Rayleigh–Type Renormalized Flows in Weighted Geometries Energy Dissipation, Invariance, and Applications to PDE Dynamics

Rayleigh–Type Renormalized Flows in Weighted Geometries Energy Dissipation, Invariance, and Applications to PDE Dynamics Mateus R. de Maria Abstract We introduce a class of Rayleigh–type renormalized flows driven by weighted geometries and non–uniform density structures. Given a strictly positive weight w=κ+ 1 and the renormalized derivative Dκf:= 1 w d dx(wf), we consider the self–adjoint dissipation operator Aκ:= D∗ κDκon L2(dµ)with dµ = w dx. Our main object is the normalized Rayleigh flow ∂tf=−Aκf+ρ(f)f, ρ(f)=⟨Aκf, f⟩µ,∥f(t)∥L2(dµ)≡1, which can be defined canonically as a normalization of the linear semigroup e−tAκ. We prove invariance of the L2(dµ)constraint, positivity preservation under standard Markovian assumptions, and a sharp Lyapunov dissipation identity d dtE(f(t))=−2∥Aκf(t)−ρ(f(t))f(t)∥2 µ≤0,E(f)=∥Dκf∥2 µ. Stationary states are characterized as eigenfunctions of Aκ, and spectral gap mechanisms yield stability and mode selection in basic compact settings. The framework provides a unified viewpoint connecting weighted PDE dissipation, constrained gradient dynamics, and eigenmode extraction under heterogeneous densities. All statements are formulated for compact one-dimensional domains; higher-dimensional extensions are discussed in the conclusion. 1 Contents 1 Introduction and motivation 3 1.1 Rayleigh quotients, normalization, and dissipation .............. 3 1.2 Weighted geometries and renormalized derivatives .............. 3 1.3 Main object: the normalized Rayleigh flow .................. 4 1.4 What is new (and what is not) ......................... 4 1.5 Outline of the paper .............................. 4 2 Weighted operators and renormalized derivatives 4 2.1 Definition of the renormalized derivative ................... 4 2.2 Basic properties ................................. 5 2.3 Integration by parts and adjoint structure .................. 5 2.4 Standing hypotheses .............................. 5 3 Definition of the renormalized Rayleigh flow 5 3.1 Phase space and admissible class ....................... 6 3.2 Rayleigh quotient and flow equation ...................... 6 3.3 Invariant sets: L2(µ)constraint and positivity ................ 6 3.4 Normalization of the linear semigroup ..................... 7 4 Energy functionals and monotonicity 7 4.1 Natural energy and Rayleigh quotients .................... 7 4.2 Lyapunov structure and sharp dissipation identity .............. 7 4.3 Interpretation as constrained gradient dynamics ............... 8 5 Well–posedness and regularity 8 5.1 Canonical construction via semigroup normalization ............. 8 5.2 Derivation of the PDE for t > 0........................ 8 5.3 Uniqueness and positivity ........................... 9 6 Stationary states and stability 9 6.1 Stationary states are eigenfunctions ...................... 10 6.2 Omega–limit points are stationary (compactness route) ........... 10 6.3 Spectral gap stability .............................. 10 6.4 Positivity and mode selection ......................... 11 7 Model examples 11 7.1 One–dimensional weighted diffusion ...................... 11 7.2 A smooth periodic weight ........................... 11 7.3 Piecewise constant weights ........................... 11 7.4 Takeaway .................................... 11 8 Informational and computational perspectives 12 8.1 Rayleigh flows as constrained optimization dynamics ............ 12 8.2 Discrete schemes and spectral extraction ................... 12 8.3 Links with replicator–type dynamics (finite dimensional intuition) ..... 12 8.4 Outlook: information geometry and mirror–type normalizations ...... 13 2 9 Summary and future directions 13 9.1 What was established .............................. 13 9.2 Conceptual interpretation ........................... 13 9.3 Directions for extension ............................. 13 A Appendix A: Technical lemmas and canonical well–posedness 14 A.1 Semigroup smoothing .............................. 14 A.2 Canonical well–posedness for the normalized Rayleigh flow ......... 14 B Appendix B: Variational derivations and alternative formulations 15 B.1 Constrained gradient flow on the L2(µ)–sphere ................ 15 B.2 Lagrange multiplier derivation ......................... 15 B.3 Alternative normalizations ........................... 15 1 Introduction and motivation In this section: (i) we recall Rayleigh quotients and normalized dissipation, (ii) we motivate the weighted differential geometry behind Dκ, (iii) we summarize the scope and organization of the paper. 1.1 Rayleigh quotients, normalization, and dissipation Rayleigh quotients sit at the intersection of spectral theory, variational principles, and dissipative PDE. Given a non–negative self–adjoint operator Aon a Hilbert space, the quantity R(f) = ⟨Af, f⟩ ⟨f, f⟩ measures the energy level of a state relative to its size, and extremizers are eigenfunctions. A classical idea is to evolve fby a dissipative dynamics that decreases ⟨Af, f⟩while preventing trivial decay of amplitude. This leads to normalized flows on constraint manifolds (typically spheres), where dissipation is geometrically projected onto the tangent space. In heterogeneous media, however, the operator Aitself should encode the geometry induced by the density/weight of the medium. This is the setting of the present paper. 1.2 Weighted geometries and renormalized derivatives Let w(x)=κ(x) + 1 be a strictly positive weight and define the weighted measure dµ = w dx. A natural derivative adapted to this geometry is Dκf:= 1 w(wf)′, which is the one–dimensional prototype of a weighted divergence structure. We associate to it the dissipation operator Aκ:= D∗ κDκon L2(I, dµ), so that the energy E(f)=∥Dκf∥2 µis intrinsic to the weighted geometry. 3 1.3 Main object: the normalized Rayleigh flow Our main object is the Rayleigh–type normalized flow ∂tf=−Aκf+ρ(f)f, ρ(f)=⟨Aκf, f⟩µ,∥f(t)∥µ≡1.(1) This is the steepest descent of Econstrained to the L2(dµ)–sphere, hence it keeps the trajectory on a fixed–norm manifold while dissipating energy. A key feature is that (1) admits a canonical construction: if g(t) = e−tAκf0solves the linear dissipation, then f(t) = g(t) ∥g(t)∥µ solves (1) for t > 0(see Chapter 5and Appendix A). 1.4 What is new (and what is not) Not new. Normalized Rayleigh flows and constrained gradient dynamics are classical tools in spectral analysis and PDE. Weighted self–adjoint operators and Dirichlet forms are also standard in analysis on weighted spaces. New contribution. The contribution of this paper is structural and unifying: •we encode the heterogeneity at the differential level via the renormalized derivative Dκ, so that dissipation, normalization, and invariances become transparent and compatible with the weighted geometry; •we emphasize the normalized–semigroup representation as the canonical definition of the nonlinear Rayleigh flow, yielding a clean bridge between weighted PDE dissipation and eigenmode extraction under heterogeneous densities. 1.5 Outline of the paper Chapter 3defines the flow and its invariant sets. Chapter 4establishes the Lyapunov structure and sharp dissipation identity. Chapter 5develops well–posedness via normalized semigroups and recovers the PDE for t>0. Chapter 6characterizes equilibria as eigenfunctions and discusses stability via spectral gaps. Chapters 7–8present model examples and computational viewpoints. Appendices collect technical lemmas and variational derivations. 2 Weighted operators and renormalized derivatives In this section: (i) we define the renormalized derivative Dκand its basic calculus, (ii) we record weighted integration by parts and the adjoint structure, (iii) we state standing analytic hypotheses used throughout. 2.1 Definition of the renormalized derivative Let κ=κ(x)be a non–negative weight function and set w(x) := κ(x) + 1. We define Dκf:= 1 w d dxwf. 4 2.2 Basic properties For sufficiently regular f, g, Dκ(fg) = g Dκf+fdg dx. When κ≡0(hence w≡1), Dκreduces to the classical derivative. Also Dκf=f′+w′ wf. 2.3 Integration by parts and adjoint structure Let dµ =w dx. Under periodic or Neumann–type boundary conditions (no boundary terms), ZI (Dκf)g dµ =−ZI fdg dx dµ. (2) Hence, formally on L2(dµ), D∗ κ=−d dx, Aκ:= D∗ κDκ=−d dxw−1(w·)′. 2.4 Standing hypotheses (H1) (Weight)w=κ+ 1 ∈W1,∞(I)and 0< wmin ≤w(x)≤wmax <∞. We set dµ =w dx and H=L2(I, dµ). (H2) (Boundary conditions) Periodic (or Neumann–type) boundary conditions so that integration by parts yields no boundary terms for the form aκ(f, g) = ⟨Dκf, Dκg⟩µ. (H3) (Self–adjoint dissipation) The quadratic form aκwith domain H1(I)is closed and non–negative, hence defines a non–negative self–adjoint operator Aκ:= D∗ κDκ on H, and −Aκgenerates a contraction semigroup e−tAκ. (H4) (Smoothing) For each t>0,e−tAκ(H)⊂ D(Aκ)and Aκe−tAκis bounded (Lemma A.1). (H5) (Positivity, when used) When positivity statements are invoked, assume e−tAκ is positivity preserving on H(and positivity improving when mode selection is claimed). Under (H1)–(H4) the normalized Rayleigh flow is well-defined for all f0∈H\ {0}via normalized semigroup trajectories (Definition 5.1). 3 Definition of the renormalized Rayleigh flow In this section: (i) we set the weighted phase space and the operator Aκ, (ii) we define the Rayleigh quotient and the renormalized Rayleigh flow, (iii) we record invariant sets and the normalized–semigroup representation. 5 3.1 Phase space and admissible class Let Ibe a compact one–dimensional domain (e.g. I=T). Assume the standing hypotheses of Section 2.4. Set w=κ+ 1,dµ =w dx, and H:= L2(I, dµ),⟨f, g⟩µ:= ZI fg dµ, ∥f∥2 µ:= ⟨f, f⟩µ. Let Dκf=w−1(wf)′and Aκ:= D∗ κDκ. Constraint manifold. Sµ:= {f∈H:∥f∥µ= 1}. When positivity is desired, we restrict further to S+ µ:= {f∈Sµ:f≥0a.e.}. 3.2 Rayleigh quotient and flow equation We use the global Rayleigh quotient ρ(f) := ⟨Aκf, f⟩µ ⟨f, f⟩µ , f ∈ D(A1/2 κ)\ {0}.(3) On Sµ,ρ(f)=⟨Aκf, f⟩µ. Definition 3.1 (Renormalized Rayleigh flow).The renormalized Rayleigh flow associated with Aκis ∂tf=−Aκf+ρ(f)f, f(0) = f0∈H, f0≡ 0.(4) 3.3 Invariant sets: L2(µ)constraint and positivity Proposition 3.2 (L2(µ)norm is preserved).Let fbe a sufficiently regular solution of (4). Then d dt∥f(t)∥2 µ= 0. In particular, if ∥f0∥µ= 1 then f(t)∈Sµfor all tin the interval of existence. Proof. Take the H–inner product of (4) with f: 1 2 d dt∥f∥2 µ=⟨∂tf, f⟩µ=−⟨Aκf, f⟩µ+ρ(f)⟨f, f⟩µ. By (3), ρ(f)⟨f, f⟩µ=⟨Aκf, f⟩µ. Proposition 3.3 (Positivity preservation (structural)).Assume e−tAκis positivity preserving on H. If f0≥0a.e., then any sufficiently regular solution of (4)satisfies f(t)≥0 a.e. for all t. Idea. Let α(t) = Rt 0ρ(f(s)) ds and set g(t)=e−α(t)f(t). Then ∂tg=−Aκg. Positivity of the semigroup implies g(t)≥0, hence f(t)≥0. 6 3.4 Normalization of the linear semigroup Let gsolve ∂tg=−Aκg, g(0) = f0∈H, f0≡ 0.(5) Define f(t) := g(t) ∥g(t)∥µ .(6) Then f(t)∈Sµfor all t≥0, and for t > 0it solves (4) (see Theorem 5.3). This gives the canonical construction of solutions. Remark 3.4 (Alternative constraints).If one requires preservation of Rf dµ, a different normalization is needed. We record such variants in Appendix B. 4 Energy functionals and monotonicity In this section: (i) we introduce the intrinsic weighted energy E(f), (ii) we prove a sharp dissipation identity, (iii) we interpret the flow as constrained gradient descent. 4.1 Natural energy and Rayleigh quotients We work in H=L2(I, dµ)with dµ =w dx and the self–adjoint operator Aκ=D∗ κDκ. Define E(f):=∥Dκf∥2 µ=⟨Aκf, f⟩µ, f ∈ D(A1/2 κ).(7) Recall ρfrom (3). 4.2 Lyapunov structure and sharp dissipation identity Consider the Rayleigh flow ∂tf=−Aκf+ρ(f)f. (8) Proposition 4.1 (Sharp dissipation identity).Let f(t)be a sufficiently regular solution of (8). Then d dtE(f(t))=−2∥Aκf(t)−ρ(f(t))f(t)∥2 µ≤0.(9) In particular, E(f(t)) is non–increasing along the flow. Proof. Differentiate E(f) = ⟨Aκf, f⟩µ: d dtE(f(t)) = 2⟨Aκf(t), ∂tf(t)⟩µ. Insert (8): d dtE=−2∥Aκf∥2 µ+ 2ρ(f)⟨Aκf, f⟩µ. Using ρ(f)⟨f, f⟩µ=⟨Aκf, f⟩µfrom (3), expand ∥Aκf−ρ(f)f∥2 µ=∥Aκf∥2 µ−2ρ(f)⟨Aκf, f⟩µ+ρ(f)2∥f∥2 µ, and note ρ(f)2∥f∥2 µ=ρ(f)⟨Aκf, f⟩µ. Rearranging yields (9). 7 Corollary 4.2 (Finite dissipation).If f(t)is global and E(f(t)) is bounded below, then for any τ > 0, Z∞ τ ∥Aκf(t)−ρ(f(t))f(t)∥2 µdt < ∞. In particular, there exists tn→ ∞ such that ∥Aκf(tn)−ρ(f(tn))f(tn)∥µ→0. 4.3 Interpretation as constrained gradient dynamics On Sµ, the driving force is the projection of Aκfonto TfSµ={u:⟨u, f⟩µ= 0}, namely Aκf−ρ(f)f. Thus (8) is steepest descent of Eunder the constraint ∥f∥µ≡1. Remark 4.3 (Stationary points).Stationary points satisfy Aκf=ρ(f)f, i.e. eigenfunctions (Chapter 6). 5 Well–posedness and regularity In this section: (i) we define the canonical solution via normalized semigroups, (ii) we derive the PDE for t>0and basic regularity, (iii) we record uniqueness and positivity under standard assumptions. 5.1 Canonical construction via semigroup normalization Assume (H1)–(H4) from Section 2.4. For f0∈H\ {0}define g(t) := e−tAκf0, f(t) := g(t) ∥g(t)∥µ . Then f(t)∈Sµfor all t≥0. The key point is that g(t)= 0 for all t > 0(see Lemma 5.2 below or Appendix A). Definition 5.1 (Normalized semigroup solution).We call f(t)defined by f(t) = e−tAκf0 ∥e−tAκf0∥µ , f0∈H\ {0},(10) the normalized semigroup solution of the Rayleigh flow. We set f(0) := f0/∥f0∥µ. Lemma 5.2 (Non–vanishing of the linear orbit).If f0= 0, then e−tAκf0= 0 for every t>0. Proof. Since Aκis self–adjoint and non–negative, e−tAκis defined by spectral calculus. Because exp(−tλ)>0for all λ≥0, the operator e−tAκhas trivial kernel. Hence e−tAκf0= 0implies f0= 0. 5.2 Derivation of the PDE for t>0 Theorem 5.3 (PDE for t>0).Let f0∈H\{0}and let f(t)be the normalized semigroup solution (10). Then for every t>0we have f(t)∈ D(Aκ)and f∈C1((0,∞); H). Moreover, for t > 0the trajectory satisfies the Rayleigh flow equation (4): ∂tf(t) = −Aκf(t)+ρ(f(t)) f(t), ρ(f) = ⟨Aκf, f⟩µ ⟨f, f⟩µ . 8 Proof. By (H4), for each t > 0the semigroup orbit satisfies g(t)∈ D(Aκ)and g′(t) = −Aκg(t)in H. Set n(t) := ∥g(t)∥µ>0for t > 0and f(t) = g(t)/n(t). Then fis differentiable for t > 0and f′(t) = g′(t) n(t)−n′(t) n(t)f(t). Since d dt 1 2n(t)2=⟨g′(t), g(t)⟩µ=−⟨Aκg(t), g(t)⟩µ, we get n′(t) n(t)=−⟨Aκg(t), g(t)⟩µ ∥g(t)∥2 µ =−⟨Aκf(t), f(t)⟩µ ∥f(t)∥2 µ =−ρ(f(t)), using g=nf and linearity of Aκ. Substituting g′=−Aκgand g=nf yields f′(t) = −Aκf(t)+ρ(f(t))f(t). Remark 5.4 (One-shot well–posedness reference).Appendix Arecords the semigroup smoothing input (Lemma A.1) and a clean one–shot well–posedness statement (Theorem A.2) that can be cited directly. 5.3 Uniqueness and positivity Proposition 5.5 (Uniqueness among normalized solutions for t > 0).Let f, ˜ f∈C1((0,∞); H)∩ C((0,∞); D(Aκ)) satisfy (4)for all t > 0, and assume ∥f(t)∥µ≡ ∥ ˜ f(t)∥µ≡1. Fix τ > 0 and suppose f(τ) = ˜ f(τ). Then f(t) = ˜ f(t)for all t≥τ. Proof. For t≥τdefine α(t) := Rt τρ(f(s)) ds and g(t) := e−α(t)f(t). A direct computation gives g′(t)=−Aκg(t). The same holds for ˜g(t)built from ˜ f. Since f(τ) = ˜ f(τ), we have g(τ) = ˜g(τ), and uniqueness for the linear equation implies g(t) = ˜g(t)for t≥τ, hence f(t) = ˜ f(t). Proposition 5.6 (Positivity preservation for normalized semigroup solutions).Assume in addition that e−tAκis positivity preserving on H. If f0≥0a.e., then the normalized semigroup solution satisfies f(t)≥0a.e. for all t≥0. Proof. If f0≥0and the semigroup is positivity preserving, then g(t) = e−tAκf0≥0for all t≥0. Normalization by the positive scalar ∥g(t)∥µpreserves nonnegativity. Remark 5.7 (Energy law).For t>0the solution is regular enough to justify Proposition 4.1, hence E(f(t)) is non–increasing. 6 Stationary states and stability In this section: (i) we characterize stationary states as eigenfunctions of Aκ, (ii) we show that omega–limit points are stationary under compactness, (iii) we state spectral gap stability and positivity–based mode selection. 9 References [1] L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, 2008. [2] L. C. Evans, Partial Differential Equations, 2nd ed., AMS, 2010. [3] C. Villani, Topics in Optimal Transportation, AMS, 2003. [4] F. Otto, The geometry of dissipative evolution equations, Comm. PDE 26 (2001), 101–174. [5] K.-T. 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