where $\mathcal{N}$ is a normalization factor. Now the action is linear in $\delta X_r$ and the path integral over this variable can be performed resulting in a functional Dirac's $\delta$. We arrive at:

$$Z = \int \mathcal{D}[\xi^i] \mathcal{D}[X_{cl}^i] \delta \left( \frac{\delta^2 S_0}{\delta X^i \delta X^j} \bigg|_{\bar{X}} \delta X_{cl}^j(t) - \int dt' G_R^{ij}(t, t'; v) \delta X_{cl}^j(t') - \xi^i(t) \right) \times \exp \left[ -\frac{1}{2} \int \xi^i (G_{\text{sym}}^{-1})^{ij} \xi^j \right]. \quad (F.20)$$

From this expression, one concludes that the fluctuation $\delta X_{cl}(t)$ around the classical particle trajectory $\bar{X}^i(t)$ obeys the Generalized Langevin equation

$$\frac{\delta^2 S_0}{\delta X^i \delta X^j} \bigg|_{\bar{X}} \delta X_{cl}^j(t) - \int dt' G_R^{ij}(t, t'; v) \delta X_{cl}^j(t') = \xi^i(t) \quad (F.21)$$

where the memory kernel $G_R(t, t')$ is the retarded correlator of the force operator $\mathcal{F}$ and with Gaussian noise variance is the symmetric correlator of the same operator.

## F.2 A double-field Ward identity

In the formalism of the previous subsection, it is straightforward to obtain the consistency relation (F.1) between the “classical” and “quantum” terms arising from the interaction with the fluid. By definition, and with the relation (F.17), we have

$$\langle \mathcal{F}^i \rangle_{\bar{X}} = \frac{1}{2} \left\langle \left( \mathcal{F}_+^i + \mathcal{F}_-^i \right) \exp \left\{ i \int dt \bar{X}^i(t) \left( \mathcal{F}_+^i - \mathcal{F}_-^i \right) \right\} \right\rangle_{\text{ens}} \quad (F.22)$$

Specifying a uniform velocity background trajectory,

$$\bar{X}^i(t) = v^i t \quad (F.23)$$

and taking a derivative of (F.22) with respect to $v^j$ we obtain:

$$\frac{\partial \langle \mathcal{F}^i \rangle_{\vec{v}}}{\partial v^j} = \frac{i}{2} \left\langle \int dt' t' \left( \mathcal{F}_+^j(t') - \mathcal{F}_-^j(t') \right) \left( \mathcal{F}_+^i(t) + \mathcal{F}_-^i(t) \right) \right\rangle_{\vec{v}} = \int dt' t' G_R^{ij}(t, t'; \vec{v}) \quad (F.24)$$

where in the last equality we have used the relation (F.11) between the retarded correlator and the double-field path-ordered correlators. Going to Fourier space, $G_R(t, t') = \int (d\omega/2\pi)e^{-i\omega(t-t')}G_R(\omega)$, this becomes:

$$\frac{\partial \langle \mathcal{F}^i \rangle_{\vec{v}}}{\partial v^j} = -i \frac{d G_R^{ij}(\omega; v)}{d\omega} \bigg|_{\omega=0} \quad (F.25)$$

From this general relation, projecting over $v^j$ and noting that $d/dv = (v^j/v)\partial/\partial v^j$ we obtain (F.1).

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
References

[1] E. Witten, *Anti-de Sitter space, thermal phase transition and confinement in gauge theories*, *Adv. Theor. Math. Phys.* **2** (1998) 505 [hep-th/9803131] [INSPIRE].

[2] J.M. Maldacena and C. Núñez, *Towards the large $N$ limit of pure $N = 1$ super Yang-Mills*, *Phys. Rev. Lett.* **86** (2001) 588 [hep-th/0008001] [INSPIRE].

[3] I.R. Klebanov and M.J. Strassler, *Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities*, JHEP **08** (2000) 052 [hep-th/0007191] [INSPIRE].

[4] T. Sakai and S. Sugimoto, *Low energy hadron physics in holographic QCD*, *Prog. Theor. Phys.* **113** (2005) 843 [hep-th/0412141] [INSPIRE].

[5] S. Kuperstein and J. Sonnenschein, *Non-critical, near extremal $AdS_6$ background as a holographic laboratory of four dimensional YM theory*, JHEP **11** (2004) 026 [hep-th/0411009] [INSPIRE].

[6] J. Erlich, E. Katz, D.T. Son and M.A. Stephanov, *QCD and a holographic model of hadrons*, *Phys. Rev. Lett.* **95** (2005) 261602 [hep-ph/0501128] [INSPIRE].

[7] L. Da Rold and A. Pomarol, *Chiral symmetry breaking from five dimensional spaces*, Nucl. Phys. **B** **721** (2005) 79 [hep-ph/0501218] [INSPIRE].

[8] U. Gürsoy and E. Kiritsis, *Exploring improved holographic theories for QCD: Part I*, JHEP **02** (2008) 032 [arXiv:0707.1324] [INSPIRE].

[9] U. Gürsoy, E. Kiritsis and F. Nitti, *Exploring improved holographic theories for QCD: Part II*, JHEP **02** (2008) 019 [arXiv:0707.1349] [INSPIRE].

[10] S.S. Gubser and A. Nellore, *Mimicking the QCD equation of state with a dual black hole*, *Phys. Rev. D* **78** (2008) 086007 [arXiv:0804.0434] [INSPIRE].

[11] C. Herzog, A. Karch, P. Kovtun, C. Kozcaz and L. Yaffe, *Energy loss of a heavy quark moving through $N = 4$ supersymmetric Yang-Mills plasma*, JHEP **07** (2006) 013 [hep-th/0605158] [INSPIRE].

[12] S.S. Gubser, *Drag force in AdS/CFT*, *Phys. Rev. D* **74** (2006) 126005 [hep-th/0605182] [INSPIRE].

[13] H. Liu, K. Rajagopal and U.A. Wiedemann, *Calculating the jet quenching parameter from AdS/CFT*, *Phys. Rev. Lett.* **97** (2006) 182301 [hep-ph/0605178] [INSPIRE].

[14] H. Liu, K. Rajagopal and U.A. Wiedemann, *Wilson loops in heavy ion collisions and their calculation in AdS/CFT*, JHEP **03** (2007) 066 [hep-ph/0612168] [INSPIRE].

[15] J. Casalderrey-Solana and D. Teaney, *Heavy quark diffusion in strongly coupled $N = 4$ Yang-Mills*, *Phys. Rev. D* **74** (2006) 085012 [hep-ph/0605199] [INSPIRE].

[16] S.S. Gubser, *Momentum fluctuations of heavy quarks in the gauge-string duality*, Nucl. Phys. **B** **790** (2008) 175 [hep-th/0612143] [INSPIRE].

[17] J. Casalderrey-Solana and D. Teaney, *Transverse Momentum Broadening of a Fast Quark in a $N = 4$ Yang-Mills Plasma*, JHEP **04** (2007) 039 [hep-th/0701123] [INSPIRE].

[18] J. de Boer, V.E. Hubeny, M. Rangamani and M. Shigemori, *Brownian motion in AdS/CFT*, JHEP **07** (2009) 094 [arXiv:0812.5112] [INSPIRE].
[19] D.T. Son and D. Teaney, Thermal Noise and Stochastic Strings in AdS/CFT, JHEP 07 (2009) 021 [arXiv:0901.2338] [INSPIRE].

[20] G. Giecold, E. Iancu and A. Mueller, Stochastic trailing string and Langevin dynamics from AdS/CFT, JHEP 07 (2009) 033 [arXiv:0903.1840] [INSPIRE].

[21] C. Hoyos-Badajoz, Drag and jet quenching of heavy quarks in a strongly coupled $N = 2^*$ plasma, JHEP 09 (2009) 068 [arXiv:0907.5036] [INSPIRE].

[22] U. Gürsoy, E. Kiritsis, G. Michalogiorgakis and F. Nitti, Thermal Transport and Drag Force in Improved Holographic QCD, JHEP 12 (2009) 056 [arXiv:0906.1890] [INSPIRE].

[23] U. Gürsoy, E. Kiritsis, L. Mazzanti and F. Nitti, Langevin diffusion of heavy quarks in non-conformal holographic backgrounds, JHEP 12 (2010) 088 [arXiv:1006.3261] [INSPIRE].

[24] A. Magana, J. Mas, L. Mazzanti and J. Tarrio, Probes on D3-D7 quark-gluon Plasmas, JHEP 07 (2012) 058 [arXiv:1205.6176] [INSPIRE].

[25] E. Kiritsis, L. Mazzanti and F. Nitti, Dressed spectral densities for heavy quark diffusion in holographic plasmas, J. Phys. G 39 (2012) 054003 [arXiv:1111.1008] [INSPIRE].

[26] P.M. Hohler and M.A. Stephanov, Bulk spectral function sum rule in QCD-like theories with a holographic dual, JHEP 06 (2011) 130 [arXiv:1103.0977] [INSPIRE].

[27] S. Caron-Huot, P.M. Chesler and D. Teaney, Fluctuation, dissipation and thermalization in non-equilibrium AdS$_5$ black hole geometries, Phys. Rev. D 84 (2011) 026012 [arXiv:1102.1073] [INSPIRE].

[28] J.M. Maldacena, Wilson loops in large $N$ field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].

[29] S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large $N$ gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [INSPIRE].

[30] Y. Kinar, E. Schreiber and J. Sonnenschein, $Q\bar{Q}$ potential from strings in curved space-time: Classical results, Nucl. Phys. B 566 (2000) 103 [hep-th/9811192] [INSPIRE].

[31] Y. Kinar, E. Schreiber, J. Sonnenschein and N. Weiss, Quantum fluctuations of Wilson loops from string models, Nucl. Phys. B 583 (2000) 76 [hep-th/9911123] [INSPIRE].

[32] U. Gürsoy, E. Kiritsis, L. Mazzanti, G. Michalogiorgakis and F. Nitti, Improved Holographic QCD, Lect. Notes Phys. 828 (2011) 79 [arXiv:1006.5461] [INSPIRE].

[33] A.E. Lawrence and E.J. Martinec, Black hole evaporation along macroscopic strings, Phys. Rev. D 50 (1994) 2680 [hep-th/9312127] [INSPIRE].

[34] U. Gürsoy, E. Kiritsis, L. Mazzanti and F. Nitti, Deconfinement and Gluon Plasma Dynamics in Improved Holographic QCD, Phys. Rev. Lett. 101 (2008) 181601 [arXiv:0804.0899] [INSPIRE].

[35] U. Gürsoy, E. Kiritsis, L. Mazzanti and F. Nitti, Holography and Thermodynamics of 5D Dilaton-gravity, JHEP 05 (2009) 033 [arXiv:0812.0792] [INSPIRE].

[36] U. Gürsoy, E. Kiritsis, L. Mazzanti and F. Nitti, Improved Holographic Yang-Mills at Finite Temperature: Comparison with Data, Nucl. Phys. B 820 (2009) 148 [arXiv:0903.2859] [INSPIRE].

[37] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific, Singapore (2004).