}rcl@{}} (\mathcal{Q}_{t} f)(x) := {\int}_{X} \left[f(y)-f(x) \right]\alpha_{t}(x,\mathrm{d} y). \end{array} $$

We now make three additional assumptions under which the change-of-measure formula holds.

$$ \begin{array}{@{}rcl@{}} \bullet \quad &&\text{ there is a } \gamma > 0 \text{ such that } |{y-x}| \leqslant \gamma \text{ for all } x\in\mathcal{X},t \in (0,T),{\kern4.3pc}\\ &&\text{ and } \alpha_{t}(x,\cdot)-\ae \textit{y}, \end{array} $$
(A.1)
$$ \begin{array}{@{}rcl@{}} \bullet \quad &&\lim_{n \rightarrow \infty} \mathbb{P}\left( \tau_{n} < t\right) = 0 \text{ for all } t\in(0,T), n \in \mathbb{N}, {\kern10.7pc}\\ &&\text{ where } \tau_{n} := \inf \left\{t \colon \alpha_{t}(X(t),\mathcal{X} ) \geqslant n \right\}, \end{array} $$
(A.2)
$$ \begin{array}{@{}rcl@{}} {\kern-9.6pc}\bullet \quad \mathbb{E}\left[Z^{\beta}(t)\right]< \infty \text{ for all } t\in(0,T),\beta>0, \text{ where } \end{array} $$
(A.3)
$$ \begin{array}{@{}rcl@{}} Z^{\beta}(t) &:=& \exp\left( \beta |{X(0)}|\right) + \exp\left( \beta |{X(t)}|\right)\\ &&+ {{\int}_{0}^{t}} \exp\left( \beta |{X(s)}|\right)\beta |{X(s)}| \mathrm{d} s + {{\int}_{0}^{t}} \exp\left( \beta |{X(s)}|\right)\beta \alpha_{s} \left( X(s),\mathcal{X}\right) \mathrm{d} s. \end{array} $$

The next result is a variation on [26, Appendix 1, Lem. 5.1]:

Proposition A.1

Let \(f \colon [0,T) \times \mathcal {X} \rightarrow \mathbb {R}\) be bounded, absolutely continuous in t and measurable in x, with measurable, uniformly bounded derivative tf(t,x). Then under Assumptions (A.2) & (A.1),

$$ M^{f}(t) := f\left( t,X(t)\right) - f\left( 0,X(0)\right) - {{\int}_{0}^{t}}\!\left( \left( \partial_{s} + \mathcal{Q}_{s}\right)f\right)\left( s,X(s)\right) \mathrm{d} s $$

is a Martingale in the filtration \(\left (\mathcal {F}_{t}\right )_{t\geqslant 0}\) generated by X(t).

Proof

In the case that f does not depend on time and \(\sup _{t,x} \alpha _{t}(x,\mathcal {X}) < \infty \) the result follows from [20, Ch. 4 Sect. 7]. The additional term s is added for time-dependent test functions due to a chain rule. By approximating by the process stopped at τn and using Assumptions (A.2)&(A.3) one can remove the boundedness assumption on α. □

Lemma A.2

Under Assumptions(A.1), (A.2) & (A.3), the conclusion of PropositionA.1 is valid whenf(t,x) = ζ(t) ⋅ x and when f(t,x) = eζ(t)⋅x, in both cases for \(\zeta \in C^{1}_{\mathrm {b}}\left ([0,T);\mathcal {X}^{\ast }\right )\) where \(\mathcal {X}^{\ast }\) is the Banach dual of \(\mathcal {X}\).

Proof

The exponential case is proved here; the linear case is similar. Let \(\theta _{n} \in C^{\infty }(\mathbb {R})\) be such that 𝜃n(y) = y for \(y\leqslant n\), \(\theta _{n} \leqslant n+1\) and \(0\leqslant \theta _{n}^{\prime }\leqslant 1\). Take an arbitrary \(\zeta \in C^{1}_{\mathrm {b}}\left ([0,\infty );\mathcal {X}^{\ast }\right )\) and set \(f_{n}(t,x) = \exp \!\left (\theta _{n}\!\left (\zeta (t)\cdot x \right )\right )\) so that Proposition A.1 can be applied to \(M^{f_{n}}(t)\).

It follows from the definitions that for all t and x,

$$ \lim_{n} f_{n}(t,x) = f(t,x) \quad \text{and} \quad \lim_{n} \partial_{t} f_{n}(t,x) = \partial_{t} f(t,x). $$

Because of Assumption (A.1) \(\mathcal {Q}_{t} f\) is well defined and one can prove by dominated convergence that \(\lim _{n} (\mathcal {Q}_{t} f_{n})(t,x)=(\mathcal {Q}_{t} f)(t,x)\) for all t,x. Preparatory to further applications of dominated convergence we estimate

$$ \begin{array}{@{}rcl@{}} f_{n}(t,x) &\leqslant& \exp\left( \|{\zeta}\|_{\infty} |{x}|\right), \\ |{\partial_{t} f_{n}(t,x)}| &\leqslant& \exp\left( \|{\zeta}\|_{\infty} |{x}|\right)\|{\dot \zeta}\|_{\infty}|{x}|, \qquad\text{and}\\ |{(\mathcal{Q}_{t} f_{n})(t,x)}| &\leqslant& \exp\left( \|{\zeta}\|_{\infty} |{x}|\right) \left( \exp\left( \|{\zeta}\|_{\infty} \gamma\right)+1\right)\alpha_{t}(x,\mathcal{X}). \end{array} $$

With these estimates and Assumption (A.3) one checks \(\lim _{n} M^{f_{n}}(t) = M^{f}(t)\) almost surely. Again using Assumption (A.3) one can find a β > 0 such that \(|{M^{f_{n}}(t)}| \leqslant Z^{\beta }(t)\) almost surely. By the conditional expectation form of the dominated convergence theorem, for s < t,

$$ M^{f}(s) = \lim_{n} M^{f_{n}}(s) = \lim_{n} \mathbb{E}\left\lbrack M^{f_{n}}(t) \vert \mathcal{F}_{s}\right\rbrack = \mathbb{E}\left\lbrack\lim_{n} M^{f_{n}}(t) \vert \mathcal{F}_{s}\right\rbrack = \mathbb{E}\left\lbrack M^{f}(t) \vert \mathcal{F}_{s}\right\rbrack. $$

Finally, for the exponential change of measure we will need a bounded time interval.

Theorem A.3

Let \(T<\infty \), \(\zeta \in {C_{b}^{1}}(0,T;\mathbb {R}^{\mathcal {R}})\), and let Assumptions(A.1), (A.2) and (A.3) all hold. Suppose \(\mathbb {P}_{\zeta }\) is the law of some process with paths in Ω and having initial distribution μ. Under \(\mathbb {P}_{\zeta }\), X is a Markov process with generator

$$ (\mathcal{Q}_{\zeta,t}f)(x)= {\int}_{\mathcal{X}} \left[f(y)-f(x) \right] {\mathrm{e}}^{\zeta(t)\cdot y - \zeta(t)\cdot x}\alpha_{t}(x,\mathrm{d} y) $$

if and only if

$$ \log\frac{d\mathbb{P}_{\zeta}}{d\mathbb{P}}(X)= \zeta(T)\cdot X(T) - \zeta(0)\cdot X(0) - {{\int}_{0}^{T}}\!{\mathrm{e}}^{-\zeta(t)\cdot X(t)} \left( \partial_{t} + \mathcal{Q}_{t}\right)e^{\zeta(t)\cdot X(t)}\mathrm{d} t. $$
(A.4)

Proof

We only need to show the direction “⇐=”; the converse then follows immediately from the uniqueness of the generator. To this end define \(\widehat {\mathbb {P}}_{\zeta }\) by (A.4) and let the associated expectation operator be \(\widehat {\mathbb {E}}_{\zeta }\). We sketch a number of steps, similar to [26, Appendix 1, Sect. 7] and [38], by which it is shown that under \(\widehat {\mathbb {P}}_{\zeta }\) X is Markov with generator \(\mathcal {Q}_{\zeta , t}\).

  1. 1.

    Define for t ∈ (0,T), the process

    $$ E(t):=\exp\!\left( \zeta(t)\cdot X(t) - \zeta(0)\cdot X(0) - {{\int}_{0}^{t}}\!{\mathrm{e}}^{-\zeta(s)\cdot X(s)} \left( \partial_{s} + \mathcal{Q}_{s}\right)e^{\zeta(s)\cdot X(s)}\mathrm{d} s \right) $$

    and recall \(E(T) = \lim _{t\nearrow T}E(t)\). By Lemma A.2 above, E(t) is a strictly positive, mean-one \(\mathbb {P}\)-Martingale. One then shows that \(\left .\frac {\mathrm {d} \widehat {\mathbb {P}}_{\zeta }}{\mathrm {d} \mathbb {P}}\right \vert _{\mathcal {F}_{t}} = E(t)\) and \(\left .\frac {\mathrm {d} \mathbb {P}}{\mathrm {d}\widehat {\mathbb {P}}_{\zeta }}\right \vert _{\mathcal {F}_{t}} = \frac {1}{E(t)}\).

  2. 2.

    For any \(Y \in L^{1}({\Omega }, \mathcal {F})\), using the definition of conditional expectation and the results from the previous point, it follows that \(\widehat {\mathbb {E}}_{\zeta }\left [Y\middle \vert \mathcal {F}_{t}\right ] = \mathbb {E}\left [Y E(T)/E(t)\middle \vert \mathcal {F}_{t}\right ]\).

  3. 3.

    Next one can use the result from point 2 to show via conditional expectations under \(\mathbb {P}\) and the \(\mathbb {P}\)-Markov property that for \(t \geqslant s\) and any bounded and measurable \(f:\mathcal {X}\to \mathbb {R}\), we have \(\widehat {\mathbb {E}}_{\zeta }\left [f(X(t)) \middle \vert \mathcal {F}_{s}\right ] =\widehat {\mathbb {E}}_{\zeta }\left [f(X(t)) \middle \vert \sigma (X(s))\right ]\), and so X is \(\widehat {\mathbb {P}}_{\zeta }\)-Markov.

  4. 4.

    Finally, the propagators \((P^{\zeta }_{s,t}f)(x) := \widehat {\mathbb {E}}_{\zeta }\left [f(X(t)) \middle \vert X(s)= x \right ]\) then satisfy \(\frac {\mathrm {d}}{\mathrm {d}s}(P_{s,t}f)(x) = -(\mathcal {Q}_{\zeta ,s} P_{s,t}f)(x)\). This implies that under \(\widehat {\mathbb {P}}_{\zeta }\), X has the same finite dimensional distributions as the process with generator \(\mathcal {Q}_{\zeta , t}\) and thus \(\widehat {\mathbb {P}}_{\zeta } = \mathbb {P}_{\zeta }\).

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Patterson, R.I.A., Renger, D.R.M. Large Deviations of Jump Process Fluxes. Math Phys Anal Geom 22, 21 (2019). https://doi.org/10.1007/s11040-019-9318-4

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Keywords

  • Chemical reaction networks
  • Markov jump processes
  • Large deviations

Mathematics Subject Classification 2010

  • 60F10
  • 60J25
  • 80A30
  • 82C22