Codex @codex  0

The Gronwall inequality applied to part (a) givesContinuity implies $Z_{\tau }=0$ on $\{\tau <\infty \}$. Thus on $\{\tau \le t\}$ the stopped reciprocal equals $1/ϵ$. Positivity givesLet $ϵ\downarrow 0$, using $z_0>0$, to obtain $\mathbb{P}(\tau \le t)=0$. Taking the union over integer $t$ proves $\tau =\infty$ almost surely. Hence the strict order preservation for scalar Lipschitz diffusions conclusion iswhich is stronger than the requested weak inequality. The single-event argument prevents a gap between fixed-time ordering and simultaneous ordering at all times.

Literally, the printed hypothesis with positive $C$ is impossible for pairs at distance less than one, even when $U$ is constant. Use the corrected log-Lipschitz modulus $\mu$ from part (e). Local existence for the continuous finite-dimensional vector field follows from the Peano existence theorem. Boundedness gives $|X(t)-X(s)|\le \Vert U{\Vert }_{\infty }|t-s|$. Thus a trajectory cannot escape to infinity in finite time; at a finite endpoint it has a limit, and local existence there extends it. Solutions exist globally.

For uniqueness, let $X,Y$ have the same initial point and set $\delta (t)=|X(t)-Y(t)|$. This absolutely continuous function satisfies, almost everywhere,This is the Osgood uniqueness criterion, rather than the usual linear Gronwall inequality, because the velocity need not be Lipschitz continuous. To see the argument directly, put $w_{\epsilon }=\delta +\epsilon$. As long as $w_{\epsilon }\le 1$, monotonicity of $\mu$ givesFor $z_{\epsilon }=1-\log w_{\epsilon }$, this becomes $z_{\epsilon }^{\prime }\ge -C{z}_{\epsilon }$. Integration givesFor any fixed finite interval, this upper bound tends uniformly to zero as $\epsilon \downarrow 0$. Taking $\epsilon$ small prevents a first exit through $w_{\epsilon }=1$, so the bound remains valid throughout that interval. Thus $\delta =0$. Backward time has the same proof. Every initial point has exactly one global trajectory.

The decisive fact is ${\int }_0^{1}dr/\mu (r)=\infty$. For different nearby initial points, the same comparison givesuntil the separation reaches one. This quantitative continuous dependence makes the flow a homeomorphism, although it need not be a smooth diffeomorphism.

Put $z=(X,V)$ and $b(t,z)=(V,F(t,X,V))$. On a small closed time interval and a closed ball around the initial point, let $M$ bound $b$ and let $K$ be its spatial Lipschitz constant. The mapsends the corresponding closed ball of continuous paths into itself when the time length times $M$ is at most the ball radius. If the time length times $K$ is less than one, it is a contraction in the supremum norm. The Banach fixed-point theorem gives a local solution and uniqueness; differentiating its integral equation gives the ordinary differential equation. Overlapping local solutions agree by this uniqueness.

For continuation, the growth assumption gives, on any bounded time interval,for forward time, with the analogous reversed-time bound. The Gronwall inequality bounds the trajectory on that interval. A finite terminal time is impossible: within the resulting compact ball the field is bounded, so the path has a limit at the endpoint, and the local construction restarts there. This proves the global characteristic flow under linear growth. No differentiability of the flow with respect to its initial point is needed for this existence and uniqueness proof.

The reference to part (f) within this part is a printed self-reference; the needed product estimate is part (e). Since both masses are one, subtracting the Bobylev identities gives, for $\xi \ne 0$,The normalized angular average and part (e) giveFor completeness, the Duhamel principle for this scalar equation yields $d(t)\le e^{-t}d(0)+{\int }_0^t{e}^{-(t-s)}d(s)ds$. Apply the Gronwall inequality to $e^{t}d(t)$ to obtain the Fourier nonexpansion for Maxwell molecules, $\boxed{d(f_t,g_t)\le d(f_0,g_0)}$. This estimate is nonexpansion; by itself it does not prove strict decay or convergence to a specified equilibrium.

Use the same Brownian motion for both solutions: this is a synchronous coupling. For $D_t=X_t-Y_t$, the Itô formula givesOn any fixed finite horizon the stochastic integral has mean zero. Indeed, boundedness of $\sigma$ and the assumed second-moment bounds make the expectation of its squared integrand integrable in time. With $h(t)=\mathbb{E}{D}_t^2$, the contraction condition yields, for every $s\le t$,The allowed Gronwall inequality gives the mean-square contraction of synchronously coupled diffusionsThe identical estimate can also be obtained by localizing the nonnegative local supermartingale $e^{kt}{D}_t^2$ and using Fatou's lemma, a formulation useful when the drift has linear growth.

There is a genuine compatibility issue in the printed global assumptions. If $|b|\le B$, then for $d=|x-y|$,It cannot be at most $-k{d}^2$ for all $d>4B/k$. Thus a globally bounded drift cannot satisfy the stated strict contraction on all of the real line. The stochastic estimate above is the requested conditional calculation. For a non-vacuous application, global boundedness of the drift must be relaxed while retaining suitable existence and moment hypotheses; it is not silently changed here.

Multiply the viscous scalar conservation law by $-u_{xx}$, rather than estimating $F^{\prime \prime }$ after differentiating. Integration by parts yieldsThus $d\Vert u_x{\Vert }_2^2/dt\le (M^2/\epsilon )\Vert u_x{\Vert }_2^2$. Applying the Gronwall inequality givesOnly the assumed bound on $F^{\prime }$ is used; a global bound on $F^{\prime \prime }$ is not needed for this energy estimate.

Let $M=\Vert F^{\prime }{\Vert }_{\infty }$. The sharper energy estimate uses the divergence structure of the viscous scalar conservation law. Multiply by $u$ and integrate over the line. With $G(s)={\int }_0^{s}r{F}^{\prime }(r)dr$,because the decay makes $G(u)$ tend to zero at both ends. Integration by parts in the diffusion term therefore givesOne may take $C_0=0$, uniformly in $\epsilon$. This does not require $F(0)=0$.

If a bound explicitly involving $M$ and $\epsilon$ is desired, retaining the transport term and using the Cauchy-Schwarz inequality and the elementary inequality $ab\le (a^2+b^2)/2$ givesThe Gronwall inequality then gives the valid but weaker choice $C_0=M^2/\epsilon$. The exact cancellation explains why its divergence is unnecessary.

We use the homogeneous version of the Sobolev embedding theorem in three dimensions:Here homogeneous Sobolev space ${\dot{H}}^1$ is the completion of compactly supported smooth functions in the L2 norm of the gradient, identified with its $L^6$ representative. Apply this Sobolev inequality both to $\varphi$ and to its spatial derivatives.

LetThe Plancherel theorem identifies the L2 norm of the Hessian matrix with $\Vert \varphi {\Vert }_{{\dot{H}}^2}$, because ${\sum }_{i,j}{\xi }_i^2{\xi }_j^2=|\xi {|}^4$. In particular,Differentiating the defocusing semilinear wave equation gives $\square ({\partial }_i\varphi )=3{\varphi }^2{\partial }_i\varphi$. By the Holder inequality with exponents $3$ and $6$, the preceding Sobolev inequality, and conservation of the positive wave energy,Use the inhomogeneous wave energy estimate simultaneously for the three spatial derivatives. It givesThe Gronwall inequality therefore yields $B(t)\le D\exp (CEt)$. We may take the continuous, locally bounded functionThe constant is universal; the dependence on the initial data is only through $D$ and $E$. The H2 bound for the defocusing cubic wave equation holds on every existing smooth interval, without assuming the global conclusion. If $D=0$ or $E=0$, the zero solution satisfies the same estimate.

The opposite sign is the focusing semilinear wave equation ${\varphi }_{tt}-\Delta \varphi ={\varphi }^3$. Begin with a spatially constant solution, reducing the partial differential equation to the ordinary differential equation $a^{\prime \prime }=a^3$. Substitution of $a(t)=\beta (t-1{)}^{-\alpha }$ givesFor a nonzero profile, equality of powers and coefficients gives $\alpha =1$ and ${\beta }^2=2$. ChooseTo obtain compact support, take a smooth cutoff function $\chi$ equal to one on $B(0,2)$ and zero outside $B(0,3)$, and prescribeThese are smooth, compactly supported Cauchy data. By finite propagation speed, the local solution agrees with $a(t)$ throughout $|x|<2-t$ for $0\le t<\min (1,T_{\ast })$, where $T_{\ast }$ is its maximal forward smooth existence time.

For completeness, the semilinear domain of dependence assertion follows by comparing two solutions: their difference $z$ obeys $z_{tt}-\Delta z=bz$, with $b={\varphi }^2+\varphi a+a^2$. On any compact time interval before $t=1$ on which the solutions are smooth, $b$ is bounded in the backward light cone. Add $z^2/2$ to the shrinking-ball wave energy; its derivative is bounded above by $C$ times that energy, with the same nonpositive boundary flux. Zero initial difference and the Gronwall inequality give $z=0$ there.

If $T_{\ast }>1$, the identity at the origin would imply $\varphi (t,0)=\sqrt{2}/(1-t)$ as $t\uparrow 1$, contradicting smoothness at $t=1$. ThusIf the solution loses regularity earlier, that is already finite-time blowup. The usual smooth continuation criterion for semilinear wave equations precludes a finite maximal time with all continuation norms bounded. This localized ordinary differential equation blowup for a wave equation therefore supplies the required compactly supported examples.

Let $E\to M$ have rank $r$. A vector bundle covariant derivative is a real-linear operatorPairing its one-form slot with a vector field $X$ gives ${\nabla }_{X}s$, with ${\nabla }_{fX}s=f{\nabla }_{X}s$ and ${\nabla }_X(fs)=X(f)s+f{\nabla }_{X}s$.

In the horizontal description, a connection on a vector bundle $A$ is a smooth splitting $T_{a}E=V_{a}E\oplus H_{a}E$, where $V_{a}E=\ker (d\pi {)}_a$, and the horizontal choice is compatible with fibre addition and scalar multiplication. This linearity is essential; an arbitrary nonlinear horizontal splitting is not a vector-bundle connection. Identify a vertical vector at $a\in E_x$ with the element of $E_x$ obtained by differentiating the curve $a+tv$. Let $K_A:T_{a}E\to E_x$ be vertical projection followed by this identification. The induced vector bundle covariant derivative isIn a local frame of a vector bundle $e=(e_1,\dots ,e_r)$, a linear horizontal splitting has the formfor a matrix-valued connection one-form $\Gamma$. Thus for $s=ev$, ${\nabla }^{A}s=e(dv+\Gamma v)$, which directly verifies the Leibniz rule.

Conversely, the Leibniz rule makes $\nabla$ local: if a section vanishes near a point, multiply it by a smooth cutoff function supported in that neighborhood and equal to one near the point to see that its derivative vanishes there too. Local sections can therefore be differentiated by any extension agreeing near the point. Given $\nabla$, define its local connection one-form by $\nabla e_b={\sum }_a{e}_a{\Gamma }_b^a$. The Leibniz rule gives $\nabla (ev)=e(dv+\Gamma v)$. Under $e^{\prime }=eg$, $v^{\prime }=g^{-1}v$, one hasThe tangent fibre coordinate transforms by $w^{\prime }=g^{-1}w-g^{-1}(dg)(\xi )g^{-1}v$, so $w=-\Gamma (\xi )v$ transforms precisely to $w^{\prime }=-{\Gamma }^{\prime }(\xi )v^{\prime }$. Consequently these local horizontal subspaces of a vector bundle connection glue to a smooth linear splitting inducing the given $\nabla$. Its local formula also shows uniqueness. This proves the horizontal connection associated to a covariant derivative construction.

A horizontal lift $\tilde{\gamma }$ of $\gamma$ is a smooth curve in $E$ with $\pi \circ \tilde{\gamma }=\gamma$ and $\dot{\tilde{\gamma }}(t)\in H_{\tilde{\gamma }(t)}E$. In a local frame write $\tilde{\gamma }(t)=e(\gamma (t))v(t)$. Horizontality is exactly the linear ordinary differential equationWe use existence and uniqueness for an ODE with smooth coefficients, and the continuation theorem that a solution extends across a finite interval when it remains bounded inside the coordinate domain. For a linear system $\dot{v}=-C(t)v$ with continuous coefficients bounded by $L$ on a compact time interval, the Gronwall inequality bounds $|v(t)|$ by its initial norm times $e^{L|t-t_0|}$, preventing finite-time fibre escape. Cover a compact subinterval of $I$ by finitely many bundle charts and partition it so each piece lies in one chart. Solve successively, changing fibre frames at the partition points. Coordinate invariance of the horizontal equation and uniqueness make the pieces agree. Exhausting $I$ by compact intervals containing $0$ produces a unique horizontal lift on all of $I$ for every prescribed $a_0$. This is global continuation of linear parallel transport.

The lift equation is linear in its initial value. Hence the endpoint map $P_{\gamma }:E_{\gamma (0)}\to E_{\gamma (1)}$ is linear. Solving the same equation backwards, or transporting along $t\mapsto \gamma (1-t)$, gives its inverse, soThis is parallel transport. For two parallel lifts $s_1(t),s_2(t)$, the assumed metric connection identity extends along the embedded curve and becomesHere $D_t$ is the covariant derivative along a curve. The permitted extension of sections and the tangent vector along the embedded curve justifies applying the bundle identity; equivalently it follows in each local frame. Thereforeso parallel transport is an isometry. This is the principle that parallel transport preserves a fibre metric.

Normalize $B_0=1$ and put ${\tilde{S}}_t=e^{-rt}{S}_t$. Its dynamics are $d{\tilde{S}}_t={\tilde{S}}_t\sigma (t,S_t)d{W}_t$. Bounded volatility makes this stochastic exponential a true martingale on each finite horizon, by the Novikov condition. It also gives a finite second moment: stopping the Itô formula for $S^2$ and applying the Gronwall inequality yields $\mathbb{E}{S}_T^2\le S_0^2{e}^{(2r+L^2)T}$ when $\sigma \le L$.

The discounted payoff $\xi =e^{-rT}(S_T-K{)}^{+}$ is thus square-integrable. Define the nonnegative martingaleThe Brownian martingale representation theorem says that every square-integrable martingale in the Brownian filtration has a representation $M_t=M_0+{\int }_0^t{h}_{u}d{W}_u$ with predictable $h$ and $\mathbb{E}{\int }_0^T{h}_u^{2}du<\infty$. Since $\tilde{S}>0$ and $\sigma >0$, chooseThen discounted gains satisfy $d{M}_t={\pi }_{t}d{\tilde{S}}_t$. Consequently $X_t=B_t{M}_t={\varphi }_t{B}_t+{\pi }_t{S}_t$ is self-financing, nonnegative, and hence an admissible trading strategy. At maturity $X_T=(S_T-K{)}^{+}$, proving claim replication at cost $C(T,K)$.

For minimality, discounted wealth of any admissible self-financing portfolio is a local martingale bounded below, hence a supermartingale by localization and the conditional Fatou lemma. Thus any such replication with initial wealth $x$ obeysTogether with the constructed portfolio, this provesThis is Brownian representation replication in a local volatility market. The given drift $r$ means that the original probability measure already serves as the risk-neutral measure.

For energy stability for variable-coefficient reaction diffusion, let $h_x=1/(M+1)$, let $D_h$ be the Dirichlet discrete Laplacian, and set $V_h=\mathrm{diag}(a_m)$. Use the mesh-weighted inner product $(v,w{)}_h=h_x{\sum }_{m=1}^M{v}_m{w}_m$ with the corresponding discrete L2 norm. For zero boundary values, summation by parts givesFor a solution, or a difference of two solutions, the energy method givesThe Gronwall inequality therefore provesThe constant is independent of $M$, so this is stability of a numerical method on every fixed finite time interval. The bound allows physical growth if $a_{+}>0$; stability here does not mean uniform boundedness as $t\to \infty$.

There is also a uniform maximum-norm bound. At a positive spatial maximum the second difference is nonpositive. Applying this observation to $e^{-a_{+}t}v$ and to its negative gives $\Vert v(t){\Vert }_{\infty }\le e^{a_{+}t}\Vert v(0){\Vert }_{\infty }$. With a perturbing source $r(t)$, the Duhamel principle gives the corresponding initial-data bound plus ${\int }_0^t{e}^{a_{+}(t-s)}\Vert r(s)\Vert ds$ in either norm. Thus the estimate controls accumulated residuals as well as initial perturbations.