1 Introduction↩︎

In the classical Sherrington–Kirkpatrick (SK) model, the celebrated work of Parisi [1], [2] predicted that its free energy is given by the so-called Parisi formula. Guerra [3] and Talagrand [4] established Parisi’s prediction rigorously, and Panchenko [5] extended the Parisi formula to mixed \(p\)-spin models.

Multispecies spin glasses generalize the SK model by partitioning the spins into \(D\) subsets, called species. Mathematically, the model is defined as follows.

Throughout the paper, we assume that the proportion of each block in the partition is nondegenerate as \(N \to \infty\). More precisely, letting \[N_{N,d}=|I_{N,d}|, \qquad \lambda_{N,d}=N_{N,d}/N, \qquad \boldsymbol{\lambda}_N=(\lambda_{N,1},\ldots,\lambda_{N,D}),\] we assume that \(\boldsymbol{\lambda}_N\to\boldsymbol{\lambda}=(\lambda_1,\ldots,\lambda_D)\), where \(\lambda_d>0\) for every \(d\in\{1,\ldots,D\}\) and \(\sum_{d=1}^D\lambda_d=1\).

Given the parameters \((\xi,\boldsymbol{\lambda}_N)\), the free energy of the multispecies model is defined as \[\require{physics} \label{eq:hopfapp-positive-free-energy} F_N(\xi,\boldsymbol{\lambda}_N) = \frac{1}{N}\mathbb{E}\log \int_{\Sigma_N}\exp(H_N(\sigma))\dd{P_N(\sigma)},\tag{1}\] where \(P_N=(\frac{1}{2}\delta_{-1}+\frac{1}{2}\delta_1)^{\otimes N}\) is the uniform probability measure on \(\Sigma_N\). When \(D=1\), we denote by \(F_N(\xi)\) the free energy of the one-species model with the covariance function \(\xi\).

As in the one-species case, the question of interest is to compute the limiting free energy. The first rigorous result for the limiting free energy of multispecies spin glasses was obtained by Barra, Contucci, Mingione, and Tantari [6] and Panchenko [7]. However, the method of Barra et al. [6] requires convexity of \(\xi\), so it does not cover, for example, the bipartite model, where \(D=2\) and \(\xi(x_1,x_2)=x_1x_2\).

To tackle nonconvex models, Mourrat and coauthors developed a Hamilton–Jacobi framework to study the free energy of spin-glasses with possibly nonconvex covariance. For a pedagogical discussion of this approach, see [8]. In [9], Mourrat proved an upper bound¹ for the free energy of the bipartite model, and he later proved the same type of upper bound for nonconvex vector spin glasses in [10].

On the other hand, for the lower bound, less is known in the literature. When \(\xi\) has permutation invariant coordinates, Issa proved (see Theorem 1.4 in [11]) a lower bound for the limit inferior of the free energy via a one-species model constructed from symmetrizing \(\xi\). Recently, Bates and Sohn [12] showed that for the balanced multispecies models, the limit inferior of the free energy is lower bounded by the Parisi formula of a one-species mixed model.

We recall a few examples of the balanced models.

Our goal is to match Bates and Sohn’s lower bound for the balanced models with Mourrat’s upper bound.

To find the matching upper bound, a key input is the convexity of a functional, which we will define in the next paragraph. This convexity will then imply the convexity of 9 , and will allow us to apply a Hopf upper bound, see 8 below, under the Hamilton–Jacobi framework.

For all \(p\in [1,\infty)\), introduce the path space \[\label{eq:hopfapp-scalar-path-space} \mathcal{Q}_p = \Bigl\{\mathsf{q}\in L^p([0,1);\mathbb{R}_+) \,\Big|\, \mathsf{q}\text{ is c\`adl\`ag and nondecreasing} \Bigr\}.\tag{3}\] When \(\mathsf{q}\in\mathcal{Q}_\infty:=\mathcal{Q}_1\cap L^\infty\), set \(\mathsf{q}(1):=\lim_{r\uparrow1}\mathsf{q}(r)\).

For \(\mathsf{q}\in\mathcal{Q}_\infty\), define the functional \[\label{eq:def-psi} \psi(\mathsf{q})=\mathsf{q}(1)-\Phi^{2\mathsf{q}}(0,0),\tag{4}\] where \(\Phi^\mathsf{q}\) is the solution of a Parisi PDE \[\label{eq:q-pde} \begin{cases} -\partial_\tau\Phi^\mathsf{q}(\tau,x) = \dfrac12\left( \partial_{xx}\Phi^\mathsf{q}(\tau,x) + \mathsf{q}^{-1}(\tau)(\partial_x\Phi^\mathsf{q}(\tau,x))^2 \right), & \tau\in [0,\mathsf{q}(1)), \\[1ex] \Phi^\mathsf{q}(\mathsf{q}(1),x)=\log\cosh x, & x\in\mathbb{R}. \end{cases}\tag{5}\] Here, \(\mathsf{q}^{-1}:[0,\mathsf{q}(1)]\to[0,1]\) denotes the right generalized inverse of \(\mathsf{q}\).

We now state the convexity of the functional \(\psi\).

Since \(\mathsf{q}\mapsto \mathsf{q}(1)\) is affine and \(\mathsf{q}\mapsto 2\mathsf{q}\) is linear, the convexity of \(\psi\) is equivalent to the concavity of \(\mathsf{q}\mapsto \Phi^{\mathsf{q}}(0,0)\). On the other hand, from the form of the Parisi PDE 5 , the dependence of \(\Phi^{\mathsf{q}}\) on \(\mathsf{q}\) seems to be highly nonlinear. To facilitate the analysis, we thus consider a change of variables. Formally, on a dense class of paths with sufficient regularity, the change of variables \[\Psi^{\dot{\mathsf{q}}}(s,x)=\Phi^\mathsf{q}(\mathsf{q}(s),x)\] transforms the problem into the concavity of \(\gamma\mapsto\Psi^\gamma(0,0)\). Moreover, the PDE for \(\Psi^\gamma\) is a time-changed Parisi PDE with diffusion profile \(\gamma\). The precise formulation will be given in Section 3 below, and we will transform the problem into Theorem 8.

We remark that the concavity of \(\mathsf{q}\mapsto\Phi^{\mathsf{q}}(0,0)\) should be distinguished from the strict convexity of the map \(\mu\mapsto \Phi^\mu(0,0)\) on \(\mu\in\mathrm{Pr}([0,1])\), proven by Auffinger and Chen [13]. Here, \(\mathrm{Pr}([0,1])\) denotes the set of probability measures on \([0,1]\). Indeed, if we restrict \(\mathcal{Q}_\infty\) to the set of quantile functions on \([0,1]\), then \(\Phi^\mu(0,0)\) can be viewed as a functional on \(\mu\in\mathrm{Pr}([0,1])\). The strict convexity of \(\mu\mapsto \Phi^\mu(0,0)\) was used in [13] to establish the uniqueness of the Parisi measure.

While the previous exposition focuses on Ising multispecies models, the multispecies models also admit a spherical version, where the spins are constrained to lie on the sphere. In particular, Bates and Sohn’s lower bound also holds for the balanced spherical multispecies models. For the earlier works, Baik and Lee [14] determined the limiting free energy of the bipartite spherical model. Bates and Sohn [15] proved a Parisi formula for multispecies mixed spherical models under a convexity assumption in the upper bound. Subag [16], [17] developed a TAP approach for multispecies spherical models and used it to compute the limiting free energy for pure multispecies spherical models under appropriate convergence assumptions. Although the present paper does not treat the spherical case, the method is expected to extend to that setting. This is left for future work.

Organization↩︎

Section 2 applies Theorem 5 to prove Theorem 3. Section 3 introduces the time-changed Parisi PDE and reformulates the problem into Theorem 8. Then, the proof of Theorem 8 occupies Section 4–Section 7, and Section 3.1 provides the proof heuristic and outline of Theorem 8. Finally, Section 8 proves Theorem 5.

2 Proof of Theorem 3↩︎

For a balanced pair \((\xi,\boldsymbol{\lambda})\), as mentioned in the introduction, Theorem 1.3 in [12] showed that \[\liminf_{N\rightarrow\infty} F_N(\xi,\boldsymbol{\lambda}_N) \geq \lim_{N\rightarrow\infty} F_N(\xi^{\mathrm{1sp}}). \label{eq:balanced-superior}\tag{6}\] Therefore, it suffices to show that \[\limsup_{N\rightarrow\infty} F_N(\xi,\boldsymbol{\lambda}_N) \leq \lim_{N\rightarrow\infty} F_N(\xi^{\mathrm{1sp}}). \label{rjuychsp}\tag{7}\]

Inputs from the Hamilton–Jacobi framework↩︎

We first record the two consequences of the Hamilton–Jacobi framework. Define the Hopf formula with the parameters \((\xi,\boldsymbol{\lambda})\). \[\require{physics} \mathscr H(\xi,\boldsymbol{\lambda}) = \sup_{\mathsf{p}\in \mathcal{Q}_\infty^D} \inf_{\mathsf{q}\in \mathcal{Q}_\infty^D} \left\{ \sum_{d=1}^D \lambda_d \psi(\mathsf{q}_d) - \sum_{d=1}^D \int_0^1 \mathsf{p}_d(s)\mathsf{q}_d(s)\dd{s} + \frac{1}{2} \int_0^1 \xi(\mathsf{p}(s)) \dd{s} \right\}.\] For a one-species model with the parameter \(\xi\), we define \[\require{physics} \mathscr H(\xi) = \sup_{\mathsf{p}\in \mathcal{Q}_\infty} \inf_{\mathsf{q}\in \mathcal{Q}_\infty} \left\{ \psi(\mathsf{q}) - \int_0^1 \mathsf{p}(s)\mathsf{q}(s)\dd{s} + \frac{1}{2} \int_0^1 \xi(\mathsf{p}(s)) \dd{s} \right\}.\]

With the notation above, we now state the two consequences where the sign and normalization are converted to the usual conventions.

1. Given a multispecies model with parameters \((\xi,\boldsymbol{\lambda})\), we have \[\limsup_{N\rightarrow\infty} F_N(\xi,\boldsymbol{\lambda}_N) \leq \frac{1}{2}\xi(\boldsymbol{\lambda}) - \mathscr H(\xi,\boldsymbol{\lambda}).\label{eq:HJ1}\tag{8}\] The upper bound 8 is a specialized version of Chen–Xia’s reformulation (Theorem 4.14 in [18]) of Mourrat’s upper bound (Theorem 3.4 in [10]), combined with their Hopf representation (Theorem 4.7(3) in [18]). These results are applicable because we can identify the multispecies Ising model as a vector spin model by sending a spin \(\sigma_d\) in species \(d\) to the vector \(\sigma_d e_d\in\mathbb{R}^D\). Then, the vector overlap matrix is diagonal, with diagonal entries \(R_{N,d}\), and the covariance is obtained from \[\hat{\xi}(A)=\xi(A_{11},\ldots,A_{DD}),\] where \(A\) is a \(D\)-by-\(D\) symmetric matrix. Thus, on diagonal paths, \(\hat{\xi}\) is just the original multispecies covariance function.

   Therefore, we can apply Theorem 4.14 in [18] to bound the limit superior of the free energy with the Lipschitz viscosity solution of the Hamilton–Jacobi equation stated in that paper. Then, we apply Theorem 4.7(3) in [18] to identify that Lipschitz viscosity solution with their Hopf formula with \(t=1/2\), \(\mu=\mathsf{0}\). Here, we replace their \(\psi\) with the specialization for multispecies Ising models \[\sum_{d=1}^D \lambda_d \psi(\mathsf{q}_d), \label{eq:multispecies-initial-cond}\tag{9}\] so the convexity of 9 is provided by Theorem 5.

2. Given a one-species model with the covariance function \(\xi\), we have \[\lim_{N\rightarrow\infty} F_N(\xi) = \frac{1}{2}\xi(1) - \mathscr H(\xi).\label{eq:HJ2}\tag{10}\] For (HJ2), Theorem 4.7(3) in [18] identifies the one-species Hopf formula with the unique Lipschitz viscosity solution, while Theorem 1.1 in [19] identifies that solution with the limiting free energy. Their covariance class allows power series beginning at \(k=1\); see (1.24) in [19].

Proof of 6↩︎

We can restrict the supremum in 8 to the paths \(\hat{\mathsf{p}}= (\lambda_1\mathsf{p},\ldots,\lambda_D\mathsf{p})\) for all \(\mathsf{p}\in \mathcal{Q}_\infty\). Moreover, the homogeneity of each \(\xi_k\) yields \[\xi(\hat{\mathsf{p}}) = \xi(\lambda_1\mathsf{p},\ldots,\lambda_D\mathsf{p}) = \sum_{k\geq 1} \biggl( \sum_{\substack{\boldsymbol{p}\in\mathsf P_k}} \Delta_{\boldsymbol{p}}^2\prod_{d=1}^D \lambda_d^{p_d} \biggr) \mathsf{p}^k = \sum_{k\geq 1}\xi_k(\boldsymbol{\lambda})\mathsf{p}^k = \xi^{\mathrm{1sp}}(\mathsf{p}),\] so we obtain the lower bound \[\require{physics} \mathscr H(\xi,\boldsymbol{\lambda}) \geq \sup_{\mathsf{p}\in \mathcal{Q}_\infty} \inf_{\mathsf{q}\in \mathcal{Q}_\infty^D} \left\{ \sum_{d=1}^D \lambda_d \psi(\mathsf{q}_d) - \int_0^1 \mathsf{p}(s) \sum_{d=1}^D \lambda_d\mathsf{q}_d(s)\dd{s} + \frac{1}{2} \int_0^1 \xi^{\mathrm{1sp}}(\mathsf{p}(s)) \dd{s} \right\}. \label{eq:Hopf462}\tag{11}\] Now, note that we assume \(\lambda_d>0\) for every \(d\in\{1,\ldots,D\}\) and \(\sum_{d=1}^D \lambda_d = 1\), which defines a probability measure on \(\{1,\ldots,D\}\). Applying Jensen’s inequality with this measure to \(\psi\) yields \[\psi\Bigl(\sum_{d=1}^D \lambda_d \mathsf{q}_d\Bigr) \leq \sum_{d=1}^D \lambda_d \psi(\mathsf{q}_d). \label{eq:Jensen-psi}\tag{12}\] Combining 12 with the inclusion \[\overline{\mathcal{Q}}_{D,\infty} \mathrel{\vcenter{:}}= \Bigl\{ \bar\mathsf{q} \,\Big|\, \bar\mathsf{q} = \sum_{d=1}^D \lambda_d \mathsf{q}_d, \, \mathsf{q}\in \mathcal{Q}_\infty^D \Bigr\} \subseteq \mathcal{Q}_\infty,\] we obtain \[\require{physics} \begin{align} \inf_{\mathsf{q}\in \mathcal{Q}_\infty^D} \left\{ \sum_{d=1}^D \lambda_d \psi(\mathsf{q}_d) - \int_0^1 \mathsf{p}(s) \sum_{d=1}^D \lambda_d\mathsf{q}_d(s)\dd{s} \right\} &\geq \inf_{\bar\mathsf{q}\in \overline{\mathcal{Q}}_{D,\infty}} \left\{ \psi(\bar\mathsf{q})-\int_0^1 \mathsf{p}(s)\bar\mathsf{q}(s)\dd{s} \right\} \nonumber \\ &\geq \inf_{\mathsf{q}\in \mathcal{Q}_\infty} \left\{ \psi(\mathsf{q})-\int_0^1 \mathsf{p}(s)\mathsf{q}(s)\dd{s} \right\}. \label{eq:Hopf-inf} \end{align}\tag{13}\] Combining 8 , 11 and 13 and noting that \(\xi(\boldsymbol{\lambda})=\xi^{\mathrm{1sp}}(1)\), we obtain \[\limsup_{N\rightarrow\infty} F_N(\xi,\boldsymbol{\lambda}_N) \leq \frac{1}{2}\xi^{\mathrm{1sp}}(1) - \mathscr H(\xi^{\mathrm{1sp}}) = \lim_{N\rightarrow\infty} F_N(\xi^{\mathrm{1sp}}),\] where the equality above is provided by 10 .

3 Time-changed Parisi PDE↩︎

As mentioned in the introduction, we want to perform a change of variables on a dense class with sufficient regularity to simplify the analysis. From now on, fix \(\alpha\in(0,1)\). Introduce the following two classes of paths. \[\begin{align} \mathcal{S}_\alpha &= \Bigl\{ \mathsf{q}\in C^{1+\alpha/2}([0,1]) \,\Big|\, \mathsf{q}(0)=0,\;\inf_{[0,1]}\dot{\mathsf{q}}>0 \Bigr\}, \tag{14} \\ \mathcal{U}_\alpha &= \Bigl\{\gamma\in C^{\alpha/2}([0,1]) \,\Big|\, \inf_{[0,1]}\gamma>0\Bigr\}. \tag{15} \end{align}\] Note that \(\mathsf{q}\mapsto \dot{\mathsf{q}}\) maps \(\mathcal{S}_\alpha\) to \(\mathcal{U}_\alpha\). Introduce the change of variables \[\label{eq:gamma-change-of-variables} \Psi^{\dot{\mathsf{q}}}(s,x)=\Phi^{\mathsf{q}}(\mathsf{q}(s),x), \qquad (s,x)\in[0,1]\times\mathbb{R}.\tag{16}\] Given \(\gamma\in\mathcal{U}_\alpha\), \(\Psi^\gamma\) solves a time-changed Parisi PDE. \[\label{eq:gamma-pde} \begin{cases} -\partial_s\Psi^\gamma(s,x) = \dfrac12\gamma(s)\bigl(\partial_{xx}\Psi^\gamma(s,x) + s\, (\partial_x\Psi^\gamma(s,x))^2\bigr), & (s,x)\in [0,1)\times\mathbb{R}, \\[0.5ex] \Psi^\gamma(1,x)=\log\cosh x, & x\in\mathbb{R}. \end{cases}\tag{17}\] Interestingly, 17 already appeared in (16) of Parisi’s original paper [2].

Since the map \(\mathsf{q}\mapsto \dot{\mathsf{q}}\) is linear, to prove the concavity of \(\mathsf{q}\mapsto\Phi^\mathsf{q}(0,0)\), it suffices to prove the concavity of the map \(\gamma\mapsto\Psi^\gamma(0,0)\) defined on \(\mathcal{U}_\alpha\). To achieve this, a natural strategy is to compute its second Fréchet derivative. The standard theory of parabolic PDEs in Hölder spaces provides the needed regularity, and we refer to Krylov [20] for a pedagogical treatment of this theory.

Based on the previous discussions, the following theorem implies Theorem 5.

Auffinger and Chen’s raywise concavity↩︎

Theorem 1 in [21] gives concavity only along the one-dimensional ray generated by the fixed profile \(\gamma_0\). To explain this, we briefly recall their result. Let \(\xi(x) = \sum_{k=2}^\infty \Delta_k^2 x^k\) be the covariance function of a one-species mixed model. Let \(\mathsf{q}\) be a quantile function on \([0,1]\) and \(\mathsf{q}^{-1}\) be its generalized right inverse, or the corresponding CDF. Let \(\Phi_{\beta}\) be the solution of the Parisi PDE \[\label{eq:Parisi-pde} \begin{cases} -\partial_t\Phi_{\beta}(t,x) = \dfrac{\beta^2\xi''(t)}{2}\left( \partial_{xx}\Phi_{\beta}(t,x) + \mathsf{q}^{-1}(t)(\partial_x\Phi_{\beta}(t,x))^2 \right), & t\in [0,1), \\[2ex] \Phi_{\beta}(1,x)=\log\cosh x, & x\in\mathbb{R}. \end{cases}\tag{18}\] Theorem 1 in [21] proved that for fixed \(\xi\) and \(\mathsf{q}\), the map \(\beta^2\mapsto\Phi_{\beta}(0,0)\) is concave. Assuming for this discussion that \(\mathsf{q}\) is smooth and strictly increasing, then \(\Phi_\beta(\mathsf{q}(s),x)\) solves 17 with \(\gamma=\beta^2\gamma_0\) and \(\gamma_0(s)=\dot{\mathsf{q}}(s)\xi''(\mathsf{q}(s))\). In particular, \[\Phi_\beta(0,0) = \Psi^{\beta^2\gamma_0}(0,0).\] Therefore, their result may be viewed as the raywise concavity of \[c\longmapsto \Psi^{c\gamma_0}(0,0).\]

3.1 Proof heuristics and outline of Theorem 8↩︎

The proof is guided by the preceding interpretation of Auffinger–Chen’s argument. Rather than working in discretized time, we work directly with the time-changed PDE 17 and its transition operators. First, Schauder’s estimates in Hölder spaces and the Banach implicit function theorem give the \(C^2\)-regularity of the solution map \[\gamma\longmapsto \Psi^\gamma-\log\cosh ,\] which proves Theorem 8(1). Differentiating 17 with respect to \(\gamma\) then gives linearized equations for the first and second Fréchet derivatives.

The main step is to apply the Feynman–Kac formula to these variation equations and rewrite the second derivative in a min-kernel form. This representation reduces the sign of \(D^2\Psi^\gamma\) to the spatial symmetries preserved by the transition operator. We leave the details to Section 7.

Organization of the proof of Theorem 8↩︎

In Section 4 and Section 5, we introduce the required machinery from the standard parabolic PDE theory and known properties of the solution of the Parisi PDE. These allow us to prove Theorem 8(1) in Section 5.1. Then, the PDEs for the Fréchet derivatives are provided in Section 5.2. Using these PDEs and the Feynman–Kac formula, we derive the min-kernel representation in Section 6. Then, the details of the cone preservation argument are in Section 7, and this leads to the proof of Theorem 8(2) and Theorem 8(3) in Section 7.1.

4 PDE preliminaries↩︎

This section collects the conventions used in the rest of the paper. The notation in Section 2 is independent of this section.

4.1 Notation and conventions↩︎

Unless otherwise stated, functions defined on only one factor of \([0,1]\times\mathbb{R}\) are identified with their canonical lifts: if \(a:[0,1]\to\mathbb{R}\), then \(a(s,x)=a(s)\), and if \(f:\mathbb{R}\to\mathbb{R}\), then \(f(s,x)=f(x)\). We write \(\|\cdot\|_\infty\) for the relevant supremum norm.

For \(t\in[0,1]\), \(C([0,t];C_b^2(\mathbb{R}))\) denotes the space of continuous maps \[r\mapsto u(r,\cdot)\in C_b^2(\mathbb{R}),\] equipped with the supremum norm. We write \(C_b^{1,2}([0,t]\times\mathbb{R})\) for the space of functions that are \(C^1\) in time and \(C^2\) in space, with all derivatives up to these orders bounded and continuous. Classical solutions are understood in this sense.

For \(\gamma\in \mathcal{U}_\alpha\), let \(\Psi^\gamma\) be the solution of the Parisi PDE, and set \[g(x)=\log\cosh x,\qquad \boldsymbol{\Psi}(\gamma)=\Psi^\gamma-g.\] For fixed \(\gamma\in \mathcal{U}_\alpha\), define \[\begin{align} L_s^\gamma u(s,x) &= \frac{1}{2}\partial_{xx}u(s,x) + s\,\partial_x\Psi^\gamma(s,\cdot)\,\partial_x u(s,x), \tag{19} \\ \mathscr{D}_s u(s,x) &= \partial_s u(s,x) + \gamma(s)L_s^\gamma u(s,x), \tag{20} \end{align}\] for all \((s,x)\in[0,1]\times\mathbb{R}\).

Finally, set \[v^\gamma = \partial_{xx}\Psi^\gamma+s(\partial_x\Psi^\gamma)^2.\]

4.2 Parabolic PDEs in Hölder spaces↩︎

This section recalls the parabolic Hölder theory used throughout the paper, which includes standard facts of parabolic Hölder norms, the maximum principle, and Schauder estimates. Most of the contents are based on Krylov’s lecture note [20].

For a time interval \(I\subset[0,1]\), write \[Q_I:=I\times\mathbb{R}.\] Below, when \(I=[0,1]\) or \(Q_I=[0,1]\times\mathbb{R}\), we omit \(I\) and \(Q_I\) from the notation.

4.2.1 Parabolic Hölder norms↩︎

We write \(C_b^{\alpha/2}(I)\) for the Banach space of bounded continuous functions \(\gamma:I\to\mathbb{R}\) such that \[[\gamma]_{\alpha/2;I} := \sup_{\substack{s,t\in I\\s\neq t}} \frac{|\gamma(t)-\gamma(s)|}{|t-s|^{\alpha/2}} <\infty , \qquad \norm{\gamma}_{\alpha/2;I} := \norm{\gamma}_{\infty;I} + [\gamma]_{\alpha/2;I}.\] On compact intervals this agrees with the usual Hölder space \(C^{\alpha/2}(I)\).

For \(m\in\mathbb{N}_0\), we write \(C_b^m(\mathbb{R})\) for the space of functions whose derivatives \(\partial_x^j f\), \(0\le j\le m\), are bounded and continuous. We write \(C_b^{m+\alpha}(\mathbb{R})\) for the subspace of \(C_b^m(\mathbb{R})\) such that \(\partial_x^m f\) is \(\alpha\)-Hölder, with norm \[\norm{u}_{m+\alpha;\mathbb{R}} := \sum_{j=0}^{m} \norm*{\partial_x^j u}_{\infty;\mathbb{R}} + [\partial_x^m u]_{\alpha;\mathbb{R}},\] where \[[w]_{\alpha;\mathbb{R}} := \sup_{\substack{x,y\in\mathbb{R}\\x\neq y}} \frac{|w(x)-w(y)|}{|x-y|^\alpha}.\]

For \(u:Q_I\to\mathbb{R}\), define \[[u]_{\alpha/2,t;Q_I} := \sup_{x\in\mathbb{R}} \sup_{\substack{s,t\in I\\s\neq t}} \frac{|u(t,x)-u(s,x)|}{|t-s|^{\alpha/2}}, \qquad [u]_{\alpha,x;Q_I} := \sup_{t\in I} \sup_{\substack{x,y\in\mathbb{R}\\x\neq y}} \frac{|u(t,x)-u(t,y)|}{|x-y|^\alpha}.\] Set \[[u]_{\alpha/2,\alpha;Q_I} := [u]_{\alpha/2,t;Q_I} + [u]_{\alpha,x;Q_I}, \qquad \norm{u}_{\alpha/2,\alpha;Q_I} := \norm{u}_{\infty;Q_I} + [u]_{\alpha/2,\alpha;Q_I}.\] We write \[C_b^{\alpha/2,\alpha}(Q_I) := \bigl\{ u\in C_b(Q_I): \norm{u}_{\alpha/2,\alpha;Q_I}<\infty \bigr\}.\]

For \(m\in\mathbb{N}_0\), we write \(C_b^{\alpha/2,m+\alpha}(Q_I)\) for the space of functions \(u:Q_I\to\mathbb{R}\) such that, for every \(0\le j\le m\), the spatial derivative \(\partial_x^j u\) exists and belongs to \(C_b^{\alpha/2,\alpha}(Q_I)\). The norm is \[\norm{u}_{\alpha/2,m+\alpha;Q_I} := \sum_{j=0}^{m} \norm*{\partial_x^j u}_{\alpha/2,\alpha;Q_I}.\]

Finally, we write \(C_b^{1+\alpha/2,2+\alpha}(Q_I)\) for the standard second-order parabolic Hölder space of functions \(u\) whose derivatives \(\partial_t^i\partial_x^j u\), \(0\le 2i+j\le2\), are bounded and continuous on \(Q_I\), and whose top parabolic derivatives \(\partial_tu\) and \(\partial_{xx}u\) belong to \(C_b^{\alpha/2,\alpha}(Q_I)\). We equip it with the norm \[\norm{u}_{1+\alpha/2,2+\alpha;Q_I} := \sum_{0\le 2i+j\le2} \norm*{\partial_t^i\partial_x^j u}_{\infty;Q_I} + [\partial_t u]_{\alpha/2,\alpha;Q_I} + [\partial_{xx}u]_{\alpha/2,\alpha;Q_I}.\]

All spaces above are Banach spaces with their indicated norms.

We will use without further comment the following standard facts.

1. For all \(v,w\in C_b^{{\alpha}/2,{\alpha}}([0,1]\times\mathbb{R})\), \[\norm{vw}_{\alpha/2,\alpha} \le \norm{v}_{\alpha/2,\alpha} \norm{w}_{\alpha/2,\alpha}.\]

2. There exists a constant \(C=C_\alpha>0\) such that for all \(v,w\in C_{b}^{1+{\alpha}/2,2+{\alpha}}([0,1]\times\mathbb{R})\), \[\norm{vw}_{1+\alpha/2,2+\alpha} \le C \norm{v}_{1+\alpha/2,2+\alpha} \norm{w}_{1+\alpha/2,2+\alpha}.\]

3. There exists \(C=C_\alpha>0\) such that for all \(u\in C_{b}^{1+{\alpha}/2,2+{\alpha}}([0,1]\times\mathbb{R})\), \[\norm{u}_{\alpha/2,\alpha} \le C\norm{u}_{1+\alpha/2,2+\alpha}.\]

We also record a lemma stating that the relevant derivatives define bounded operators between the corresponding parabolic Hölder spaces.

4.2.2 Maximum principle↩︎

We will use the weak maximum principle in Krylov’s lecture note only in the following form. Let \(\Omega\) be either \(\mathbb{R}\) or \((0,\infty)\). Fix \(T>0\). Define \[Q_T^\Omega:=(0,T)\times\Omega.\] Define the terminal-time parabolic boundary by \[\partial'_T Q_T^\Omega := \bigl((0,T)\times\partial\Omega\bigr) \cup \bigl(\{T\}\times\overline{\Omega}\bigr).\]

Let \[\mathscr L_t u = a_2(t,x)\partial_{xx}u + a_1(t,x)\partial_xu, \label{eq:LinearOp}\tag{21}\] where \(a_2,a_1\) are bounded and continuous on \([0,T]\times\overline{\Omega}\), and \(a_2(t,x)\ge\kappa>0\). Let \(a_0\) be bounded and continuous on \([0,T]\times\overline{\Omega}\).

4.2.3 Schauder estimates↩︎

The following terminal-time estimate is the form of Krylov’s whole-space Schauder theorem used below.

4.3 Preliminaries for the Parisi PDE↩︎

This section collects the basic facts about the Parisi PDE used throughout the paper. Most of these facts are standard, and we recall them here only to fix conventions.

The following lemma connects the time-changed Parisi PDE 17 to the conventional Parisi PDE.

4.3.1 Regularity of the Parisi solution↩︎

The next lemma collects the standard spatial regularity, symmetry, and derivative bounds for the Parisi solution that will be used throughout the sequel.

5 Regularity of the Parisi solution map↩︎

The purpose of this section is to prove Theorem 8(1), which goes through Schauder estimates plus the Banach implicit function theorem.

Recall that \[g(x)=\log\cosh x, \qquad \boldsymbol{\Psi}(\gamma)=\Psi^\gamma-g\] for \(\gamma\in\mathcal{U}_\alpha\).

For \(t\in [0,1]\), we introduce the Banach subspace of \(C_{b}^{1+{\alpha}/2,2+{\alpha}}([0,1]\times\mathbb{R})\) defined by \[\mathcal{X}_t := \Bigl\{ u\in C_{b}^{1+{\alpha}/2,2+{\alpha}}([0,1]\times\mathbb{R}): u(t,\cdot)=0 \Bigr\} \subseteq C_{b}^{1+{\alpha}/2,2+{\alpha}}([0,1]\times\mathbb{R}).\]

Before introducing the nonlinear residual map, we verify that the normalized Parisi solution \(\boldsymbol{\Psi}(\gamma)\) actually lies in \(\mathcal{X}_1\).

After subtracting the terminal datum \(g=\log\cosh\), the Parisi PDE can be written as the zero set of a nonlinear residual map on \(\mathcal{X}_1\). This allows us to apply the Banach implicit function theorem.

We next verify that the isomorphism condition required to apply the implicit function theorem holds.

5.1 Proof of Theorem 8(1)↩︎

The requirements of applying the Banach implicit function theorem (cf. Theorem 5.9 in [24]) are fulfilled by (1) and (2) from Proposition 13 and Proposition 15. This yields that for every \(\gamma_0\in\mathcal{U}_\alpha\), there exist a neighbourhood \(\mathcal{V}\subset\mathcal{U}_\alpha\) of \(\gamma_0\) and a unique \(C^2\) map \[\mathcal{V}\ni\gamma\longmapsto u(\gamma)\in\mathcal{X}_1\] such that \[\mathcal{R}(\gamma,u(\gamma))=0, \qquad u(\gamma_0)=\Psi^{\gamma_0}-\log\cosh=\boldsymbol{\Psi}(\gamma_0).\] Therefore, \(\boldsymbol{\Psi}\) is \(C^2\) in the Fréchet sense, as desired.

5.2 PDEs for the Fréchet derivatives↩︎

Since \(g\) is independent of \(\gamma\), we define, for \(h,h_1,h_2\in C^{\alpha/2}([0,1])\), \[V_h^\gamma = D\Psi^\gamma[h] := D\boldsymbol{\Psi}(\gamma)[h],\] and \[W_{h_1,h_2}^\gamma := D^2\Psi^\gamma[h_1,h_2] := D^2\boldsymbol{\Psi}(\gamma)[h_1,h_2].\]

6 Min-kernel representation of \(D^2\Psi^\gamma\)↩︎

The goal of this section is to rewrite \(D^2\Psi^\gamma\) in a form suitable to study its sign. In view of Corollary 1 and the standard Feynman–Kac formula, the right framework is the study of the transition operator for diffusion processes. While this topic is standard, we recall the relevant facts in Section 6.1. The derivation of the representation is provided in Section 6.2. Finally, we provide the implications of that representation in Section 6.3.

6.1 Tools for the kernel representation↩︎

This section collects the tools for the computation to derive the min-kernel representation of \(D^2\Psi^\gamma\) in Section 6.2.

6.1.1 Standard properties of the transition operators↩︎

We first record the standard transition-operator facts used in the derivation of the min-kernel representation. Recall from 19 and 20 that for all \((s,x)\in [0,1]\times \mathbb{R}\), \[\begin{align} L_s^\gamma u(s,x) &= \frac{1}{2} \partial_{xx}u(s,x) + s\,\partial_x\Psi^\gamma(s,x)\partial_x u(s,x), \\ \mathscr{D}_s u(s,x) &= \partial_s u(s,x) + \gamma(s)L_s^\gamma u(s,x). \end{align}\] Fix \(0\le s< \sigma\le1\). The family of operators \(\boldsymbol{\mathcal{L}}_{s,\sigma}^\gamma = (\gamma(\rho)L_\rho^\gamma)_{\rho\in [s,\sigma]}\) has the associated diffusion process \(\boldsymbol{X}^{\gamma,s,\sigma,x}=(X_\rho^{\gamma,s,\sigma,x})_{\rho\in [s,\sigma]}\) defined by the following SDE \[\require{physics} \label{eq:X-sde} \begin{cases} \dd X_\rho^{\gamma,s,\sigma,x} = \gamma(\rho)\rho\, \partial_x\Psi^\gamma(\rho,X_\rho^{\gamma,s,\sigma,x})\,\dd \rho + \sqrt{\gamma(\rho)}\,\dd B_\rho, & \rho\in[s,\sigma], \\[0.5ex] X_s^{\gamma,s,\sigma,x}=x, & x\in\mathbb{R}. \end{cases}\tag{26}\] By Lemma 3, the drift of this diffusion process is bounded and globally Lipschitz in \(x\), uniform in \(s\) and \(\sigma\). Moreover, the diffusion process has the associated transition operator family \((P_{r,\rho}^\gamma)_{s\leq r\leq \rho\leq\sigma}\) defined by \[\label{eq:def-P} P_{r,\rho}^\gamma f(x) \mathrel{\vcenter{:}}= \mathbb{E}\bigl[f\bigl(X_\rho^{\gamma,r,\rho,x}\bigr)\bigr].\tag{27}\] From the definition 27 , we immediately see the following properties of \((P_{r,\rho}^\gamma)_{s\leq r\leq \rho\leq\sigma}\).

In addition to these elementary properties of the transition family, we will use the following standard facts without further comment.

1. Given terminal condition \(\varphi\in C_b^2(\mathbb{R})\) and source \(b\in C([s,\sigma];C_b^2(\mathbb{R}))\), the backward Cauchy problem \[\begin{cases} -\partial_\rho U(\rho,x)=\gamma(\rho)L_\rho^\gamma U(\rho,x)+b(\rho,x), &(\rho,x)\in[s,\sigma)\times\mathbb{R}, \\[1ex] U(\sigma,x)=\varphi(x), &x\in\mathbb{R} \end{cases} \label{eq:backwardCauchy}\tag{28}\] has a unique bounded classical solution given by the Feynman–Kac formula (cf. §5.7, Theorem 7.6 of Karatzas and Shreve [25]) \[\require{physics} U(\rho,x) = P_{\rho,\sigma}^\gamma\varphi(x) + \int_\rho^\sigma P_{\rho,r}^\gamma[b(r,\cdot)](x)\dd r. \label{eq:FK}\tag{29}\] In particular if \(U(\rho,x)=P_{\rho,\sigma}^\gamma\varphi(x)\), then \(\mathscr{D}_\rho u=0\) on \([s,\sigma)\times\mathbb{R}\).

2. If \(f\in C_b^{1,2}([s,\sigma]\times\mathbb{R})\), then for \(s\le \rho<r\le \sigma\), we have the terminal time differentiation formula \[\partial_r P_{\rho,r}^\gamma[f(r,\cdot)](x) = P_{\rho,r}^\gamma[ \mathscr{D}_r f(r,\cdot)](x). \label{eq:TD}\tag{30}\] This can be derived from applying Dynkin’s formula to \((r,X_r^{\gamma,s,\sigma,x})_{r\in [s,\sigma]}\), whose generator is \((\mathscr{D}_r)_{r\in [s,\sigma]}\).

6.1.2 Properties of auxiliary functions↩︎

We now introduce two auxiliary functions that will enter the kernel representation: \[v^\gamma(s,x) = \partial_{xx}\Psi^\gamma(s,x) + s\bigl(\partial_x\Psi^\gamma(s,x)\bigr)^2, \label{eq:def-vgamma-new}\tag{31}\] and, for \(0\le \rho\le \sigma\le 1\), \[w_{\rho,\sigma}^\gamma(x) = P_{\rho,\sigma}^\gamma \bigl[ (\partial_x\Psi^\gamma(\sigma,\cdot))^2 \bigr](x). \label{eq:def-wgamma-new}\tag{32}\] Now, we record a few identities for the two auxiliary functions that will be convenient for the computation.

The first one is a differentiation identity for \(v^\gamma\).

Lemma 5 implies the following corollary recording the derivative identities for \(P_{\rho,r}^\gamma[v^\gamma(r,\cdot)]\).

Finally, we record the identity for \(\mathscr{D}_\rho(L_\rho^\gamma w_{\rho,\sigma}^\gamma)\) that will be needed later.

6.2 Min-kernel representation↩︎

Since the argument is computational, we first carry out the calculation and then state Proposition 18 at the end of the section.

Recall also the abbreviations introduced in Section 5.2 of the directional derivatives: \[V_h^\gamma = D\Psi^\gamma[h], \qquad W_{h_1,h_2}^\gamma = D^2\Psi^\gamma[h_1,h_2],\] for all \(h,h_1,h_2\in C^{\alpha/2}([0,1])\).

Corollary 1 and the Feynman–Kac formula yields \[\require{physics} V_h^\gamma(\rho,x) = \frac{1}{2} \int_\rho^1 h(\sigma) P_{\rho,\sigma}^\gamma \bigl[ v^\gamma(\sigma,\cdot) \bigr](x)\dd{\sigma} \label{eq:V-FK}\tag{45}\] and \[\require{physics} \begin{gather} W_{h_1,h_2}^\gamma(s,x) = \int_s^1 P_{s,\rho}^\gamma \Bigl[ h_1(\rho)L_\rho^\gamma V_{h_2}^\gamma(\rho,\,\cdot\,) \Bigr](x)\dd \rho \\ + \int_s^1 P_{s,\rho}^\gamma \Bigl[ h_2(\rho)L_\rho^\gamma V_{h_1}^\gamma(\rho,\,\cdot\,) \Bigr](x) \dd \rho + \int_s^1 P_{s,\rho}^\gamma \Bigl[ \gamma(\rho)\rho\, \partial_xV_{h_1}^\gamma(\rho,\,\cdot\,) \partial_xV_{h_2}^\gamma(\rho,\,\cdot\,) \Bigr](x) \dd \rho. \label{eq:W-FK} \end{gather}\tag{46}\]

Using 45 and the identities from Corollary 2, the integrands in the first two terms of 46 are \[\require{physics} \begin{align} \int_s^1 P_{s,\rho}^\gamma \Bigl[ h_1(\rho)L_\rho^\gamma V_{h_2}^\gamma (\rho,\,\cdot\,) \Bigr](x) \dd{\rho} &= - \frac{1}{2} \int_s^1 \int_{\Delta_{s,\sigma}} h_1(\rho)h_2(r) P_{s,\rho}^\gamma \bigl[ L_\rho^\gamma w_{\rho,\sigma}^\gamma \bigr] (x) \dd{r} \dd{\rho} \dd{\sigma}, \tag{47} \\ \int_s^1 P_{s,\rho}^\gamma \Bigl[ h_2(\rho)L_\rho^\gamma V_{h_1}^\gamma (\rho,\,\cdot\,) \Bigr](x) \dd{\rho} &= - \frac{1}{2} \int_s^1 \int_{\Delta_{s,\sigma}} h_2(\rho)h_1(r) P_{s,\rho}^\gamma \bigl[ L_\rho^\gamma w_{\rho,\sigma}^\gamma \bigr] (x) \dd{r} \dd{\rho} \dd{\sigma}, \tag{48} \end{align}\] with \(\Delta_{s,\sigma}=\bigl\{(\rho,r) \,\big|\, s\leq \rho\leq r\leq \sigma\bigr\}\) for \(\sigma\in [s,1]\). By Fubini and adopting the change of variables \(\rho=\rho_1\) and \(r=\rho_2\) for 47 and the change of variables \(\rho=\rho_2\) and \(r=\rho_1\) for 48 , the sum of 47 and 48 equals \[\require{physics} - \frac{1}{2} \int_s^1 \iint_{[s,\sigma]^2} h_1(\rho_1)h_2(\rho_2) P_{s,\rho_1\wedge\rho_2}^\gamma \bigl[ L_{\rho_1\wedge \rho_2}^\gamma w_{\rho_1\wedge \rho_2,\sigma}^\gamma \bigr] (x) \dd{\rho_1} \dd{\rho_2} \dd{\sigma}. \label{eq:first-second-answer}\tag{49}\]

We now treat the third term in 46 . Using 45 , the third term in 46 equals \[\require{physics} \begin{gather} \int_s^1 P_{s,\rho}^\gamma \Bigl[ \gamma(\rho)\rho\, \partial_xV_{h_1}^\gamma(\rho,\,\cdot\,) \partial_xV_{h_2}^\gamma(\rho,\,\cdot\,) \Bigr](x) \dd{\rho} \\ = \frac{1}{4} \int_s^1 \iint_{[\rho,1]^2} h_1(\rho_1)h_2(\rho_2) P_{s,\rho}^\gamma \Bigl[ \gamma(\rho)\rho\, \partial_x P_{\rho,\rho_1}^\gamma \bigl[ v^\gamma(\rho_1,\,\cdot\,) \bigr] \partial_x P_{\rho,\rho_2}^\gamma \bigl[ v^\gamma(\rho_2,\,\cdot\,) \bigr] \Bigr](x) \dd{\rho_1}\dd{\rho_2} \dd{\rho}. \label{eq:quadratic-term} \end{gather}\tag{50}\] The domain in 50 admits another parametrization \[\Bigl\{ (\rho,\rho_1,\rho_2) \,\Big|\, s\leq \rho\leq 1,\, (\rho_1,\rho_2)\in [\rho,1]^2 \Bigr\} = \Bigl\{ (\rho,\rho_1,\rho_2) \,\Big|\, s\leq \rho\leq \rho_1\wedge\rho_2,\, (\rho_1,\rho_2)\in [s,1]^2 \Bigr\},\] so Fubini yields \[\require{physics} \begin{gather} \eqref{eq:quadratic-term} = \frac{1}{4} \iint_{[s,1]^2} h_1(\rho_1)h_2(\rho_2) \\ \int_s^{\rho_1\wedge\rho_2} P_{s,\rho}^\gamma \Bigl[ \gamma(\rho)\rho\, \partial_x P_{\rho,\rho_1}^\gamma \bigl[ v^\gamma(\rho_1,\,\cdot\,) \bigr] \partial_x P_{\rho,\rho_2}^\gamma \bigl[ v^\gamma(\rho_2,\,\cdot\,) \bigr] \Bigr](x) \dd{\rho} \dd{\rho_1}\dd{\rho_2}. \label{eq:quadratic-term-reparametrization} \end{gather}\tag{51}\] Applying Corollary 2 with \(r=\rho_1\vee\rho_2\), the inner integral in 51 becomes \[\require{physics} \begin{gather} \int_s^{\rho_1\wedge\rho_2} P_{s,\rho}^\gamma \Bigl[ \gamma(\rho)\rho\, \partial_x P_{\rho,\rho_1\wedge\rho_2}^\gamma \bigl[ v^\gamma(\rho_1\wedge\rho_2,\,\cdot\,) \bigr] \partial_x P_{\rho,\rho_1\vee\rho_2}^\gamma \bigl[ v^\gamma(\rho_1\vee\rho_2,\,\cdot\,) \bigr] \Bigr](x) \dd{\rho} \\ = - \int_s^{\rho_1\wedge\rho_2} \int_{\rho_1\vee\rho_2}^1 P_{s,\rho}^\gamma \Bigl[ \gamma(\rho)\rho\, \partial_x P_{\rho,\rho_1\wedge\rho_2}^\gamma \bigl[ v^\gamma(\rho_1\wedge\rho_2,\,\cdot\,) \bigr] \partial_x w_{\rho,\sigma}^\gamma \Bigr](x) \dd{\sigma} \dd{\rho}. \label{eq:inner-integral} \end{gather}\tag{52}\] Plug 52 back to 50 . Then, applying Fubini with the reparametrization \[\rho_1,\rho_2\in[s,1],\;\rho_1\vee\rho_2\le\sigma\le1 \quad\text{if and only if}\quad s\le\sigma\le1,\;\rho_1,\rho_2\in[s,\sigma],\] yields \[\require{physics} \begin{gather} \eqref{eq:quadratic-term} = - \frac{1}{4} \int_s^1 \iint_{[s,\sigma]^2} h_1(\rho_1)h_2(\rho_2) \\ \int_s^{\rho_1\wedge\rho_2} P_{s,\rho}^\gamma \Bigl[ \gamma(\rho)\rho\, \partial_x P_{\rho,\rho_1\wedge\rho_2}^\gamma \bigl[ v^\gamma(\rho_1\wedge\rho_2,\,\cdot\,) \bigr] \partial_x w_{\rho,\sigma}^\gamma \Bigr](x) \dd{\rho} \dd{\rho_1}\dd{\rho_2} \dd{\sigma}. \label{eq:third-term-answer} \end{gather}\tag{53}\]

Combining 49 and 53 yields the following min-kernel representation, stated as a proposition.

6.3 Corollaries of the min-kernel representation↩︎

We now state a few corollaries of Proposition 18 that will be useful in the next sections.

The first one is a factorization of the min-kernel representation.

By Corollary 4, the sign of the second Fréchet derivative \(D^2\Psi^\gamma\) is governed by the left endpoint \(A^{\gamma,s,\sigma,x}(s)\) and the density \(\partial_\rho A^{\gamma,s,\sigma,x}(\rho)\). The next corollary reduces the problem to studying the auxiliary function \(w_{s,\sigma}^\gamma\), which will be done in the next section.

7 Cone preservation of the transition kernel↩︎

As explained in Section 6.3, the sign of the second Fréchet derivative \(D^2\Psi^\gamma\) is governed by the left endpoint \(A^{\gamma,s,\sigma,x}(s)\) and the density \(\partial_\rho A^{\gamma,s,\sigma,x}(\rho)\). In view of Corollary 5, this reduces to understanding the function \[w_{\rho,\sigma}^\gamma(x) = P_{\rho,\sigma}^\gamma\bigl[ (\partial_x\Psi^\gamma(\sigma,\cdot))^2 \bigr](x).\] By Lemma 3(2)–(3), \(x\mapsto (\partial_x\Psi^\gamma(\sigma,x))^2\) is even and nonnegative on \(\mathbb{R}\) and nondecreasing on \([0,\infty)\), which motivates the cone-preservation property of the transition operator \(P_{\rho,\sigma}^\gamma\). More precisely, define the cone \[\mathcal{C} = \Bigl\{ f\in C_b^{3+\alpha}(\mathbb{R}) \,\Big|\, f \text{ is even, nonnegative, and nondecreasing on }[0,\infty) \Bigr\}. \label{eq:def-cone}\tag{59}\] The purpose of this section is to prove that the transition operator preserves the cone.

We now apply Corollary 5 and Proposition 19 to obtain the following result, which will immediately imply Theorem 8(2) and Theorem 8(3).

7.1 Proof of Theorem 8(2) and Theorem 8(3)↩︎

Let \(h_1,h_2\in C^{\alpha/2}([0,1])\). Recall from Corollary 4 that \[\require{physics} \begin{gather} D^2\Psi^\gamma[h_1,h_2](s,0) = - \int_s^1 \biggl( A^{\gamma,s,\sigma,0}(s) \Bigl( \iint_{[s,\sigma]^2} h_1(\rho_1)h_2(\rho_2) \dd{\rho_1}\dd{\rho_2} \Bigr) \\ + \int_s^\sigma \Bigl( \iint_{[\rho,\sigma]^2} h_1(\rho_1)h_2(\rho_2) \dd\rho_1\dd\rho_2 \Bigr) \partial_\rho A^{\gamma,s,\sigma,0}(\rho) \dd{\rho} \biggr) \dd{\sigma}. \label{eq:DPsi-formula} \end{gather}\tag{60}\] Moreover, Corollary 6 implies that both \(A^{\gamma,s,\sigma,0}(s)\) and \(\partial_\rho A^{\gamma,s,\sigma,x}(\rho)\) are nonnegative.

If \(h_1=h_2=h\), then 60 is nonpositive, which proves Theorem 8(2).

If \(h_1\geq 0\) and \(h_2\geq 0\), then 60 is nonpositive, which proves Theorem 8(3).

8 Proof of Theorem 5↩︎

We recall the path spaces \(\mathcal{Q}_p\) from 3 .

We next prove the \(L^1\)-stability of \(\psi\).

We now prove Theorem 5.

Acknowledgements↩︎

I warmly thank Jhih-Huang Li at National Taiwan University for his hospitality during my 2023 visit, and Jean-Christophe Mourrat and Victor Issa at ENS Lyon for their hospitality during my 2025 visit. I am grateful to Wai-Kit Lam for bringing Auffinger–Chen’s work [21] to my attention during the former visit. I thank Victor Issa for suggesting the problem and for providing several insights, both during and after my visit to ENS Lyon. I warmly thank Eliran Subag and Justin Ko for helpful advice on the early drafts of the paper.

I gratefully acknowledge support from Eliran Subag’s grants: ISF grant No. 2055/21 and ERC grant No. 101165541.

During the preparation of this manuscript, I was made aware that Hong-Bin Chen was independently investigating the convexity of the initial condition \(\psi\).

References↩︎

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1. Note that Mourrat and coauthors’ convention of the free energy has a sign difference with the usual convention. For consistency of the introduction, we align with the usual convention.↩︎